Tutorial 3 : Selected problem of Assignment 3

Leon Li

3/10/2018

Q1 ) ( HN 3 Ex.

4) Reference : Fourier Analysis : an Introduction(by Stein - Shak archi

)

I et T : Eaa ] → IR be a 21T - periodic integrable function

with Fourier coefficients an,

bn EIR.

For each of rsl.

define an infinite series of functions by

co

fix ) : = aot E r" ( ancosnxtbnsinnx )

h -

- I

(a) Show that frlx ) defines a 21T - periodic continuous Function.

lb ) Show that fray = ¥,Prksflx - 2) dz

I - r'

is called the Poisson kernel.

wherePrk ) =

I - zrcusztt '

(c) If in addition T B continuous at x,

then

tiff fix = fix )

Sol : (a) Recall that rnlcosnxtisinnx) =Me" "

!.

rn cos n×=

He " "

t Tei=

rneinx + me- inx

2 2

rneinx - rue- inx

Similarly rhsinnx =

zi

Formally : frlx )

⇒aot Ene

,

rnlancosnxtbnsinnx )

= aot Ff,

fan . rn (e'

'

"

tze.

ing + bnrnqeinxz.ae-

inxgy= II. r"'

Cne" "

, a Cn EE

:.

suffice to show E. or " '

Cne" "

converge uniformly on IR :

V-nc-2.hn/=IFafiItlyle-

in "

dyI EM

,I MEIR

.

Vx EIR,

I r' " Cneinx ) fMrm

s a

1

and r' "

= 1+2 rn Ctcs as rel

.

.

. By M - test,

frm = II. V' "

Cne" "

converges uniformly on R

and hence is 21T - periodic continuous as so is true for all

rhlcneinx.

(b) Trixie E. rt "

one' "

= E. r' "

( ÷, ,

fly,e-

imdy) einx

= ÷, fifth(E. r' "

e- inYeinx)dy y

( by uniform convergence of It,I

' E.Itlyir " 'e

-

in ein 4)dy )

= IT,

flx- 2) II

or " ' ein 2) fdz ) l by change of variable z=x - y )

= ITS tix . alII.r' " ein YdzTherefore

, it suffices to establish the following identity:

Lemma " Prk II!! + r

.= II. r' " ' ein -

Pf of Lemma : Let w := rei ? then RHS = If Wnt IF,

-w"

= Iw t To . If =l I - b) t Tull - w )

=

I - Iwf( I - w ) ( I - To ) ( I - w ) LI - Tu )

=I - r

'

I - r.

Cl - rei 2) ( I - re- i 2 )

=

I - Zrcosz + rt - FA

:.

Trix ) = IT,

Prk ) fix - z ) dz

(c) We first prove the following three properties of { Prk ) ).

i

( PL ) Hoshi,

IT Prksdz =L

I P2 ) 7- KEIR s . t . to era , IT,

lprlzildz

EkI P 3)

Vosssit

. fine. I

lprkildz= O

SEKI SIT

Proof of IPD : IT Prksdz = IT, II.r' " ein ) dz

= ¥ IE r" ' eimdz I by absolute convergence of ¥

, f II. Ir " '

e' ' Hdz)

= It,

dz C- : tinto, fine " ' da = o ) =L - #

Proof of ( P2 ) Note that Prk ) = Iw⇒ 70 for all of Kl

and - EZE IT

i. Choose K =L,

then ( PI ) implies IT,

lprkildz =L = k - I #

P root of 1133 ) For each

0484T,

let Csis I - cos

8>0,

then

I - 2r cos z + r'

= ( I - h 't 2 r ( I - cos z ) Z Ot 2. ( I - cos D= Cg SO

there I,

SEKI ET. .

: I

PrestEll - r

' ) . Is

c

'

. tiff ftp.fz.HPrksldz E t.IT. # Xi- ri 's ) . zits ) = o - #

Proof of trim frm = TH ) it f 13 continuous at x :

Given E > O , by continuity of fat x,

there exists 870 such that

I fix . z ) - That CE,

t 12158

Try to estimate fr CX ) - FIX) : fr H ) - fix ,

= Ifa

Prlzsfcxesdz - IT,

Prk , -5kHz (by lb) and (P2 ) )

= Ia fta

Prk ) ( TH - z ) - fan ) dz

:.

Hrh -

ThatE Ia Ii lprksltflx . zi - tlxsldz

= IT fuglprhllflx-4-fcxsldztzt.ge/z,,!PrlzslHcx-zs--scxsldz

= I t I

F-or I : fr ,

I E Iq f lprksl .

Edz E ¥,

' lprksldzf.FI,

kits I by ( P2 ) )

For I : ILS 24744ftp.klldz LE,

⇒ roll . Frosh I

844511 ( by LP 3 ) )

:. tf road

,Ifrlxi - THI E It If ELITE = El ÷ tl )

.

'

. fifty -4M - full = O,

i.e . ftp.frlxs-flx ) - E

Rink : I I ) (Pl ) -

433) says that { Prk )3r→

,- is a

"

good kernel "

which in particular ensures that (c) is true.

(2) Dirichlet kernel { Dark) ) n → assatisfies D. CPD but NOT LP2 ) :

In fact,

Property IV in Lecture Note Ch.

Isays the contrary :

Is > O, ) IDnksldz → as as n → as

12158

Therefore,

estimate in I fails,

unless f has some decay condition at x

e.g .f is Lipschitz continuous at x

.