delta function, the fourier transform, and fourier integral theorem

Gauge Institute Journal, Volume 8, No.2, May 2012 H. Vic Dannon

Delta Function, the Fourier Transform,

and Fourier Integral Theorem H. Vic Dannon

[email protected] September, 2010

Abstract The Fourier Integral Theorem guarantees that the

Fourier Transform and its Inverse are well defined operations, so

that the inversed transform is the originally transformed function.

It is believed to hold in the Calculus of Limits under some highly

restrictive sufficient conditions. In fact,

The Theorem does not hold in the Calculus of Limits

under any conditions,

because evaluating the Fourier Integral requires the integration of

( )k

ik x

k

e dξ=∞

−

=−∞∫ k ,

that diverges at x . ξ=

Only in Infinitesimal Calculus, the integral is the Hyper-real

Delta Function

( )1( )

2

kik x

k

x e ξδ ξπ

=∞−

=−∞

− = ∫ dk ,

1


and the Fourier Integral Theorem states the sifting property for

the Delta Function

( ) ( ) ( )f x fξ

ξ

ξ δ ξ ξ=∞

=−∞

= −∫ x d .

In infinitesimal Calculus we can integrate over singularities, and

the Fourier Integral Theorem holds

1( ) ( )

2

kik ikx

k

f x f eξ

ξ

ξ

ξ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ d e dk ,

where the Integrals are Hyper-real.

The highly restrictive conditions for the Fourier Integral Theorem,

in the Calculus of Limits, are irrelevant to the simplest functions,

such as constants, and useless for singular functions.

In particular, the singular violates these conditions ( )xδ

the Hyper-real Delta is not defined in the Calculus of

Limits, and is not Piecewise Continuous.

( )xδ

is not defined, and is not Piecewise Continuous in any

bounded interval.

'( )xδ

But in Infinitesimal Calculus, satisfies the Hyper-real

Fourier integral Theorem

( )xδ

1( ) ( )

2

kik ikx

k

x eξ

ξ

ξ

δ δ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ d e dkξ .

2


Also, the constant function ( ) 1f x ≡ violates the sufficient

conditions’ requirement of absolute integrability, 1x

x

dx=∞

=−∞

= ∞∫ .

But in Infinitesimal Calculus, ( ) 1f x ≡ satisfies the Hyper-real

Fourier integral Theorem

11

2

kik ikx

k

e d e dξ

ξ

ξ

ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ k .

Keywords: Infinitesimal, Infinite-Hyper-Real, Hyper-Real

Function, Infinitesimal Calculus, Delta Function, Fourier

Transform, Fourier Integral Theorem

2000 Mathematics Subject Classification 26E15; 26E20;

26A06; 97I40; 97I30; 44A10;

3


Contents

Introduction

1. Hyper-real Line

2. Integral of a Hyper-real Function

3. Delta Function

4. and the Fourier Transform ( )xδ

5. Fourier Integral Theorem

6. Cosine representation of and the Fourier integral ( )xδ

Theorem

7. and the Fourier Transform ( , )x yδ

8. and the Fourier transform ( , , )x y zδ

9. Delta Sequence 2

2

sin( )( )n

n nxx

nxδ

π

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

4


Introduction By Fourier Integral Theorem

1( ) ( )

2

kik ikx

k

f x f eξ

ξ

ξ

ξ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ d e dk

( )1( )

2

kik x

k

f e dkξ

ξ

ξ

ξ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ d

Thus, the integral ( )12

kik x

k

e ξ

π

=∞− −

=−∞∫ dk sifts through the values of

the function ( )f ξ , and picks its value at x .

Cauchy (1816), and Poisson (1815) derived the Fourier Integral

Theorem by using the sifting property of the integral.

In the derivation of his Zeta Function, Riemann (1859) uses this

sifting property repeatedly, without using a function notation for

the integral ( )12

kik x

k

e ξ

π

=∞− −

=−∞∫ dk . The derivations are in [Dan4,

p.84, p.90, p.97].

However, ( ) 1ik xx e ξξ − −= ⇒ = ,

and the integral ( )12

kik x

k

e ξ

π

=∞− −

=−∞∫ dk diverges.

5


Avoiding the singularity at does not recover the Fourier

Integral Theorem, because without the singularity the Fourier

integral equals zero.

xξ =

Thus, the Fourier Integral Theorem cannot be written in the

Calculus of Limits.

In Infinitesimal Calculus [Dan4], the singularity can be integrated

over, and defines the Delta Function

( )1( )

2

kik x

k

x e ξδ ξπ

=∞−

=−∞

− = ∫ dk .

Then, the Fourier Integral theorem states the sifting property for

the Delta Function

( ) ( ) ( )f x fξ

ξ

ξ δ ξ ξ=∞

=−∞

= −∫ x d ,

and for any hyper-real function ( )f x , the Fourier Transform pairs

converge, and the Fourier Integral Theorem holds.

In the Calculus of Limits, the Delta Function cannot be defined,

and its sifting property does not apply.

That sifting property allows for a Hyper-real Fourier Integral

Theorem for ( ) 1f x ≡ .

While 1x

x

dx=∞

=−∞

= ∞∫ , we have 11

2

kik ikx

k

e d e dξ

ξ

ξ

ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ k .

6


1.

Hyper-real Line Each real number α can be represented by a Cauchy sequence of

rational numbers, so that . 1 2 3( , , ,...)r r r nr α→

The constant sequence ( is a constant hyper-real. , , ,...)α α α

In [Dan2] we established that,

1. Any totally ordered set of positive, monotonically decreasing

to zero sequences constitutes a family of

infinitesimal hyper-reals.

1 2 3( , , ,...)ι ι ι

2. The infinitesimals are smaller than any real number, yet

strictly greater than zero.

3. Their reciprocals (1 2 3

1 1 1, , ,...ι ι ι ) are the infinite hyper-reals.

4. The infinite hyper-reals are greater than any real number,

yet strictly smaller than infinity.

5. The infinite hyper-reals with negative signs are smaller

than any real number, yet strictly greater than −∞ .

6. The sum of a real number with an infinitesimal is a

non-constant hyper-real.

7. The Hyper-reals are the totality of constant hyper-reals, a

family of infinitesimals, a family of infinitesimals with

7


negative sign, a family of infinite hyper-reals, a family of

infinite hyper-reals with negative sign, and non-constant

hyper-reals.

8. The hyper-reals are totally ordered, and aligned along a

line: the Hyper-real Line.

9. That line includes the real numbers separated by the non-

constant hyper-reals. Each real number is the center of an

interval of hyper-reals, that includes no other real number.

10. In particular, zero is separated from any positive real

by the infinitesimals, and from any negative real by the

infinitesimals with negative signs, . dx−

11. Zero is not an infinitesimal, because zero is not strictly

greater than zero.

12. We do not add infinity to the hyper-real line.

13. The infinitesimals, the infinitesimals with negative

signs, the infinite hyper-reals, and the infinite hyper-reals

with negative signs are semi-groups with

respect to addition. Neither set includes zero.

14. The hyper-real line is embedded in , and is not

homeomorphic to the real line. There is no bi-continuous

one-one mapping from the hyper-real onto the real line.

∞

8


15. In particular, there are no points on the real line that

can be assigned uniquely to the infinitesimal hyper-reals, or

to the infinite hyper-reals, or to the non-constant hyper-

reals.

16. No neighbourhood of a hyper-real is homeomorphic to

an ball. Therefore, the hyper-real line is not a manifold. n

17. The hyper-real line is totally ordered like a line, but it

is not spanned by one element, and it is not one-dimensional.

9


2.

Integral of a Hyper-real Function

In [Dan3], we defined the integral of a Hyper-real Function.

Let ( )f x be a hyper-real function on the interval [ , . ]a b

The interval may not be bounded.

( )f x may take infinite hyper-real values, and need not be

bounded.

At each

a x≤ ≤ b ,

there is a rectangle with base 2

[ ,dx dxx x− +2], height ( )f x , and area

( )f x dx .

We form the Integration Sum of all the areas for the x ’s that

start at x , and end at x b , a= =

[ , ]

( )x a b

f x dx∈∑ .

If for any infinitesimal dx , the Integration Sum has the same

hyper-real value, then ( )f x is integrable over the interval [ , . ]a b

Then, we call the Integration Sum the integral of ( )f x from ,

to x , and denote it by

x a=

b=

10


( )x b

x a

f x dx=

=∫ .

If the hyper-real is infinite, then it is the integral over [ , , ]a b

If the hyper-real is finite,

( ) real part of the hyper-realx b

x a

f x dx=

=

=∫ .

2.1 The countability of the Integration Sum

In [Dan1], we established the equality of all positive infinities:

We proved that the number of the Natural Numbers,

Card , equals the number of Real Numbers, , and

we have

2CardCard =

2 2( ) .... 2 2 ...CardCardCard Card= = = = = ≡ ∞ .

In particular, we demonstrated that the real numbers may be

well-ordered.

Consequently, there are countably many real numbers in the

interval [ , , and the Integration Sum has countably many terms. ]a b

While we do not sequence the real numbers in the interval, the

summation takes place over countably many ( )f x dx .

11


The Lower Integral is the Integration Sum where ( )f x is replaced

by its lowest value on each interval 2 2

[ ,dx dxx x− + ]

2.2 2 2[ , ]

inf ( )dx dxx t xx a b

f t dx− ≤ ≤ +∈

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠∑

The Upper Integral is the Integration Sum where ( )f x is replaced

by its largest value on each interval 2 2

[ ,dx dxx x− + ]

2.3 2 2[ , ]

sup ( )dx dxx t xx a b

f t dx− ≤ ≤ +∈

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∑

If the integral is a finite hyper-real, we have

2.4 A hyper-real function has a finite integral if and only if its

upper integral and its lower integral are finite, and differ by an

infinitesimal.

12


3.

Delta Function In [Dan5], we have defined the Delta Function, and established its

properties

1. The Delta Function is a hyper-real function defined from the

hyper-real line into the set of two hyper-reals 1

0,dx

⎧ ⎫⎪⎪⎨⎪⎪ ⎪⎩ ⎭

⎪⎪⎬⎪. The

hyper-real is the sequence 0 0,0, 0,... . The infinite hyper-

real 1dx

depends on our choice of dx .

2. We will usually choose the family of infinitesimals that is

spanned by the sequences 1n

,2

1

n,

3

1

n,… It is a

semigroup with respect to vector addition, and includes all

the scalar multiples of the generating sequences that are

non-zero. That is, the family includes infinitesimals with

negative sign. Therefore, 1dx

will mean the sequence n .

Alternatively, we may choose the family spanned by the

sequences 1

2n,

1

3n,

1

4n,… Then, 1

dx will mean the

13


sequence 2n . Once we determined the basic infinitesimal

, we will use it in the Infinite Riemann Sum that defines

an Integral in Infinitesimal Calculus.

dx

3. The Delta Function is strictly smaller than ∞

4. We define, 2 2,

1( ) ( )dx dxx x

dxδ χ⎡ ⎤−⎢ ⎥⎣ ⎦

≡ ,

where 2 2

2 2,

1, ,( )

0, otherwisedx dx

dx dxxxχ⎡ ⎤−⎢ ⎥⎣ ⎦

⎧ ⎡ ⎤⎪ ∈ −⎢ ⎥⎪ ⎣ ⎦= ⎨⎪⎪⎩.

5. Hence,

for , 0x < ( ) 0xδ =

at 2dx

x = − , jumps from to ( )xδ 01dx

,

for 2 2

,dx dxx ⎡ ⎤∈ −⎢ ⎥⎣ ⎦ , 1

( )xdx

δ = .

at , 0x =1

(0)dx

δ =

at 2dx

x = , drops from ( )xδ1dx

to . 0

for , . 0x > ( ) 0xδ =

( ) 0x xδ =

6. If 1n

dx = , 1 1 1 1 1 12 2 4 4 6 6

[ , ] [ , ] [ , ]( ) ( ),2 ( ), 3 ( )...x x xδ χ χ χ− − −= x

7. If 2n

dx = , 2 2 2

1 2 3( ) , , ,...

2 cosh 2cosh 2 2cosh 3x

x x xδ =

14


8. If 1n

dx = , 2 3[0, ) [0, ) [0, )( ) ,2 , 3 ,...x x xx e e eδ χ χ χ− − −

∞ ∞ ∞=

9. . ( ) 1x

x

x dxδ=∞

=−∞

=∫

10. ( )1( )

2

kik x

k

x e ξδ ξπ

=∞−

=−∞

− = ∫ dk

15


4.

( )xδ and the Fourier Transform

4.1 { }( ) 1xδ =F

Proof: For any infinitesimal dx , the Integration Sum for the

function

2 2

1( ) [ , ]i x i x dx dxx e e

dxω ωδ χ− −= −

has only the unique hyper-real term

2 2 2 2

1[ , ] [ ,i x i xdx dx dx dxe dx e

dxω ωχ χ− −− = − ]

dx

.

Therefore, the Fourier Transform

{ }( ) ( )x

i x

x

x x e ωδ δ=∞

−

=−∞

= ∫F

exists.

Since 2 2

[ ,i x dx dxe ω χ− − ] is bounded by 1 , it is a finite hyper-real.

Therefore, the Fourier Transform equals to the constant part of

this hyper-real.

Since the constant hyper-real in 2 2

[ ,dx dx− ] is zero, the constant

hyper-real part of 2 2

[ ,i x dx dxe ω χ− − ] is

16


0 1ie ω− = .

That is

{ }( ) 1xδ =F .

Consequently,

4.2 = the inverse Fourier Transform of the unit function 1 ( )xδ

12

i xe dω

ω

ω

ωπ

=∞

=−∞

= ∫

, 2 ixe dν

π

ν

ν=∞

=−∞

= ∫ 2ω π= ν

Thus,

4.3 0

12

i x

x

e ddx

ωω

ω

ωπ

=∞

=−∞ =

=∫1

= an infinite hyper-real

0

0i x

x

e dω

ω

ω

ω=∞

=−∞ ≠

=∫

Proof: 1(0)

dxδ = .

0

( ) 0x

xδ≠

= .

17


5.

Fourier Integral Theorem

The Fourier Integral Theorem is the Fundamental Theorem of the

Fourier Transform Theory.

It guarantees that the Fourier Transform and its Inverse are well

defined operations, so that inversion yields the originally

transformed function.

It is well known to hold in the Calculus of Limits under given

conditions.

In fact, it does not hold in the Calculus of Limits under any

conditions.

That failure is due to the inadequacy of the Calculus of Limits for

dealing with singularities

5.1 Fourier Integral Theorem does not hold in the

Calculus of Limits

Proof: By the Fourier Integral Theorem

1( ) ( )

2

kik ikx

k

f x f eξ

ξ

ξ

ξ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ d e dk

18


( )1( )

2

kik x

k

f e dkξ

ξ

ξ

ξ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ d .

However,

( ) 1ik xx e ξξ −= ⇒ = ,

and the integral

( )12

kik x

k

e dξ

π

=∞− −

=−∞∫ k ,

diverges.

That is, the Fourier Integral Theorem cannot be written in the

Calculus of Limits.

Avoiding the singularity at does not recover the Theorem,

because without the singularity the integral equals zero.

xξ =

Furthermore,

5.2 Calculus of Limits Conditions are not sufficient

for the Fourier Integral Theorem

Proof: The Calculus of Limits Conditions are

1. Piecewise Continuity of ( )f x , and '( )f x in any bounded

interval.

2. convergence of ( )x

x

f x dx=∞

=−∞∫

19


3. At a discontinuity point, ( )f x is replaced by

( )1( 0) ( 0)

2f x f x+ + − .

It is clear from 5.1 that even an infinitely differentiable ( )f x will

not resolve the singularity of the Delta Function.

In particular, the Calculus of Limits Conditions are not sufficient

for the Fourier Integral Theorem.

In Infinitesimal Calculus, the Fourier Integral Theorem holds for

any Hyper-Real function

5.3 Fourier Integral Theorem for hyper-real ( )f x

If ( )f x is hyper-real function,

Then, the Fourier Integral Theorem holds.

1( ) ( )

2

kik ikx

k

f x f eξ

ξ

ξ

ξ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ d e dk

Proof: In Infinitesimal Calculus, the Integration Sum

( ) ( )f x dξ

ξ

ξ δ ξ ξ=∞

=−∞

−∫

yields ( )f x . That is,

( ) ( ) ( )f x f xξ

ξ

ξ δ ξ ξ=∞

=−∞

= −∫ d .

20


Since Delta is the Inverse Fourier Transform of the function 1,

(xδ − )ξ equals the Integration Sum

( )12

kik x

k

e dξπ

=∞−

=−∞∫ k ,

which vanishes at any x , and equals ξ≠1dx

at x . ξ=

Substituting in the Integration Sum for ( )f x ,

( )1( ) ( )

2

kik x

k

f x f e dξ

ξ

ξ

ξ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ k d .

By changing the Summation order,

1( ) ( )

2

kik ikx

k

f x f e d e dkξ

ξ

ξ

ξ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ .

Then, the Fourier transform of ( )f x

( )x

i x

x

f x e dxα=∞

−

=−∞∫ ,

converges to a hyper-real function , some of its values may be

infinite hyper-reals, like the Delta Function.

( )F α

And the Inverse Fourier Transform of ( )F α

1( )

2i xF e d

αα

α

α απ

=∞−

=−∞∫

21


converges to the hyper-real function ( )f x .

5.4 If ( )f x is hyper-real function,

Then,

the hyper-real integral ( )x

i x

x

f x e dxα=∞

−

=−∞∫ converges to ( )F α

the hyper-real integral 1( )

2i xF e d

αα

α

απ

=∞

=−∞∫ α converges to ( )f x

The value of the Hyper-real Fourier Integral Theorem is

demonstrated by the following examples:

5.5 The Hyper-real Delta is not defined in the Calculus of

Limits, and is not Piecewise Continuous. is not defined, and

is not Piecewise Continuous in any bounded interval.

( )xδ

'( )xδ

But satisfies the Hyper-real Fourier integral Theorem ( )xδ

1( ) ( )

2

kik ikx

k

x eξ

ξ

ξ

δ δ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ d e dkξ

Proof: By the sifting property of the Hyper-real Delta

22


( ) 1ike dξ

ξ

ξ

δ ξ ξ=∞

−

=−∞

=∫ .

Substituting this representation of 1 into 1( )

2

kikx

k

x eδπ

=∞

=−∞

= ∫ dk ,

1( ) ( )

2

kik ikx

k

x e dξ e dkξ

ξ

ξ

δ δ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ .

5.6 The Hyper-real ( ) 1f x ≡ , does not satisfy ( )x

x

f x dx=∞

=−∞

< ∞∫ .

But satisfies the Hyper-real Fourier integral Theorem

11

2

kik ikx

k

e d e dξ

ξ

ξ

ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ k

Proof:

( )

( )

1 11

2 2

k kik ikx ik x

k k

x

e d e dk e dkdξ ξ

ξ ξ

ξ ξ

δ ξ

ξ ξπ π

=∞ =∞ =∞ =∞− −

=−∞ =−∞ =−∞ =−∞

−

⎛ ⎞⎟⎜ ⎟⎜ ⎟ = =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ ∫ ∫ .

23


6.

Cosine representation of and

the Fourier Integral Transform

( )xδ

6.1 Cosine representation of Delta

0

1( ) cos( )x x

ω

ω

δ ωπ

=∞

=

= ∫ dω

0

2 cos(2 )x dν

ν

πν ν=∞

=

= ∫

Proof: 1( )

2i xx e

ωω

ω

δ ωπ

=∞

=−∞

= ∫ d

0

0

12

i x i xe d e dω ω

ω ω

ω ω

ω ωπ

= =∞

=−∞ =

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪= +⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭∫ ∫

0 0

12

i x i xe d e dω ω

ω ω

ω ω

ω ωπ

=∞ =∞−

= =

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪= +⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭∫ ∫

0

1cos( )x d

ω

ω

ω ωπ

=∞

=

= ∫

24


6.2 Cosine representation of the Fourier Integral

Theorem

0

1( ) ( )cos ( )f x f x

ω ξ

ω ξ

ξ ω ξ ξπ

=∞ =∞

= =−∞

= −∫ ∫ d dω

0

2 ( )cos2 ( )f x d dν ξ

ν ξ

ξ πν ξ ξ=∞ =∞

= =−∞

= −∫ ∫ ν ν, 2ω π=

Proof: ( ) ( ) ( )f x f xξ

ξ

ξ δ ξ ξ=∞

=−∞

= −∫ d

0

1( ) cos ( )f x d d

ξ ω

ξ ω

ξ ω ξπ

=∞ =∞

=−∞ =

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪= −⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭∫ ∫ ω ξ

0

1( )cos ( )f x d d

ω ξ

ω ξ

ξ ω ξ ξ ωπ

=∞ =∞

= =−∞∫ ∫= − .

25


7.

( , )x yδ and the Fourier Transform

7.1 2-Dimesional Delta Function

( ) 1 1,

2 2

yx

yx

x y

i yi xx yx y e d e d

ωωωω

ω ω

δ ωπ π

=∞=∞

=−∞ =−∞

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎟⎜⎜ ⎟⎟= ⎜⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎟⎜⎝ ⎠⎝ ⎠∫ ∫ ω

νν

, 22yx

yx

x y

i yi xx ye d e d

ννπ νπ ν

ν ν

ν ν=∞=∞

=−∞ =−∞

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎟⎜⎜ ⎟⎟= ⎜⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎟⎜⎝ ⎠⎝ ⎠∫ ∫

2

2x x

y y

ω πω π

==

7.2 2-Dimesional Fourier Transform

{ }( , ) ( , ) x y

y xi x i y

y x

f x y f x y e dxdyω ω=∞ =∞

− −

=−∞ =−∞

= ∫ ∫F

2 ( )( , ) x y

y xi x y

y x

f x y e dxdyπ ν ν=∞ =∞

− +

=−∞ =−∞

= ∫ ∫ , 2

2x x

y y

ω πω π

==

νν

7.3 2-Dimesional Inverse Fourier Transform

{ } ( )12

1( , ) ( , )

(2 )

y x

x y

y x

i x yx y x y x yF F e

ω ωω ω

ω ω

ω ω ω ω ω ωπ

=∞ =∞+−

=−∞ =−∞

= ∫ ∫F d d

26


, 2 ( )(2 ,2 )y x

x y

y x

i x yx y xF e d y

ν νπ ν ν

ν ν

πν πν ν ν=∞ =∞

+

=−∞ =−∞

= ∫ ∫2

2x x

y y

ω πνω πν

==

d

7.4 2-Dimesional Fourier Integral Theorem

( )

2

1( , ) ( , )

(2 )

y x

x y x y

y x

i i i x yx yf x y f e d d e d d

ω ω η ξω ξ ω η ω ω

ω ω η ξ

ξ η ξ η ω ωπ

=∞ =∞ =∞ =∞− − +

=−∞ =−∞ =−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ ∫ ∫

( )( )1 1( , )

2 2

yx

yx

x y

i yi xx yf e d d e d

ωη ξ ωω ηω ξ

η ξ ω ω

ξ η ω ξ ω ηπ π

=∞=∞ =∞ =∞−−

=−∞ =−∞ =−∞ =−∞

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎟ ⎜⎜ ⎟⎟= ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜ ⎟⎜ ⎟ ⎟⎜⎝ ⎠ ⎝ ⎠∫ ∫ ∫ ∫ d

d

νν

2 ( )2 ( )( , )yx

yx

x y


ωη ξ ωπ ν ηπ ν ξ

η ξ ω ω

ξ η ν ξ ν η=∞=∞ =∞ =∞

−−

=−∞ =−∞ =−∞ =−∞

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎟ ⎜⎜ ⎟⎟= ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜ ⎟⎜ ⎟ ⎟⎜⎝ ⎠ ⎝ ⎠∫ ∫ ∫ ∫ ,

2

2x x

y y

ω πω π

==

27


8. ( , , )x y zδ and the Fourier Transform

8.1 3-Dimesional Delta Function

( ) 1 1 1, ,

2 2 2

yx z

yx z

x y z

i yi x i zx yx y z e d e d e d

ωω ωωω ω

ω ω ω

δ ω ωπ π π

=∞=∞ =∞

=−∞ =−∞ =−∞

⎛ ⎞⎛ ⎞ ⎛⎟⎜⎟ ⎟⎜ ⎜⎟⎟ ⎟⎜⎜ ⎜⎟⎟ ⎟= ⎜⎜ ⎜⎟⎟ ⎟⎜ ⎟⎜ ⎜⎟ ⎟⎜ ⎟⎜ ⎜⎟ ⎟⎟⎜⎝ ⎠ ⎝⎝ ⎠∫ ∫ ∫ zω

⎞

⎠

e d e d e d

νν νπ νπ ν π ν

ν ν ν

ν ν=∞=∞ =∞

=−∞ =−∞ =−∞

⎛ ⎞⎛ ⎞ ⎛⎟⎜⎟ ⎟⎜ ⎜⎟⎟ ⎟⎜⎜ ⎜⎟⎟ ⎟= ⎜⎜ ⎜⎟⎟ ⎟⎜ ⎟⎜ ⎜⎟ ⎟⎜ ⎟⎜ ⎜⎟ ⎟⎟⎜⎝ ⎠ ⎝⎝ ⎠∫ ∫ ∫

2

2

2

x x

y y

z z

ω πνω πνω πν

===

22 2yx z

yx z

x y z

i yi x i zx y zν

⎞

⎠,

8.2 3-Dimesional Fourier Transform

{ }( , , ) ( , , ) x y z

z y xi x i y i z

z y x

f x y z f x y z e dxdydzω ω ω=∞ =∞ =∞

− − −

=−∞ =−∞ =−∞

= ∫ ∫ ∫F

2 ( )( , , ) x y z

z y xi x y z

z y x

f x y z e dxdydzπ ν ν ν=∞ =∞ =∞

− + +

=−∞ =−∞ =−∞

= ∫ ∫ ∫ , 2

2

2

x x

y y

z z

ω πω πω πν

===

νν

8.3 3-Dimesional Inverse Fourier Transform

28


{ } ( )13

1( , , ) ( , , )

(2 )

yz x

x y z

z y x

i x y zx y z x y z x y zF F e

ωω ωω ω ω

ω ω ω

ω ω ω ω ω ω ω ω ωπ

=∞=∞ =∞+ +−

=−∞ =−∞ =−∞

= ∫ ∫ ∫F d d d

zd

νν νπ ν ν ν

ν ν ν

πν πν πν ν ν ν=∞=∞ =∞

+ +

=−∞ =−∞ =−∞

= ∫ ∫ ∫2

2

2

x x

y y

z z

ω πνω πνω πν

===

2 ( )(2 , 2 , 2 )yz x

x y z

z y x

i x y zx y z x yF e d d ,

8.4 3-Dimesional Fourier Integral Theorem

3

1( , , ) ( , , )

(2 )

yz x

x y z

z y x

i i if x y z f e d d d

ωω ω ζ η ξω ξ ω η ω ζ

ω ω ω ζ η ξ

ξ η ζ ξ η ζπ

=∞=∞ =∞ =∞ =∞ =∞− − −

=−∞ =−∞ =−∞ =−∞ =−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ×⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ ∫ ∫ ∫ ∫

( )x y zi x i y i zx y ze d dω ω ω ω ω ω+ +× d

( )( )1 1( , , )

2 2

yx

yx

x y


ωζ η ξ ωω ηω ξ

ζ η ξ ω ω

ξ η ζ ω ξ ω ηπ π

=∞=∞ =∞ =∞ =∞−−

=−∞ =−∞ =−∞ =−∞ =−∞

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎟ ⎜⎜ ⎟⎟= ×⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜ ⎟⎜ ⎟ ⎟⎜⎝ ⎠ ⎝ ⎠∫ ∫ ∫ ∫ ∫ d

( )12

z

z

z

i zze d

ωω ζ

ω

ω ζπ

=∞−

=−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟×⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ d

d

dζνν

2 ( )2 ( )( , , )yx

yx

x y


νζ η ξ νπ ν ηπ ν ξ

ζ η ξ ν ν

ξ η ζ ν ξ ν η=∞=∞ =∞ =∞ =∞

−−

=−∞ =−∞ =−∞ =−∞ =−∞

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎟ ⎜⎜ ⎟⎟= ×⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜ ⎟⎜ ⎟ ⎟⎜⎝ ⎠ ⎝ ⎠∫ ∫ ∫ ∫ ∫

2 ( )z

z

z

i zze d

νπ ν ζ

ν

ν=∞

−

=−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟×⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ,

2

2

2

x x

y y

z z

ω πω πω πν

===

29


9.

Delta Sequence 2

sin( )( )n

n nxx

nxδ

π

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

We show that the Hyper-real Delta Function is represented by the

Delta Sequence 2

sin( )( )n

n nxx

nxδ

π

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠.

The nth component of the Hyper-real Delta is 2

sin( )n nxnxπ

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠. That is,

2sin( )

( )n nx

xnx

δπ

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠.

9.1 Each 2

sin( )( )n

n nxx

nxδ

π

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

has the sifting property , ( ) 1x

nx

x dxδ=∞

=−∞

=∫

is a continuous Hyper-real function,

peaks at to 0x = 1(0)n nπ

δ = .

Proof:

30


12

2 2

2 2 20

,by Spiegel2

sin ( ) 1 sin ( )( ) 2 1

x x x

nx x x

n

n nx nxx dx dx dx

nn x x

π

δπ π

=∞ =∞ =∞

=−∞ =−∞ =

⎡ ⎤⎣ ⎦

= =∫ ∫ ∫ = .

To see that 1(0)n nπ

δ = , we use the infinitesimal 21n

. Then,

2

211

1

sin( )( ) nn n

n

nδ

π

⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠ .

Since for any infinitesimal ο , sin( )1

οο

= , we have

21( )n n

nδ

π= ,

and 1(0)n nπ

δ = .

Therefore,

9.2 The sequence represents the Hyper-real Delta

2 2 2

2 2 2

sin ( ) sin (2 ) sin (3 )( ) , , ,...

2 3

x x xx

x x xδ

π π π= .

9.3

plots in Maple the 1000th component, that peaks at 1000318

π≈ .

31


9.4

plots in Maple the component, that peaks at 610610

318,310π

≈ .

32


To show the relation between the infinitesimal dx , and this

Hyper-real , we note ( )xδ

9.5 If dx is given by 1ni n= ,

Then This Hyper-real ( )xδ

H peaks to 1dxπ

.

H may be written symbolically by 2

sin( )1( )

xdxxdx

xdx

δπ

⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

33


References

[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all Infinities,

and the Continuum Hypothesis” in Gauge Institute Journal Vol.6, No. 2, May

2010;

[Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal Vol.6,

No. 4, November 2010;

[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute Journal

Vol.7, No. 4, November 2011;

[Dan4] Dannon, H. Vic, “Riemann’s Zeta Function: the Riemann Hypothesis

Origin, the Factorization Error, and the Count of the Primes”, in Gauge

Institute Journal of Math and Physics, Vol.5, No. 4, November 2009.

A 2011 revision, published by the Gauge Institute, follows the development of

the Infinitesimal Calculus [Dan3], that is necessary to establish Riemann

results.

[Dan5] Dannon, H. Vic, “The Delta Function” in Gauge Institute Journal

Vol.8, No. 1, February 2012;

[Riemann] Riemann, Bernhard, “On the Representation of a Function by a

Trigonometric Series”.

(1) In “Collected Papers, Bernhard Riemann”, translated from

the 1892 edition by Roger Baker, Charles Christenson, and

Henry Orde, Paper XII, Part 5, Conditions for the existence of a

definite integral, pages 231-232, Part 6, Special Cases, pages

232-234. Kendrick press, 2004

(2) In “God Created the Integers” Edited by Stephen Hawking,

Part 5, and Part 6, pages 836-840, Running Press, 2005.

34


[Spiegel1] Spiegel, Murray, “Fourier Analysis with applications to Boundary

value Problems” Schaum’s Outline Series, McGraw-Hill, 1974.

p.80 has the sufficient conditions for the Fourier integral Theorem

[Spiegel2] Spiegel, Murray, “Mathematical Handbook of Formulas and

Tables” Schaum’s Outline Series, McGraw-Hill, 1968. p. 96, #15.36

35

delta function, the fourier transform, and fourier integral theorem

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