The Fourier transforms of Lipschitz functions on certain domains

Internat. J. Math. & Math. Sci.

VOL. 20 NO. 4 (1997) 817-822

817

THE FOURIER TRANSFORMS OF LIPSCHITZ

FUNCTIONS

ON CERTAIN

DOMAINS

M.S. YOUNIS

Department

ofMathematics

YarmoukUniversity Irbid,

JORDAN

(Received December 13,

1995)

ABSTRACT. The Fourier transforms of certainLipschitz functionsare discussed andcomparedwith the Hankel transforms ofthese functions and with their Fouriertransforms onthe Euclidean

Cartan

Motiongroup

M(r),

n

>

2

KEY WORDS

AND PItRASES:

Fourier transforms, Fourier series, Lipschitz functions, absolute

convergence

1991 AMS

SUBJECT

CLASSIFICATION CODES:

Primary 42B05, 42B10; Secondary 43A30,

22E30

1.

INTRODUCTION

The famous Bemstein’s Theorem states that ifthe function

f(x)

belongs to the Lipschitz class

Lip(a),

c

>

onthecirclegroup

T

[0,

27r],

then its Fourier series isabsolutelyconvergent

(see [10],

Vol 1, p.

240).

Thistheorem hasbeenextended and generalizedinvarious directions. Thepurposeof thepresentwork is tostudythe Fouriertransforms ofvariousLipschitzfunctionsandtocomparesome

ofthe results obtained here with the Hankel transforms ofthese functions and with their Fourier

transformsonthe group

M(n).

The mainfeature ofourtreatment in this work is a

change

ofvariablesfrom Cartesiantopolar

coordinates This

appears

to bethe bindingthread whichrelates some results concerning theFourier

transformsof certainLipschitzfunctions withthe Hankel

(Fourier-Bessal)

transforms ofthesefunctions

Our

paperisorganizedasfollows.

In

section2 the

necessary

definitions and notations aregiven.

In

section 3 we study the Fourier transforms

(coefficients)

of various Lip-classes by changing from Cartesian to polar coordinates with special emphasis on radial functions. Section 4 contains a brief

comparison between some of our results with those obtained for the Hankel transforms ofcertain

Lipschitz

functionsandwiththe Fouriertransforms of these functions whendefined on

M

(n)

2.

DEFINITIONS AND

NOTATIONS

This section will be brief since mostof our definitionsand notations arestandard

In

the sequel

T,

T

,

R

and

R

denotethe circlegroup, then-dimensional toms, thereal line, andthe n-dimensional Euclideanspace respectively. Thefunction

f

of severalvariables is written as

f(z)

f(zl,z2, ...,z,,)

and

(x.V)

standsforthe inner

product

The

LP-space

consistsoftheequivalence classes of functions whose p-th powerisLebesgue-integrable

withtheusualsymbol

[[.[[p

for the

LP-norm

The r-th difference in step hfor

f(x)

isgiven by

z--0

For

anonnegativeincreasing function

w(z)

theweighted

L’-space

is definedsuchthat

[fl’wdz,

<

oo.

Our

definitions arestatedforfunctionsoftwovariablesfor the sakeofbrevity Their extensionto functions in

R

isquiteobvious.

DEFINITION 2.1.

Let

f(z)- .f(zl,Z2)

belong

to

L’(R

-)

We

say that

f

is intheLipschitz space

Lip(a1,

_

p)

if

f(zx

--bhl,X2--b

h2)

f(xx,X2

-b

h2)

I]--O(hlh)

(2 1)

I(Xl

"bhi,x2 -[-

f(Xl,X2)

p

as hi,

h2

tendto zero, 1

<_

p

<

oo, 0

<

al,a2

_<

1. Another definition which will befrequentlyused

throughoutisthe following

DEFINITION

2.2. The function

f(xl,x2)

belongs

to

Lip(a1,

a2,p)if

l]f(xl

+

hl,x2

+

h2)

f(xl,x2)][,

O(h

-+-

h’’)

(2 2)

wherea], a2,h, h2, and p are the same as in

(2.1)

The function

f(x)

is called radial if

The Fouriertransform

f

(y)

of

f(x)

is definedtobe

/

(, )

/

/

(

dx

If

f(Xl’X2)

e-’(zy+zm)dxldx2"

Integrationistaken over theplane

R

.

3.

FOURIER TRANSFOIES

OF LIPSCHITZ FUNCTIONS

In

this section we examine theeffect ofthesmoothnessconditions

(2.1)

and

(2.2)

on the Fourier

transform

f

of

f

(x)

by changingtopolarcoordinates. Themeritofthisapproach appearsinsimplifying the arguments in the

proof

andinunifyingthe versionoftheoremsprovedin different areasofFourier

analysis

For

brevity, theterm"transform stands here interchangeably for theFouriertransform and the

Fourier coefficientunless anyotherwise isexplicitly mentionl.

We

start withthe following theorem.

THEOREM

3.1.

Let

f(x)

belongto

L’(R2),

1

<

p

_<

2such that

(2.2)

issatisfied. Then

P P

min(cl a2).

for

(p

+

ap/2

--1)

<

f

<

q-

(p--l)’

a=

PROOF.

By

changingtopolar coordinates,withh

h +

h2

one findsthat

f(xl

+hl,x2+h2)-f(xl,x2)

isequalto

FOURIER TRANSFORMS OF LIPSCHITZ FUNCTIONSONCERTAIN DOMANS ₈₁₉ for instance,

f

is then written as

f

(u, n),

where u E

R

+

[0,

o),

_{and n is an integer Thus the} transformof g isgivenby

0)

(e

7

ApplyingtheHausdorff-Young inequalityandtakingintoaccount

that/h

,h2-

O(h),

a--m

in(a1 ,a2)

weobtain

f

O(hq).

If0

<

u

<

-,

then

luhl

< Alsinuhl,

A

beingconstantandtherefore

X

For

fl

_<

q and argument similartothat in Titchrnarsh

([4],

Theorem84, p

115)

yields

I]ldu

O(X-+I-Mq)(NI-Mq).

(3

1)

Since

N

and

X

areof the same orderof magnitudenearinfinity, hence

(3 1)

canbe written as

Ix]

x

Thisquantityisbounded nearinfinityif_(,/2-1)’e

<

,

and theproofiscomplete.

Anyone

who is familiar with the proof of this theorem in the ordinary Cartesian form would see the brevity and

compactness oftheproof givenhere

(see

[7]

and

[8]

forexample).

For

functions in

LP(Rn)

the theorem

holds if _{(,+/n- 1)}P

<

<

q, where the left side of

(3.1)

is to be replaced

by

another containing

one integral sign for the continuous variable u and

(n-

1)

summations for the discretevariables of

f

The

proof

is quite direct and is therefore omitted.

We

notice that if

(2.1)

is applied instead

of

(22),

then the second difference

/Xh(/Xaf)=/Xh,(/Xa’)

is equivalent to the quantity

/Xh(&J’

=/Xf(rosO,

rsinO)=/Xg(r,O),

where

h’h’=

O(h+’).

At

this point one is at libertyeitherto

keep

the choice 0

<

al,

c

_<

1,ortotake equivalently 0

_<

a

_<

2. Thus we state thefollowingtheorem.

TItEOREM

3.2.

Let

the hypothesis of Theorem 3.1 hold such that

(2.1)

is satisfied Then the

conclusionofthetheorem holdsif_{(.(o /,),/__)}

<

r,

or-equivalently-if_{(,,/- )}

<

.

PROOF. In

polarcoordinates

(2.1)

takesthe form

fo

fog-

i/X2hf(r

cosO,

r

sinO)l’rdrdO

O(h’)

(32)

andfinally

for/3 _<

q

IX] _x

1

11

]

[Z

du

O(X2-Oz-2B+2B/p)

[X]

beingtheintegralpartof

X.

The rest ofthe proofis obviousand thetheoremisproved Itisclear that the condition

(2.1)

is stronger than that in

(2.2);

this is reflected on the range of

3

in thetwo theorems

For

the proofand the conclusion of Theorem 3.2 in Cartesian coordinates the interested readermayconsult oneofthereferences

[7], [8],

for example. It should be observedthatforfunctions in

LP(Tn)

one is dealing with the Fourier coefficients

proper,

so the integral sign disappears and the

productof n-summations takesplace

It

might beconvenient to notice that thewayit isstatedhere,the condition

</3

demonstratesmoreobviouslythe relation between the smoothness exponenta and thespacedimensionnincaseoffunctions of several variables.

REMARK

3.3.

We

mention herethatTheorem3.2 was proved for functionsin

LP(Rn)

and in

LP(T’)

by ratherdifferentmethods from theonejustcarried

throughout One

may referto

[5] (satz. 1)

andto

[9] (Theorem

2.15, p.

39).

4.

RADIAL FUNCTIONS AND HANKEL TRANSFORMS

Our

aimhere istocarrytheabove analysis forradialfunctions. The importance ofthisapproachlies

i.n

thefactthat it linksupseveral ideassuchastheweighted Lipschitzspaces,the Hankeltransformsand thetransformsonthegroup

M(n)

In

this case

(2.1)

and

(2.2)

arewrittenas

(4 1)

O(hO,+),)

and

[Ahg(r)[Prdr

O(h

’

+

h

).

(4.2)

If

f(x)

is a radial functionofn variables, then

(4 1)

and

(4.2)

aretobe slightly modified,the second difference in

(4.1)

is replacedwiththen-th differenceand rn-1 appearsin both estimates. Itiswell known

(see

],

p.

69)

that theFouriertransform of theradial function

g(r)

is a radial functiongiven by

(27r)n/2

g(r)r

/2

Jz

(-,)d

(4.3)

(u)-

_u<-)/

This isjustoneof severaldefinitionsoftheHankel transformavailable in the literature. The Hankel

transforms of Lipschitzfunctionsandtheir Fouriertransformson

M(n)

willbe treated elsewhere

Our

main concern here is to prove that the Hankel transform of

(g(r

+

h)-

g(r))

is of the order of

(e

--

1)(u).

This can be achieved as in the previous analysis without difficulty or directly by appealingtoanidentity of the Besselfunctions

(see

[6],

p.206,

p.217)

givenas

1

fo

2

eiZOo.+modO"

Jn(x)

Thus one obtains

1

oo

2

e,(r_h)uco+,rdO"

((-

h))

In

view ofthe boundedness of cos0 and the compactness of

[0,

2r]

this quantity isof the order of

e-’hJ,(ur).

Thus thetransformof

(g(r

+

h)

g(r))

isa constantmultiple of

(e

-’h

1)(u)

_{minus a}

FOURIER TRANSFORMSOFLIPSCHITZ FUNCTIONSON CERTAINDOMANS 821

/-Ihld

o(h),

subsequently

andfor

3

_<

q

This isboundedfor large

X

if

[(a

+

1

+

nl2)p-

1-

n12]

(44)

Thespecialcasesn 1,n 2 areespecially importantsincetheyincludetheresults corresponding to two standard definitions ofthe Hankel transformwithweightsw x

1/2,

w x

We

hintthatthe

interplay between the transforms of radial Lipschitz functions and the Hankel transforms ofthese functions isquiterichespecially whenvarious typesofthesetransformsaretaken into consideration. The

L2-theory

isof particular significancein viewofitssymmetrywhichenablesusto stateour conclusions in avariety of equivalent forms

(see

[7], [8]

forthis

purpose)

We

turnnowtoabrief discussionofthe Fouriertransforms of Lipschitzfunctions defined on

M

(n)

(see

[6],

Chapter4and

11).

We

focusattention on

M(2)

first,withthepolarcoordinates r,0, asthegroupparameters

(no

Lie

algebraisneeded

here)

Letg E

M(2).

Then the Fouriertransform of

f(g)

isgiven by

f f

B(m,n,R)

.’.’

f(r,O,)e"+’("-’)J._m(Rr)rdrdOd

.

(4.5)

do dO

In (4 5)

wehavesuppresseda constantfactorwhichdoesnotaffectthevalidity ofour arguments

ForradialLipschitzfunctions in

2(M(2))

the previous analysisshows that

RIBI#dR <

oo

(46)

(V

+

op/4

1)

and

<

oo

(4 7)

IBIndR

if_(4p+p-4)3p

<

respectively.

In

caof

M(n)

whereitsdimensionm onefindsthat

(4 6)

d

(4.7)

e valid if

P

<

(p

+

p/(

+

l)

1

and

mp

successively This contains the results for

M(2)

where m 3 It should be mentioned that the introduction ofthe estimates in

(4 6)

and

(4.7)

is not a matter of sheer repetition

(they

look almost

equivalent)

In

facttheyare

(or

atleastoneof

them)

neededinorderto extractthe precise analogueof Bemstein’s Theorem concerning the absolute

convergence

of the Fourier transforms of Lipschitz

functions on

M(n)

Finallyweremarkthatalthoughthepresenttreatmentofthesubject providedsome links between certain branchesofharmonicanalysis,weemphasizethat itwould be even more valuable whenapplied to Lipschitz functions on more general

groups

For example ifone adoptsthe

Cartan

decomposition of the

group

SL(2

R)

(see

[3],

p. 252),

then it can be easily seen that the Fourier

transforms ofradial functions on

SL(2.R)

collapse simply to a form related to the familiar Melin

transforms. This remarkappliestoradialfunctions on certaintypes of semi-simpleLiegroupsas well as toradial functions onother classes of locally compact

groups

in

general

These topicswillbe handledin aforthcomingpaper.

ACKNOWLEDGEMENT.

Publication fees for this research were partially covered by Yarmouk

University

REFERENCES

BOCHNER,

S. and

CHANDRASEKHARAN,

K,

Fourier

Transforms,

Princeton UniversityPress

(1949)

[2)

HERZ,

C., On

the mean inversion of Fourier and Hankeltransforms,

Proc. Nat.

Acad.

Sc.

U.S.A.,

40

(1951),

996-999

[3

SUGIURA,

M.,

Unitary Representationsand HarmonicAnalysis, J Wiley, NewYork

(1975)

[4] TITCHM

SH, E.C.,

Theory

of

Fourer Integrals,

2ndEd., Oxford University

Press

(1948).

[5] TREBELS,

W.,

Uber Absolute Summier Barkeit

Von

n-Dimensioneien Fourierreihen und Fourierintegralen,

J.

Approxmatmn

Theory,2

(1969),

394-399.

[6]

VILENKIN,

N

YA.,

Specialfunctionsand theory of

group

representations,Transl.

Amer.

Math.

Soc.,

22

(1968)

[7] YOUNIS, M.S.,

FouriertransformsofDini-Lipshitz functions,

Inter.

J.

Math.

&

Math.

Sc.,

9,2

(1986),

301-312.

[8]

YOUNIS,

M.S.,

Fourier

transforms

offunctions withsymmetric differences, Acta. Math.

Hung.,

51

(1958),

293-299.

[9] YOUNIS, M.S.,

Fourier transforms ofLipsehitz functions on compact groups, Ph.D. Thesis,

McMaster

University, Hamilton, Ontario, Canada

(1974).