pázmány péter catholic universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · functional...
TRANSCRIPT
![Page 1: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/1.jpg)
Functional analysis Distant Learning. Week 3.
Functional analysis
Lesson 9.
April 21, 2020
![Page 2: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/2.jpg)
Functional analysis Distant Learning. Week 3.
Review
In the WEIGHTED L2 SPACE we applied G-S orthogonalization:
{1, x , . . . , xn . . . } −→ {ϕ0, ϕ1, . . . , ϕn . . . } ON polynomials.
E.g. Legendre-, Chebishev-, Hermite-polynomials. Do you know?
Questions.
I Why are these systems of orthogonal polynomials important?
I What can we use the ON polynomials for?
![Page 3: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/3.jpg)
Functional analysis Distant Learning. Week 3.
Review
In the WEIGHTED L2 SPACE we applied G-S orthogonalization:
{1, x , . . . , xn . . . } −→ {ϕ0, ϕ1, . . . , ϕn . . . } ON polynomials.
E.g. Legendre-, Chebishev-, Hermite-polynomials. Do you know?
Questions.
I Why are these systems of orthogonal polynomials important?
I What can we use the ON polynomials for?
![Page 4: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/4.jpg)
Functional analysis Distant Learning. Week 3.
Review
In the WEIGHTED L2 SPACE we applied G-S orthogonalization:
{1, x , . . . , xn . . . }
−→ {ϕ0, ϕ1, . . . , ϕn . . . } ON polynomials.
E.g. Legendre-, Chebishev-, Hermite-polynomials. Do you know?
Questions.
I Why are these systems of orthogonal polynomials important?
I What can we use the ON polynomials for?
![Page 5: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/5.jpg)
Functional analysis Distant Learning. Week 3.
Review
In the WEIGHTED L2 SPACE we applied G-S orthogonalization:
{1, x , . . . , xn . . . } −→
{ϕ0, ϕ1, . . . , ϕn . . . } ON polynomials.
E.g. Legendre-, Chebishev-, Hermite-polynomials. Do you know?
Questions.
I Why are these systems of orthogonal polynomials important?
I What can we use the ON polynomials for?
![Page 6: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/6.jpg)
Functional analysis Distant Learning. Week 3.
Review
In the WEIGHTED L2 SPACE we applied G-S orthogonalization:
{1, x , . . . , xn . . . } −→ {ϕ0, ϕ1, . . . , ϕn . . . } ON polynomials.
E.g. Legendre-, Chebishev-, Hermite-polynomials. Do you know?
Questions.
I Why are these systems of orthogonal polynomials important?
I What can we use the ON polynomials for?
![Page 7: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/7.jpg)
Functional analysis Distant Learning. Week 3.
Review
In the WEIGHTED L2 SPACE we applied G-S orthogonalization:
{1, x , . . . , xn . . . } −→ {ϕ0, ϕ1, . . . , ϕn . . . } ON polynomials.
E.g. Legendre-, Chebishev-, Hermite-polynomials. Do you know?
Questions.
I Why are these systems of orthogonal polynomials important?
I What can we use the ON polynomials for?
![Page 8: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/8.jpg)
Functional analysis Distant Learning. Week 3.
Review
In the WEIGHTED L2 SPACE we applied G-S orthogonalization:
{1, x , . . . , xn . . . } −→ {ϕ0, ϕ1, . . . , ϕn . . . } ON polynomials.
E.g. Legendre-, Chebishev-, Hermite-polynomials. Do you know?
Questions.
I Why are these systems of orthogonal polynomials important?
I What can we use the ON polynomials for?
![Page 9: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/9.jpg)
Functional analysis Distant Learning. Week 3.
Review
In the WEIGHTED L2 SPACE we applied G-S orthogonalization:
{1, x , . . . , xn . . . } −→ {ϕ0, ϕ1, . . . , ϕn . . . } ON polynomials.
E.g. Legendre-, Chebishev-, Hermite-polynomials. Do you know?
Questions.
I Why are these systems of orthogonal polynomials important?
I What can we use the ON polynomials for?
![Page 10: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/10.jpg)
Functional analysis Distant Learning. Week 3.
Review
In the WEIGHTED L2 SPACE we applied G-S orthogonalization:
{1, x , . . . , xn . . . } −→ {ϕ0, ϕ1, . . . , ϕn . . . } ON polynomials.
E.g. Legendre-, Chebishev-, Hermite-polynomials. Do you know?
Questions.
I Why are these systems of orthogonal polynomials important?
I What can we use the ON polynomials for?
![Page 11: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/11.jpg)
Functional analysis Distant Learning. Week 3.
A detour
Theorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 12: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/12.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 13: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/13.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions.
??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 14: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/14.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 15: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/15.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 16: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/16.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) ,
with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 17: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/17.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 18: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/18.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx ,
bk =1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 19: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/19.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 20: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/20.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary.
The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 21: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/21.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system
is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 22: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/22.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 23: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/23.jpg)
Functional analysis Distant Learning. Week 3.
A detourTheorem. (Classical Fourier theorem.)
Assume f : [−π, π]→ IR satisfies the Dirichlet conditions. ??
Then ∀x ∈ [−π, π]:
f (x) =a0
2+∞∑
k=1
(ak cos(kx) + bk sin(kx)) , with
ak =1π
∫ π
−πf (x) cos(kx) dx , bk =
1π
∫ π
−πf (x) sin(kx) dx .
Corollary. The trig. system is complete in L2[−π, π].
−→ Moreover, the coefficients are known.
![Page 24: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/24.jpg)
Functional analysis Distant Learning. Week 3.
General Fourier series
![Page 25: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/25.jpg)
Functional analysis Distant Learning. Week 3.
In a Hilbert space.
(H, 〈·, ·〉) is a Hilbert space. (Can you recall the definition?)
Let (ϕk , ) ⊂ H be an ON system.
Theorem. Assume, that for some f ∈ H we have
f =∞∑
k=1
ckϕk .
Then ck = 〈f , ϕk 〉. I.e. the coefficients can be recovered from f .
Remark. If (ϕn) ⊂ H is complete, then every f ∈ H: ∃(cn)
f =∞∑
n=1
cnϕn.
![Page 26: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/26.jpg)
Functional analysis Distant Learning. Week 3.
In a Hilbert space.
(H, 〈·, ·〉) is a Hilbert space. (Can you recall the definition?)
Let (ϕk , ) ⊂ H be an ON system.
Theorem. Assume, that for some f ∈ H we have
f =∞∑
k=1
ckϕk .
Then ck = 〈f , ϕk 〉. I.e. the coefficients can be recovered from f .
Remark. If (ϕn) ⊂ H is complete, then every f ∈ H: ∃(cn)
f =∞∑
n=1
cnϕn.
![Page 27: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/27.jpg)
Functional analysis Distant Learning. Week 3.
In a Hilbert space.
(H, 〈·, ·〉) is a Hilbert space. (Can you recall the definition?)
Let (ϕk , ) ⊂ H be an ON system.
Theorem. Assume, that for some f ∈ H we have
f =∞∑
k=1
ckϕk .
Then ck = 〈f , ϕk 〉. I.e. the coefficients can be recovered from f .
Remark. If (ϕn) ⊂ H is complete, then every f ∈ H: ∃(cn)
f =∞∑
n=1
cnϕn.
![Page 28: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/28.jpg)
Functional analysis Distant Learning. Week 3.
In a Hilbert space.
(H, 〈·, ·〉) is a Hilbert space. (Can you recall the definition?)
Let (ϕk , ) ⊂ H be an ON system.
Theorem. Assume, that for some f ∈ H we have
f =∞∑
k=1
ckϕk .
Then ck = 〈f , ϕk 〉. I.e. the coefficients can be recovered from f .
Remark. If (ϕn) ⊂ H is complete, then every f ∈ H: ∃(cn)
f =∞∑
n=1
cnϕn.
![Page 29: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/29.jpg)
Functional analysis Distant Learning. Week 3.
In a Hilbert space.
(H, 〈·, ·〉) is a Hilbert space. (Can you recall the definition?)
Let (ϕk , ) ⊂ H be an ON system.
Theorem. Assume, that for some f ∈ H we have
f =∞∑
k=1
ckϕk .
Then ck = 〈f , ϕk 〉.
I.e. the coefficients can be recovered from f .
Remark. If (ϕn) ⊂ H is complete, then every f ∈ H: ∃(cn)
f =∞∑
n=1
cnϕn.
![Page 30: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/30.jpg)
Functional analysis Distant Learning. Week 3.
In a Hilbert space.
(H, 〈·, ·〉) is a Hilbert space. (Can you recall the definition?)
Let (ϕk , ) ⊂ H be an ON system.
Theorem. Assume, that for some f ∈ H we have
f =∞∑
k=1
ckϕk .
Then ck = 〈f , ϕk 〉. I.e. the coefficients can be recovered from f .
Remark. If (ϕn) ⊂ H is complete, then every f ∈ H: ∃(cn)
f =∞∑
n=1
cnϕn.
![Page 31: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/31.jpg)
Functional analysis Distant Learning. Week 3.
In a Hilbert space.
(H, 〈·, ·〉) is a Hilbert space. (Can you recall the definition?)
Let (ϕk , ) ⊂ H be an ON system.
Theorem. Assume, that for some f ∈ H we have
f =∞∑
k=1
ckϕk .
Then ck = 〈f , ϕk 〉. I.e. the coefficients can be recovered from f .
Remark. If (ϕn) ⊂ H is complete, then every f ∈ H:
∃(cn)
f =∞∑
n=1
cnϕn.
![Page 32: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/32.jpg)
Functional analysis Distant Learning. Week 3.
In a Hilbert space.
(H, 〈·, ·〉) is a Hilbert space. (Can you recall the definition?)
Let (ϕk , ) ⊂ H be an ON system.
Theorem. Assume, that for some f ∈ H we have
f =∞∑
k=1
ckϕk .
Then ck = 〈f , ϕk 〉. I.e. the coefficients can be recovered from f .
Remark. If (ϕn) ⊂ H is complete, then every f ∈ H: ∃(cn)
f =∞∑
n=1
cnϕn.
![Page 33: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/33.jpg)
Functional analysis Distant Learning. Week 3.
In a Hilbert space.
(H, 〈·, ·〉) is a Hilbert space. (Can you recall the definition?)
Let (ϕk , ) ⊂ H be an ON system.
Theorem. Assume, that for some f ∈ H we have
f =∞∑
k=1
ckϕk .
Then ck = 〈f , ϕk 〉. I.e. the coefficients can be recovered from f .
Remark. If (ϕn) ⊂ H is complete, then every f ∈ H: ∃(cn)
f =∞∑
n=1
cnϕn.
![Page 34: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/34.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 35: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/35.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 36: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/36.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 37: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/37.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 38: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/38.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0.
(Why?) =⇒ 〈f , ϕj〉 = limn→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 39: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/39.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?)
=⇒ 〈f , ϕj〉 = limn→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 40: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/40.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 41: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/41.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 42: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/42.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 43: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/43.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩=
??? =n∑
k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 44: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/44.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ???
=n∑
k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 45: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/45.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉
= cj .
![Page 46: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/46.jpg)
Functional analysis Distant Learning. Week 3.
Proof.
Let us define sn :=n∑
k=1
ckϕk .
Then by the Thm.’s assumption
limn→∞
‖f − sn‖ = 0.
It follows, that for all ϕj , j ≤ n
limn→∞〈f − sn, ϕj〉 = 0. (Why?) =⇒ 〈f , ϕj〉 = lim
n→∞〈sn, ϕj〉
If n ≥ j , then
〈sn, ϕj〉 =
⟨n∑
k=1
ckϕk , ϕj
⟩= ??? =
n∑k=1
ck 〈ϕk , ϕj〉 = cj .
![Page 47: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/47.jpg)
Functional analysis Distant Learning. Week 3.
Fourier series expansion
Let (ϕn) ⊂ H be a complete ON system. For any f ∈ H we define
I FOURIER COEFFICIENTS of f with respect to (ϕn) as
〈f , ϕn〉 , n = 1,2, . . .
I FOURIER SERIES EXPANSION of f with respect to (ϕn) as
∞∑n=1
〈f , ϕn〉 ϕn.
Notation. f ∼∞∑
n=1
cn ϕn, with cn = 〈f , ϕn〉.
It is a formal definition yet. Why?
![Page 48: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/48.jpg)
Functional analysis Distant Learning. Week 3.
Fourier series expansion
Let (ϕn) ⊂ H be a complete ON system. For any f ∈ H we define
I FOURIER COEFFICIENTS of f with respect to (ϕn) as
〈f , ϕn〉 , n = 1,2, . . .
I FOURIER SERIES EXPANSION of f with respect to (ϕn) as
∞∑n=1
〈f , ϕn〉 ϕn.
Notation. f ∼∞∑
n=1
cn ϕn, with cn = 〈f , ϕn〉.
It is a formal definition yet. Why?
![Page 49: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/49.jpg)
Functional analysis Distant Learning. Week 3.
Fourier series expansion
Let (ϕn) ⊂ H be a complete ON system. For any f ∈ H we define
I FOURIER COEFFICIENTS of f with respect to (ϕn) as
〈f , ϕn〉 , n = 1,2, . . .
I FOURIER SERIES EXPANSION of f with respect to (ϕn) as
∞∑n=1
〈f , ϕn〉 ϕn.
Notation. f ∼∞∑
n=1
cn ϕn, with cn = 〈f , ϕn〉.
It is a formal definition yet. Why?
![Page 50: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/50.jpg)
Functional analysis Distant Learning. Week 3.
Fourier series expansion
Let (ϕn) ⊂ H be a complete ON system. For any f ∈ H we define
I FOURIER COEFFICIENTS of f with respect to (ϕn) as
〈f , ϕn〉 , n = 1,2, . . .
I FOURIER SERIES EXPANSION of f with respect to (ϕn) as
∞∑n=1
〈f , ϕn〉 ϕn.
Notation. f ∼∞∑
n=1
cn ϕn, with cn = 〈f , ϕn〉.
It is a formal definition yet. Why?
![Page 51: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/51.jpg)
Functional analysis Distant Learning. Week 3.
Fourier series expansion
Let (ϕn) ⊂ H be a complete ON system. For any f ∈ H we define
I FOURIER COEFFICIENTS of f with respect to (ϕn) as
〈f , ϕn〉 , n = 1,2, . . .
I FOURIER SERIES EXPANSION of f with respect to (ϕn) as
∞∑n=1
〈f , ϕn〉 ϕn.
Notation. f ∼∞∑
n=1
cn ϕn, with cn = 〈f , ϕn〉.
It is a formal definition yet. Why?
![Page 52: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/52.jpg)
Functional analysis Distant Learning. Week 3.
Fourier series expansion
Let (ϕn) ⊂ H be a complete ON system. For any f ∈ H we define
I FOURIER COEFFICIENTS of f with respect to (ϕn) as
〈f , ϕn〉 , n = 1,2, . . .
I FOURIER SERIES EXPANSION of f with respect to (ϕn) as
∞∑n=1
〈f , ϕn〉 ϕn.
Notation. f ∼∞∑
n=1
cn ϕn, with cn = 〈f , ϕn〉.
It is a formal definition yet.
Why?
![Page 53: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/53.jpg)
Functional analysis Distant Learning. Week 3.
Fourier series expansion
Let (ϕn) ⊂ H be a complete ON system. For any f ∈ H we define
I FOURIER COEFFICIENTS of f with respect to (ϕn) as
〈f , ϕn〉 , n = 1,2, . . .
I FOURIER SERIES EXPANSION of f with respect to (ϕn) as
∞∑n=1
〈f , ϕn〉 ϕn.
Notation. f ∼∞∑
n=1
cn ϕn, with cn = 〈f , ϕn〉.
It is a formal definition yet. Why?
![Page 54: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/54.jpg)
Functional analysis Distant Learning. Week 3.
Sum of the Fourier series
Theorem. If (ϕn) is a complete ON system, then
f =∞∑
n=1
〈f , ϕn〉 ϕn.
I.e. the sum of the Fourer series gives back the original function.
Analogy. V is a finite dim. vector space. v1, . . . , vn ∈ V is a basis, if
I these vectors are linearly independent,
I ∀v ∈ V can be written as v =n∑
k=1
ck vk (i.e. a generator system).
In infinite dimensional Hilbert space basis ≡ complete ON system
![Page 55: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/55.jpg)
Functional analysis Distant Learning. Week 3.
Sum of the Fourier series
Theorem. If (ϕn) is a complete ON system, then
f =∞∑
n=1
〈f , ϕn〉 ϕn.
I.e. the sum of the Fourer series gives back the original function.
Analogy. V is a finite dim. vector space. v1, . . . , vn ∈ V is a basis, if
I these vectors are linearly independent,
I ∀v ∈ V can be written as v =n∑
k=1
ck vk (i.e. a generator system).
In infinite dimensional Hilbert space basis ≡ complete ON system
![Page 56: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/56.jpg)
Functional analysis Distant Learning. Week 3.
Sum of the Fourier series
Theorem. If (ϕn) is a complete ON system, then
f =∞∑
n=1
〈f , ϕn〉 ϕn.
I.e. the sum of the Fourer series gives back the original function.
Analogy. V is a finite dim. vector space. v1, . . . , vn ∈ V is a basis, if
I these vectors are linearly independent,
I ∀v ∈ V can be written as v =n∑
k=1
ck vk (i.e. a generator system).
In infinite dimensional Hilbert space basis ≡ complete ON system
![Page 57: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/57.jpg)
Functional analysis Distant Learning. Week 3.
Sum of the Fourier series
Theorem. If (ϕn) is a complete ON system, then
f =∞∑
n=1
〈f , ϕn〉 ϕn.
I.e. the sum of the Fourer series gives back the original function.
Analogy.
V is a finite dim. vector space. v1, . . . , vn ∈ V is a basis, if
I these vectors are linearly independent,
I ∀v ∈ V can be written as v =n∑
k=1
ck vk (i.e. a generator system).
In infinite dimensional Hilbert space basis ≡ complete ON system
![Page 58: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/58.jpg)
Functional analysis Distant Learning. Week 3.
Sum of the Fourier series
Theorem. If (ϕn) is a complete ON system, then
f =∞∑
n=1
〈f , ϕn〉 ϕn.
I.e. the sum of the Fourer series gives back the original function.
Analogy. V is a finite dim. vector space.
v1, . . . , vn ∈ V is a basis, if
I these vectors are linearly independent,
I ∀v ∈ V can be written as v =n∑
k=1
ck vk (i.e. a generator system).
In infinite dimensional Hilbert space basis ≡ complete ON system
![Page 59: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/59.jpg)
Functional analysis Distant Learning. Week 3.
Sum of the Fourier series
Theorem. If (ϕn) is a complete ON system, then
f =∞∑
n=1
〈f , ϕn〉 ϕn.
I.e. the sum of the Fourer series gives back the original function.
Analogy. V is a finite dim. vector space. v1, . . . , vn ∈ V is a basis, if
I these vectors are linearly independent,
I ∀v ∈ V can be written as v =n∑
k=1
ck vk (i.e. a generator system).
In infinite dimensional Hilbert space basis ≡ complete ON system
![Page 60: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/60.jpg)
Functional analysis Distant Learning. Week 3.
Sum of the Fourier series
Theorem. If (ϕn) is a complete ON system, then
f =∞∑
n=1
〈f , ϕn〉 ϕn.
I.e. the sum of the Fourer series gives back the original function.
Analogy. V is a finite dim. vector space. v1, . . . , vn ∈ V is a basis, if
I these vectors are linearly independent,
I ∀v ∈ V can be written as v =n∑
k=1
ck vk (i.e. a generator system).
In infinite dimensional Hilbert space basis ≡ complete ON system
![Page 61: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/61.jpg)
Functional analysis Distant Learning. Week 3.
Sum of the Fourier series
Theorem. If (ϕn) is a complete ON system, then
f =∞∑
n=1
〈f , ϕn〉 ϕn.
I.e. the sum of the Fourer series gives back the original function.
Analogy. V is a finite dim. vector space. v1, . . . , vn ∈ V is a basis, if
I these vectors are linearly independent,
I ∀v ∈ V can be written as v =n∑
k=1
ck vk (i.e. a generator system).
In infinite dimensional Hilbert space
basis ≡ complete ON system
![Page 62: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/62.jpg)
Functional analysis Distant Learning. Week 3.
Sum of the Fourier series
Theorem. If (ϕn) is a complete ON system, then
f =∞∑
n=1
〈f , ϕn〉 ϕn.
I.e. the sum of the Fourer series gives back the original function.
Analogy. V is a finite dim. vector space. v1, . . . , vn ∈ V is a basis, if
I these vectors are linearly independent,
I ∀v ∈ V can be written as v =n∑
k=1
ck vk (i.e. a generator system).
In infinite dimensional Hilbert space basis ≡ complete ON system
![Page 63: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/63.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality
Try to recall ”the original” one
Theorem. Let f ∈ H.
1. (ϕn) ⊂ H is an ON system. Then
∞∑n=1
c2n ≤ ‖f‖2, cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete ⇐⇒∞∑
n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 64: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/64.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality Try to recall ”the original” one
Theorem. Let f ∈ H.
1. (ϕn) ⊂ H is an ON system. Then
∞∑n=1
c2n ≤ ‖f‖2, cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete ⇐⇒∞∑
n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 65: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/65.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality Try to recall ”the original” one
Theorem.
Let f ∈ H.
1. (ϕn) ⊂ H is an ON system. Then
∞∑n=1
c2n ≤ ‖f‖2, cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete ⇐⇒∞∑
n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 66: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/66.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality Try to recall ”the original” one
Theorem. Let f ∈ H.
1. (ϕn) ⊂ H is an ON system. Then
∞∑n=1
c2n ≤ ‖f‖2, cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete ⇐⇒∞∑
n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 67: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/67.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality Try to recall ”the original” one
Theorem. Let f ∈ H.
1. (ϕn) ⊂ H is an ON system.
Then
∞∑n=1
c2n ≤ ‖f‖2, cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete ⇐⇒∞∑
n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 68: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/68.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality Try to recall ”the original” one
Theorem. Let f ∈ H.
1. (ϕn) ⊂ H is an ON system. Then
∞∑n=1
c2n ≤ ‖f‖2,
cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete ⇐⇒∞∑
n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 69: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/69.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality Try to recall ”the original” one
Theorem. Let f ∈ H.
1. (ϕn) ⊂ H is an ON system. Then
∞∑n=1
c2n ≤ ‖f‖2, cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete ⇐⇒∞∑
n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 70: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/70.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality Try to recall ”the original” one
Theorem. Let f ∈ H.
1. (ϕn) ⊂ H is an ON system. Then
∞∑n=1
c2n ≤ ‖f‖2, cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete
⇐⇒∞∑
n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 71: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/71.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality Try to recall ”the original” one
Theorem. Let f ∈ H.
1. (ϕn) ⊂ H is an ON system. Then
∞∑n=1
c2n ≤ ‖f‖2, cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete ⇐⇒
∞∑n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 72: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/72.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality Try to recall ”the original” one
Theorem. Let f ∈ H.
1. (ϕn) ⊂ H is an ON system. Then
∞∑n=1
c2n ≤ ‖f‖2, cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete ⇐⇒∞∑
n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 73: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/73.jpg)
Functional analysis Distant Learning. Week 3.
Parseval equality Try to recall ”the original” one
Theorem. Let f ∈ H.
1. (ϕn) ⊂ H is an ON system. Then
∞∑n=1
c2n ≤ ‖f‖2, cn = 〈f , ϕn〉 ϕn.
2. (ϕn) is ON and complete ⇐⇒∞∑
n=1
c2n = ‖f‖2.
The latter identity is called PARSEVAL EQUALITY.
![Page 74: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/74.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2
cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 75: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/75.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 76: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/76.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof.
1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 77: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/77.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk .
Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 78: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/78.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is
try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 79: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/79.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...
the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 80: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/80.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}.
Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 81: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/81.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 82: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/82.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 83: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/83.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2
=⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 84: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/84.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 85: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/85.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =
n∑k=1
c2k . Finally, with n→∞
√.
![Page 86: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/86.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k .
Finally, with n→∞√
.
![Page 87: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/87.jpg)
Functional analysis Distant Learning. Week 3.
∞∑n=1
c2n ≤ ‖f‖2 cn = 〈f , ϕn〉 ϕn.,
Proof. 1. Let us define sn :=n∑
k=1
ckϕk . Geometrically it is try to
finish...the projection of f onto span{ϕ1, ..., ϕn}. Thus (f − sn)⊥sn.
Then we can use the Pythagorean theorem:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2 =⇒ ‖sn‖2 ≤ ‖f‖2 ∀n.
By orthogonality ‖sn‖2 =n∑
k=1
c2k . Finally, with n→∞
√.
![Page 88: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/88.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2
⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 89: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/89.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 90: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/90.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 91: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/91.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON.
From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 92: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/92.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 93: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/93.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 94: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/94.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0.
From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 95: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/95.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =
∞∑k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 96: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/96.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 97: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/97.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B.
Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 98: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/98.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =
∞∑k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 99: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/99.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f ,
prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 100: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/100.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE.
Do it
Yourself. HW.
![Page 101: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/101.jpg)
Functional analysis Distant Learning. Week 3.∞∑
n=1
c2n = ‖f‖2 ⇐⇒ (ϕn) is complete,
2. To verify a proposition with ⇐⇒ inside has to parts.
Part A. Assume (ϕn) is ON. From the previous slide:
‖f‖2 = ‖f − sn‖2 + ‖sn‖2. (1)
From the completeness of (ϕn) follows, that f =∞∑
n=1
cnϕn, thus
limn→∞
‖f − sn‖2 = 0. From (1) we get ‖f‖2 = limn→∞
‖sn‖2 =∞∑
k=1
c2k .
Part B. Assuming ‖f‖2 =∞∑
k=1
c2k ∀f , prove (ϕn) is COMPLETE. Do it
Yourself. HW.
![Page 102: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/102.jpg)
Functional analysis Distant Learning. Week 3.
Generalized Parseval equality.
Theorem. Let (ϕn) be a complete ON system in L2(R)-ben.
f ,g ∈ L2(R) are arbitrary functions.Then
〈f ,g〉 =∞∑
k=1
ck dk ,
where c = (ck ) and d = (dk ) are the Fourier coefficients of f and g.
This relation can be also written as:
〈f ,g〉L2 = 〈c,d〉`2 .
![Page 103: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/103.jpg)
Functional analysis Distant Learning. Week 3.
Generalized Parseval equality.
Theorem. Let (ϕn) be a complete ON system in L2(R)-ben.
f ,g ∈ L2(R) are arbitrary functions.Then
〈f ,g〉 =∞∑
k=1
ck dk ,
where c = (ck ) and d = (dk ) are the Fourier coefficients of f and g.
This relation can be also written as:
〈f ,g〉L2 = 〈c,d〉`2 .
![Page 104: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/104.jpg)
Functional analysis Distant Learning. Week 3.
Generalized Parseval equality.
Theorem. Let (ϕn) be a complete ON system in L2(R)-ben.
f ,g ∈ L2(R) are arbitrary functions.
Then
〈f ,g〉 =∞∑
k=1
ck dk ,
where c = (ck ) and d = (dk ) are the Fourier coefficients of f and g.
This relation can be also written as:
〈f ,g〉L2 = 〈c,d〉`2 .
![Page 105: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/105.jpg)
Functional analysis Distant Learning. Week 3.
Generalized Parseval equality.
Theorem. Let (ϕn) be a complete ON system in L2(R)-ben.
f ,g ∈ L2(R) are arbitrary functions.Then
〈f ,g〉 =∞∑
k=1
ck dk ,
where c = (ck ) and d = (dk ) are the Fourier coefficients of f and g.
This relation can be also written as:
〈f ,g〉L2 = 〈c,d〉`2 .
![Page 106: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/106.jpg)
Functional analysis Distant Learning. Week 3.
Generalized Parseval equality.
Theorem. Let (ϕn) be a complete ON system in L2(R)-ben.
f ,g ∈ L2(R) are arbitrary functions.Then
〈f ,g〉 =∞∑
k=1
ck dk ,
where c = (ck ) and d = (dk ) are the Fourier coefficients of f and g.
This relation can be also written as:
〈f ,g〉L2 = 〈c,d〉`2 .
![Page 107: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/107.jpg)
Functional analysis Distant Learning. Week 3.
Generalized Parseval equality.
Theorem. Let (ϕn) be a complete ON system in L2(R)-ben.
f ,g ∈ L2(R) are arbitrary functions.Then
〈f ,g〉 =∞∑
k=1
ck dk ,
where c = (ck ) and d = (dk ) are the Fourier coefficients of f and g.
This relation can be also written as:
〈f ,g〉L2 =
〈c,d〉`2 .
![Page 108: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/108.jpg)
Functional analysis Distant Learning. Week 3.
Generalized Parseval equality.
Theorem. Let (ϕn) be a complete ON system in L2(R)-ben.
f ,g ∈ L2(R) are arbitrary functions.Then
〈f ,g〉 =∞∑
k=1
ck dk ,
where c = (ck ) and d = (dk ) are the Fourier coefficients of f and g.
This relation can be also written as:
〈f ,g〉L2 = 〈c,d〉`2 .
![Page 109: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/109.jpg)
Functional analysis Distant Learning. Week 3.
Special case: H = L2(R)
Corollary. For any f ∈ L2(R) it is possible to assign (cn) ∈ `2, usingany (ϕn) complete ON system.
The other direction is the following important Thm.
Theorem. (Riesz-Fisher thm.) Let (dk ) ∈ `2, i.e.∞∑
k=1
d2k <∞.
Then ∃f ∈ L2(R), such that ‖f‖2 =∞∑
k=1
d2k , and it’s Fourier coefficients
are dk .
Proof. (Hint) A ”candidate” is f :=∞∑
k=1
dkϕk . It is OK. Finish the proof.
![Page 110: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/110.jpg)
Functional analysis Distant Learning. Week 3.
Special case: H = L2(R)
Corollary. For any f ∈ L2(R) it is possible to assign (cn) ∈ `2, usingany (ϕn) complete ON system.
The other direction is the following important Thm.
Theorem. (Riesz-Fisher thm.) Let (dk ) ∈ `2, i.e.∞∑
k=1
d2k <∞.
Then ∃f ∈ L2(R), such that ‖f‖2 =∞∑
k=1
d2k , and it’s Fourier coefficients
are dk .
Proof. (Hint) A ”candidate” is f :=∞∑
k=1
dkϕk . It is OK. Finish the proof.
![Page 111: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/111.jpg)
Functional analysis Distant Learning. Week 3.
Special case: H = L2(R)
Corollary. For any f ∈ L2(R) it is possible to assign (cn) ∈ `2, usingany (ϕn) complete ON system.
The other direction is the following important Thm.
Theorem. (Riesz-Fisher thm.) Let (dk ) ∈ `2, i.e.∞∑
k=1
d2k <∞.
Then ∃f ∈ L2(R), such that ‖f‖2 =∞∑
k=1
d2k , and it’s Fourier coefficients
are dk .
Proof. (Hint) A ”candidate” is f :=∞∑
k=1
dkϕk . It is OK. Finish the proof.
![Page 112: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/112.jpg)
Functional analysis Distant Learning. Week 3.
Special case: H = L2(R)
Corollary. For any f ∈ L2(R) it is possible to assign (cn) ∈ `2, usingany (ϕn) complete ON system.
The other direction is the following important Thm.
Theorem. (Riesz-Fisher thm.) Let (dk ) ∈ `2, i.e.∞∑
k=1
d2k <∞.
Then ∃f ∈ L2(R), such that ‖f‖2 =∞∑
k=1
d2k , and it’s Fourier coefficients
are dk .
Proof. (Hint) A ”candidate” is f :=∞∑
k=1
dkϕk . It is OK. Finish the proof.
![Page 113: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/113.jpg)
Functional analysis Distant Learning. Week 3.
Special case: H = L2(R)
Corollary. For any f ∈ L2(R) it is possible to assign (cn) ∈ `2, usingany (ϕn) complete ON system.
The other direction is the following important Thm.
Theorem. (Riesz-Fisher thm.) Let (dk ) ∈ `2, i.e.∞∑
k=1
d2k <∞.
Then ∃f ∈ L2(R), such that ‖f‖2 =∞∑
k=1
d2k , and it’s Fourier coefficients
are dk .
Proof. (Hint)
A ”candidate” is f :=∞∑
k=1
dkϕk . It is OK. Finish the proof.
![Page 114: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/114.jpg)
Functional analysis Distant Learning. Week 3.
Special case: H = L2(R)
Corollary. For any f ∈ L2(R) it is possible to assign (cn) ∈ `2, usingany (ϕn) complete ON system.
The other direction is the following important Thm.
Theorem. (Riesz-Fisher thm.) Let (dk ) ∈ `2, i.e.∞∑
k=1
d2k <∞.
Then ∃f ∈ L2(R), such that ‖f‖2 =∞∑
k=1
d2k , and it’s Fourier coefficients
are dk .
Proof. (Hint) A ”candidate” is f :=∞∑
k=1
dkϕk .
It is OK. Finish the proof.
![Page 115: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/115.jpg)
Functional analysis Distant Learning. Week 3.
Special case: H = L2(R)
Corollary. For any f ∈ L2(R) it is possible to assign (cn) ∈ `2, usingany (ϕn) complete ON system.
The other direction is the following important Thm.
Theorem. (Riesz-Fisher thm.) Let (dk ) ∈ `2, i.e.∞∑
k=1
d2k <∞.
Then ∃f ∈ L2(R), such that ‖f‖2 =∞∑
k=1
d2k , and it’s Fourier coefficients
are dk .
Proof. (Hint) A ”candidate” is f :=∞∑
k=1
dkϕk . It is OK.
Finish the proof.
![Page 116: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/116.jpg)
Functional analysis Distant Learning. Week 3.
Special case: H = L2(R)
Corollary. For any f ∈ L2(R) it is possible to assign (cn) ∈ `2, usingany (ϕn) complete ON system.
The other direction is the following important Thm.
Theorem. (Riesz-Fisher thm.) Let (dk ) ∈ `2, i.e.∞∑
k=1
d2k <∞.
Then ∃f ∈ L2(R), such that ‖f‖2 =∞∑
k=1
d2k , and it’s Fourier coefficients
are dk .
Proof. (Hint) A ”candidate” is f :=∞∑
k=1
dkϕk . It is OK. Finish the proof.
![Page 117: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/117.jpg)
Functional analysis Distant Learning. Week 3.
L2 and `2
Corollary. L2(R) es `2 are isometrically isomorphic.
The linear isometry is based an any (ϕn) complete ON system,
using the Fourier coefficients: f ←→ (cn).
PLEASE STOP FOR A WHILE, AND UNDERSTAND THIS POINT.
L2(R) and `2 are the ”same”.
![Page 118: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/118.jpg)
Functional analysis Distant Learning. Week 3.
L2 and `2
Corollary. L2(R) es `2 are isometrically isomorphic.
The linear isometry is based an any (ϕn) complete ON system,
using the Fourier coefficients: f ←→ (cn).
PLEASE STOP FOR A WHILE, AND UNDERSTAND THIS POINT.
L2(R) and `2 are the ”same”.
![Page 119: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/119.jpg)
Functional analysis Distant Learning. Week 3.
L2 and `2
Corollary. L2(R) es `2 are isometrically isomorphic.
The linear isometry is based an any (ϕn) complete ON system,
using the Fourier coefficients:
f ←→ (cn).
PLEASE STOP FOR A WHILE, AND UNDERSTAND THIS POINT.
L2(R) and `2 are the ”same”.
![Page 120: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/120.jpg)
Functional analysis Distant Learning. Week 3.
L2 and `2
Corollary. L2(R) es `2 are isometrically isomorphic.
The linear isometry is based an any (ϕn) complete ON system,
using the Fourier coefficients:
f ←→ (cn).
PLEASE STOP FOR A WHILE, AND UNDERSTAND THIS POINT.
L2(R) and `2 are the ”same”.
![Page 121: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/121.jpg)
Functional analysis Distant Learning. Week 3.
L2 and `2
Corollary. L2(R) es `2 are isometrically isomorphic.
The linear isometry is based an any (ϕn) complete ON system,
using the Fourier coefficients: f ←→
(cn).
PLEASE STOP FOR A WHILE, AND UNDERSTAND THIS POINT.
L2(R) and `2 are the ”same”.
![Page 122: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/122.jpg)
Functional analysis Distant Learning. Week 3.
L2 and `2
Corollary. L2(R) es `2 are isometrically isomorphic.
The linear isometry is based an any (ϕn) complete ON system,
using the Fourier coefficients: f ←→ (cn).
PLEASE STOP FOR A WHILE, AND UNDERSTAND THIS POINT.
L2(R) and `2 are the ”same”.
![Page 123: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/123.jpg)
Functional analysis Distant Learning. Week 3.
L2 and `2
Corollary. L2(R) es `2 are isometrically isomorphic.
The linear isometry is based an any (ϕn) complete ON system,
using the Fourier coefficients: f ←→ (cn).
PLEASE STOP FOR A WHILE, AND UNDERSTAND THIS POINT.
L2(R) and `2 are the ”same”.
![Page 124: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/124.jpg)
Functional analysis Distant Learning. Week 3.
L2 and `2
Corollary. L2(R) es `2 are isometrically isomorphic.
The linear isometry is based an any (ϕn) complete ON system,
using the Fourier coefficients: f ←→ (cn).
PLEASE STOP FOR A WHILE, AND UNDERSTAND THIS POINT.
L2(R) and `2 are the ”same”.
![Page 125: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/125.jpg)
Functional analysis Distant Learning. Week 3.
Example. H = L2[−1,1]
In L2[−1,1] a complete ON system are the Legendre polynomials.
We have seen some elements of (Pn(x)):
P0(x) =1√2, P1(x) =
√32
x , P2(x) = it was a HW . . . ..
Then every f ∈ L2[−1,1] can be written as
f (x) =∞∑
n=0
cnPn(x), with cn =
∫ 1
−1f (x)Pn(x)dx .
Thus every f ∈ L2[−1,1] can be approximated by a polynomial of
degree n with KNOWN coefficients. Can you recall sg. similar?
![Page 126: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/126.jpg)
Functional analysis Distant Learning. Week 3.
Example. H = L2[−1,1]
In L2[−1,1] a complete ON system are the Legendre polynomials.
We have seen some elements of (Pn(x)):
P0(x) =1√2, P1(x) =
√32
x , P2(x) = it was a HW . . . ..
Then every f ∈ L2[−1,1] can be written as
f (x) =∞∑
n=0
cnPn(x), with cn =
∫ 1
−1f (x)Pn(x)dx .
Thus every f ∈ L2[−1,1] can be approximated by a polynomial of
degree n with KNOWN coefficients. Can you recall sg. similar?
![Page 127: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/127.jpg)
Functional analysis Distant Learning. Week 3.
Example. H = L2[−1,1]
In L2[−1,1] a complete ON system are the Legendre polynomials.
We have seen some elements of (Pn(x)):
P0(x) =1√2,
P1(x) =
√32
x , P2(x) = it was a HW . . . ..
Then every f ∈ L2[−1,1] can be written as
f (x) =∞∑
n=0
cnPn(x), with cn =
∫ 1
−1f (x)Pn(x)dx .
Thus every f ∈ L2[−1,1] can be approximated by a polynomial of
degree n with KNOWN coefficients. Can you recall sg. similar?
![Page 128: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/128.jpg)
Functional analysis Distant Learning. Week 3.
Example. H = L2[−1,1]
In L2[−1,1] a complete ON system are the Legendre polynomials.
We have seen some elements of (Pn(x)):
P0(x) =1√2, P1(x) =
√32
x , P2(x) =
it was a HW . . . ..
Then every f ∈ L2[−1,1] can be written as
f (x) =∞∑
n=0
cnPn(x), with cn =
∫ 1
−1f (x)Pn(x)dx .
Thus every f ∈ L2[−1,1] can be approximated by a polynomial of
degree n with KNOWN coefficients. Can you recall sg. similar?
![Page 129: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/129.jpg)
Functional analysis Distant Learning. Week 3.
Example. H = L2[−1,1]
In L2[−1,1] a complete ON system are the Legendre polynomials.
We have seen some elements of (Pn(x)):
P0(x) =1√2, P1(x) =
√32
x , P2(x) = it was a HW . . . ..
Then every f ∈ L2[−1,1] can be written as
f (x) =∞∑
n=0
cnPn(x), with cn =
∫ 1
−1f (x)Pn(x)dx .
Thus every f ∈ L2[−1,1] can be approximated by a polynomial of
degree n with KNOWN coefficients. Can you recall sg. similar?
![Page 130: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/130.jpg)
Functional analysis Distant Learning. Week 3.
Example. H = L2[−1,1]
In L2[−1,1] a complete ON system are the Legendre polynomials.
We have seen some elements of (Pn(x)):
P0(x) =1√2, P1(x) =
√32
x , P2(x) = it was a HW . . . ..
Then every f ∈ L2[−1,1] can be written as
f (x) =∞∑
n=0
cnPn(x), with cn =
∫ 1
−1f (x)Pn(x)dx .
Thus every f ∈ L2[−1,1] can be approximated by a polynomial of
degree n with KNOWN coefficients. Can you recall sg. similar?
![Page 131: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/131.jpg)
Functional analysis Distant Learning. Week 3.
Example. H = L2[−1,1]
In L2[−1,1] a complete ON system are the Legendre polynomials.
We have seen some elements of (Pn(x)):
P0(x) =1√2, P1(x) =
√32
x , P2(x) = it was a HW . . . ..
Then every f ∈ L2[−1,1] can be written as
f (x) =∞∑
n=0
cnPn(x), with cn =
∫ 1
−1f (x)Pn(x)dx .
Thus every f ∈ L2[−1,1] can be approximated by a polynomial of
degree n with KNOWN coefficients. Can you recall sg. similar?
![Page 132: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/132.jpg)
Functional analysis Distant Learning. Week 3.
Example. H = L2[−1,1]
In L2[−1,1] a complete ON system are the Legendre polynomials.
We have seen some elements of (Pn(x)):
P0(x) =1√2, P1(x) =
√32
x , P2(x) = it was a HW . . . ..
Then every f ∈ L2[−1,1] can be written as
f (x) =∞∑
n=0
cnPn(x), with cn =
∫ 1
−1f (x)Pn(x)dx .
Thus every f ∈ L2[−1,1] can be approximated by a polynomial of
degree n with KNOWN coefficients.
Can you recall sg. similar?
![Page 133: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/133.jpg)
Functional analysis Distant Learning. Week 3.
Example. H = L2[−1,1]
In L2[−1,1] a complete ON system are the Legendre polynomials.
We have seen some elements of (Pn(x)):
P0(x) =1√2, P1(x) =
√32
x , P2(x) = it was a HW . . . ..
Then every f ∈ L2[−1,1] can be written as
f (x) =∞∑
n=0
cnPn(x), with cn =
∫ 1
−1f (x)Pn(x)dx .
Thus every f ∈ L2[−1,1] can be approximated by a polynomial of
degree n with KNOWN coefficients. Can you recall sg. similar?
![Page 134: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/134.jpg)
Functional analysis Distant Learning. Week 3.
An example in H = L2[0,1]
This example gives an ON system in L2[0,1]-ben.
They are called Haar-functions.
They are not polynomials, but this is the simplest wavelet family .
( More details on that can be found in the in the book.)
They are defined in blocks.
Hn,k with n = 0,1,2, . . . k = 1, ...,2n.
For all indices Hn,k : [0,1]→ IR.
![Page 135: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/135.jpg)
Functional analysis Distant Learning. Week 3.
An example in H = L2[0,1]
This example gives an ON system in L2[0,1]-ben.
They are called Haar-functions.
They are not polynomials, but this is the simplest wavelet family .
( More details on that can be found in the in the book.)
They are defined in blocks.
Hn,k with n = 0,1,2, . . . k = 1, ...,2n.
For all indices Hn,k : [0,1]→ IR.
![Page 136: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/136.jpg)
Functional analysis Distant Learning. Week 3.
An example in H = L2[0,1]
This example gives an ON system in L2[0,1]-ben.
They are called Haar-functions.
They are not polynomials, but this is the simplest wavelet family .
(
More details on that can be found in the in the book.)
They are defined in blocks.
Hn,k with n = 0,1,2, . . . k = 1, ...,2n.
For all indices Hn,k : [0,1]→ IR.
![Page 137: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/137.jpg)
Functional analysis Distant Learning. Week 3.
An example in H = L2[0,1]
This example gives an ON system in L2[0,1]-ben.
They are called Haar-functions.
They are not polynomials, but this is the simplest wavelet family .
( More details on that can be found in the in the book.)
They are defined in blocks.
Hn,k with n = 0,1,2, . . . k = 1, ...,2n.
For all indices Hn,k : [0,1]→ IR.
![Page 138: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/138.jpg)
Functional analysis Distant Learning. Week 3.
An example in H = L2[0,1]
This example gives an ON system in L2[0,1]-ben.
They are called Haar-functions.
They are not polynomials, but this is the simplest wavelet family .
( More details on that can be found in the in the book.)
They are defined in blocks.
Hn,k with n = 0,1,2, . . . k = 1, ...,2n.
For all indices Hn,k : [0,1]→ IR.
![Page 139: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/139.jpg)
Functional analysis Distant Learning. Week 3.
An example in H = L2[0,1]
This example gives an ON system in L2[0,1]-ben.
They are called Haar-functions.
They are not polynomials, but this is the simplest wavelet family .
( More details on that can be found in the in the book.)
They are defined in blocks.
Hn,k
with n = 0,1,2, . . . k = 1, ...,2n.
For all indices Hn,k : [0,1]→ IR.
![Page 140: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/140.jpg)
Functional analysis Distant Learning. Week 3.
An example in H = L2[0,1]
This example gives an ON system in L2[0,1]-ben.
They are called Haar-functions.
They are not polynomials, but this is the simplest wavelet family .
( More details on that can be found in the in the book.)
They are defined in blocks.
Hn,k with n = 0,1,2, . . . k = 1, ...,2n.
For all indices Hn,k : [0,1]→ IR.
![Page 141: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/141.jpg)
Functional analysis Distant Learning. Week 3.
An example in H = L2[0,1]
This example gives an ON system in L2[0,1]-ben.
They are called Haar-functions.
They are not polynomials, but this is the simplest wavelet family .
( More details on that can be found in the in the book.)
They are defined in blocks.
Hn,k with n = 0,1,2, . . . k = 1, ...,2n.
For all indices Hn,k : [0,1]→ IR.
![Page 142: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/142.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions
For n = 0 there are two functions: H0,0 and H0,1.
H0,0(x) = 1.
H0,1(x) =
1 if 0 ≤ x < 1/2
−1 if 1/2 ≤ x ≤ 1
x ∈ [0,1]This is the so called mother wavelet
Easy to check, that ‖H0,0‖ = ‖H0,1‖ = 1 and H0,0⊥H0,1. DO IT.
![Page 143: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/143.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions
For n = 0 there are two functions: H0,0 and H0,1.
H0,0(x) = 1.
H0,1(x) =
1 if 0 ≤ x < 1/2
−1 if 1/2 ≤ x ≤ 1
x ∈ [0,1]This is the so called mother wavelet
Easy to check, that ‖H0,0‖ = ‖H0,1‖ = 1 and H0,0⊥H0,1. DO IT.
![Page 144: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/144.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions
For n = 0 there are two functions: H0,0 and H0,1.
H0,0(x) = 1.
H0,1(x) =
1 if 0 ≤ x < 1/2
−1 if 1/2 ≤ x ≤ 1
x ∈ [0,1]This is the so called mother wavelet
Easy to check, that ‖H0,0‖ = ‖H0,1‖ = 1 and H0,0⊥H0,1. DO IT.
![Page 145: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/145.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions
For n = 0 there are two functions: H0,0 and H0,1.
H0,0(x) = 1.
H0,1(x) =
1 if 0 ≤ x < 1/2
−1 if 1/2 ≤ x ≤ 1
x ∈ [0,1]
This is the so called mother wavelet
Easy to check, that ‖H0,0‖ = ‖H0,1‖ = 1 and H0,0⊥H0,1. DO IT.
![Page 146: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/146.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions
For n = 0 there are two functions: H0,0 and H0,1.
H0,0(x) = 1.
H0,1(x) =
1 if 0 ≤ x < 1/2
−1 if 1/2 ≤ x ≤ 1
x ∈ [0,1]This is the so called mother wavelet
Easy to check, that ‖H0,0‖ = ‖H0,1‖ = 1 and H0,0⊥H0,1. DO IT.
![Page 147: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/147.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions
For n = 0 there are two functions: H0,0 and H0,1.
H0,0(x) = 1.
H0,1(x) =
1 if 0 ≤ x < 1/2
−1 if 1/2 ≤ x ≤ 1
x ∈ [0,1]This is the so called mother wavelet
Easy to check, that ‖H0,0‖ = ‖H0,1‖ = 1 and
H0,0⊥H0,1. DO IT.
![Page 148: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/148.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions
For n = 0 there are two functions: H0,0 and H0,1.
H0,0(x) = 1.
H0,1(x) =
1 if 0 ≤ x < 1/2
−1 if 1/2 ≤ x ≤ 1
x ∈ [0,1]This is the so called mother wavelet
Easy to check, that ‖H0,0‖ = ‖H0,1‖ = 1 and H0,0⊥H0,1.
DO IT.
![Page 149: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/149.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions
For n = 0 there are two functions: H0,0 and H0,1.
H0,0(x) = 1.
H0,1(x) =
1 if 0 ≤ x < 1/2
−1 if 1/2 ≤ x ≤ 1
x ∈ [0,1]This is the so called mother wavelet
Easy to check, that ‖H0,0‖ = ‖H0,1‖ = 1 and H0,0⊥H0,1. DO IT.
![Page 150: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/150.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions, nth block.
For n ≥ 1 divide [0,1] into 2n equal parts with pointsk2n . Let’s define:
Hn,k (x) =
√2n if
k − 12n ≤ x <
k − 1/22n
−√
2n ifk − 1/2
2n ≤ x <k2n
0 otherwise
, n ≥ 1, 1 ≤ k ≤ 2n.
The nonzero part is the ”mother wavelet”, squished and stretched.
Easy to check, that ‖Hn,k‖ = 1 and Hn,k⊥Hn,j for j 6= k . DO IT.
![Page 151: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/151.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions, nth block.
For n ≥ 1 divide [0,1] into 2n equal parts with pointsk2n .
Let’s define:
Hn,k (x) =
√2n if
k − 12n ≤ x <
k − 1/22n
−√
2n ifk − 1/2
2n ≤ x <k2n
0 otherwise
, n ≥ 1, 1 ≤ k ≤ 2n.
The nonzero part is the ”mother wavelet”, squished and stretched.
Easy to check, that ‖Hn,k‖ = 1 and Hn,k⊥Hn,j for j 6= k . DO IT.
![Page 152: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/152.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions, nth block.
For n ≥ 1 divide [0,1] into 2n equal parts with pointsk2n . Let’s define:
Hn,k (x) =
√2n if
k − 12n ≤ x <
k − 1/22n
−√
2n ifk − 1/2
2n ≤ x <k2n
0 otherwise
, n ≥ 1, 1 ≤ k ≤ 2n.
The nonzero part is the ”mother wavelet”, squished and stretched.
Easy to check, that ‖Hn,k‖ = 1 and Hn,k⊥Hn,j for j 6= k . DO IT.
![Page 153: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/153.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions, nth block.
For n ≥ 1 divide [0,1] into 2n equal parts with pointsk2n . Let’s define:
Hn,k (x) =
√2n if
k − 12n ≤ x <
k − 1/22n
−√
2n ifk − 1/2
2n ≤ x <k2n
0 otherwise
, n ≥ 1, 1 ≤ k ≤ 2n.
The nonzero part is the ”mother wavelet”, squished and stretched.
Easy to check, that ‖Hn,k‖ = 1 and Hn,k⊥Hn,j for j 6= k . DO IT.
![Page 154: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/154.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions, nth block.
For n ≥ 1 divide [0,1] into 2n equal parts with pointsk2n . Let’s define:
Hn,k (x) =
√2n if
k − 12n ≤ x <
k − 1/22n
−√
2n ifk − 1/2
2n ≤ x <k2n
0 otherwise
, n ≥ 1, 1 ≤ k ≤ 2n.
The nonzero part is the ”mother wavelet”, squished and stretched.
Easy to check, that ‖Hn,k‖ = 1 and Hn,k⊥Hn,j for j 6= k . DO IT.
![Page 155: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/155.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions, nth block.
For n ≥ 1 divide [0,1] into 2n equal parts with pointsk2n . Let’s define:
Hn,k (x) =
√2n if
k − 12n ≤ x <
k − 1/22n
−√
2n ifk − 1/2
2n ≤ x <k2n
0 otherwise
, n ≥ 1, 1 ≤ k ≤ 2n.
The nonzero part is the ”mother wavelet”, squished and stretched.
Easy to check, that ‖Hn,k‖ = 1 and Hn,k⊥Hn,j for j 6= k . DO IT.
![Page 156: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/156.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions, nth block.
For n ≥ 1 divide [0,1] into 2n equal parts with pointsk2n . Let’s define:
Hn,k (x) =
√2n if
k − 12n ≤ x <
k − 1/22n
−√
2n ifk − 1/2
2n ≤ x <k2n
0 otherwise
, n ≥ 1, 1 ≤ k ≤ 2n.
The nonzero part is the ”mother wavelet”, squished and stretched.
Easy to check, that ‖Hn,k‖ = 1 and
Hn,k⊥Hn,j for j 6= k . DO IT.
![Page 157: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/157.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions, nth block.
For n ≥ 1 divide [0,1] into 2n equal parts with pointsk2n . Let’s define:
Hn,k (x) =
√2n if
k − 12n ≤ x <
k − 1/22n
−√
2n ifk − 1/2
2n ≤ x <k2n
0 otherwise
, n ≥ 1, 1 ≤ k ≤ 2n.
The nonzero part is the ”mother wavelet”, squished and stretched.
Easy to check, that ‖Hn,k‖ = 1 and Hn,k⊥Hn,j for j 6= k .
DO IT.
![Page 158: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/158.jpg)
Functional analysis Distant Learning. Week 3.
Haar functions, nth block.
For n ≥ 1 divide [0,1] into 2n equal parts with pointsk2n . Let’s define:
Hn,k (x) =
√2n if
k − 12n ≤ x <
k − 1/22n
−√
2n ifk − 1/2
2n ≤ x <k2n
0 otherwise
, n ≥ 1, 1 ≤ k ≤ 2n.
The nonzero part is the ”mother wavelet”, squished and stretched.
Easy to check, that ‖Hn,k‖ = 1 and Hn,k⊥Hn,j for j 6= k . DO IT.
![Page 159: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/159.jpg)
Functional analysis Distant Learning. Week 3.
E.g. Haar functions H2,k
As an example, hereare the graphs of the
H2,k
Haar functions fork = 1,2,3,4.
Remark. This ON system is complete. (Not trivial to prove. )
![Page 160: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/160.jpg)
Functional analysis Distant Learning. Week 3.
E.g. Haar functions H2,k
As an example, hereare the graphs of the
H2,k
Haar functions fork = 1,2,3,4.
Remark. This ON system is complete. (Not trivial to prove. )
![Page 161: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/161.jpg)
Functional analysis Distant Learning. Week 3.
E.g. Haar functions H2,k
As an example, hereare the graphs of the
H2,k
Haar functions fork = 1,2,3,4.
Remark. This ON system is complete. (Not trivial to prove. )
![Page 162: Pázmány Péter Catholic Universityusers.itk.ppke.hu/~vago/funkanal_9_20_online.pdf · Functional analysis Distant Learning. Week 3. Sum of the Fourier series Theorem. If (’ n)](https://reader034.vdocuments.mx/reader034/viewer/2022042810/5f9e3149c511e220d6081105/html5/thumbnails/162.jpg)
Functional analysis Distant Learning. Week 3.
E.g. Haar functions H2,k
As an example, hereare the graphs of the
H2,k
Haar functions fork = 1,2,3,4.
Remark. This ON system is complete. (Not trivial to prove. )