FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS ON

Internat. J. Math. & Math. Scl.

VOL. 15 NO. 3 (1992) 609-612 609

FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS ON

VlLENKIN

GROUPS

M.S. YOUNIS

Department

of Mathematics Yamrouk University

Irbid, Jordan

(Received

^{November30,1987andin revisedform}

January

17,

1991)

ABSTRACT. In [4]

^{we proved some} theoremson the Fourier Transforms offunctions satisfying conditions related to the Dini-Lipschitz conditions on the n-dimensional Euclidean space

R

ⁿ and the torus group

T . ^In

^{this paperwe extendthose theorems for}functions with Fourier series on Vilenkingroups.

KEY WORDS AND PHRASES.

Dini-Lipschitz Functions,Vilenkin Fourier Series.

1991

AMS SUBJECT CLASSIFICATION CODE.

42c,43.

1.

INTRODUCTION.

Let

f(z)

belong tothe

Lebesgue

spae

LP,1 <

p

_<

2 of functions onthe real line

R

oron the circle group

T

with its usual norm

H"

^p. The pth integral modulus of continuity

wp(f,h)is

definedas

In [4] (Theorem 3.3)

^weproved thatif

f(z) belongs

to

LP(R)

^{such that}

(f,h) ^{0(/(Log ),}

^o

^<

¹

then theFourierTrsfo

longs

to

L(R)

^where

pl(p +

^op-

1) < ^p’= pl(p- 1), > 11

theprint workwe sh extend

ts rt

forctionson

LP(G)

^whe

^G

^{is a}mpt

metrizab]ermension

AH

oup, i.e., Vilen oup.

2.

DEFINITIONS AND NOTATIONS.

Here

we intruceme dtions d notations that

H u Ist

on.

Ts

is by donesince wesh

my

follow

Oewr [I]

d Quek d

Yap [2]

ⁱⁿ

^ts rt.

t G

a Vilenoup. Thenits du

G

is a

&screte

co.table to.ion oup.

It

is

w o

that one c intruon

G

atable bic t of

n&ghurhs {Gn}

^{ofthe identity}

element

{e}

of

G

such that

G=GoDG

1,

DG2,...,

^d _=o

G={e}.

On the other

hd,

let

V

^{dote the latorin}

^G

^{of the}

^suboup Gn

ⁱⁿ

^G.

^{Thit is}

610 M. S. OUNIS

known that

{e}=V oCv

I,

C,

and thatI 0V

n=

if all

V,

^are finite, the inclusionisproper.

We

introduce thenumbers mo,ml, m2,...,mk such that

mo=l,m_k+l=pkmk;

keN,

pkbeing^aprime

>_

2. Then evetT

V,

hasm,^asits measureand thequotientsubgroup

Vn/V,_

has

P,

forits measuxe.

DEFINITION

2.1.

For zG,

let

(n,z)

denotethe continuouscharacter of z,i.e.

(,,z)G.

The Fouriertransform

(,)

of

f(z)L’ (G)

^isdefinedby

2(") I/()(.,)d

G where

(n,z)

isthe complex conjugate of

(n,z).

DEFINITION

2.2.

Let f(z)L

^p

(G).

The pth modulus ofcontinuity

o;p(f.k)

^isdefinedby

The Lipschitz and the Dini-Lipschitz

ciasses

in

LP(G)

^{are thosefor which}

w),(f,k)=

⁰

(m -(*)

and

’o Log rail

^-1respectively.

DEFINITION

2.3 if evexT

P/

^{is finiteask--)oo wesay that}

^G

has the boundedness property

(P).

3.

MAIN RESULTS.

Withthe previous definitions and notations in

hand,

^we now provethe

analogue

ofTheorem 3.3 in

[4].

^{Thuswestatethefollowing}

THEOREM

3.1.

Le f(z)eL)’(G),

¹

<

p

s

2, such that

wp(f,k) 0(m(*/(Log mk)’),

^o

<

^a

_<

1.

(.1)

Then

(n) 1)()

^{for q}

^P/(p- 1) _> > max(p/(p+ap- 1), 1/7).

PROOF.

Since the Fourier transform of

f(z+h)-f(z)is

^{given by}

f(n)(n,h)-1),

the

Hausdorff-Yotmg

theorem yields

(.)I

^q

(.,h)-

1 ^q

< Mwp(/,k)

^q

O(m-q/(Logm, ^k)’q).

G

Theboundedness property

(P)

for

C

gives

(see ^Oxmeweer [1], (2)).

ink+

-1

I?(-)1’ 0(m;’/(Log,.)q

ApplyingtheHolder’s inequalitywith

9 ^s

^qfor the lastestimate onearrives at

mk+l

-1

f(.)l 0(.;/(Log mF ^{) (m- /’)}

and thisleadstothe finalestimate

?(.)

0

( (m -- ^{+ /) (Log m-).}

eG ^{k 0}

FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS 611 If1

a + /V ^<

^{o and}

7 ^<

^{-I theseriesis convergent since}mf

_>

²

f,

^and this proves the theorem.

REMARK

^3.2.

We

remark here that for special choice of a,7, and

P

^like. a

I,

₇ 1,

P

2,the previous theorem gives special interesting^cases. This isquiteobviousandweshallnot deal with it anyfurther.

However,

the special case

P

2 and o

<

^c

<

1 is particularly important and deservesspecialconsideration.

4.

FUNCTIONS IN L2(G).

^The origin of this sectionis a theorem proved in Titchmarsh

([3]

Theorem85,p.

117)

^forfunctionsbelongingto

Lip(a,2)

onthe realline

R. For

further referencewe stateitas.

THEOREM

4.1.

Let

j’(z)Lip

(a,2)

^on

R.

Then the conditions

to2(f ,h) O(h’)

O<a<l, h^--,o and

[oX+xTl[[

^2du

^=0(X-2),

^asX-,oo

areequivalent.

This theorem was studiedrather extensivelyin

[5]

^and

[6]

for functions in

L2(R2),

^and

L2(T 2)

respectively, where several conditions of theorderofmagnitudefor theFourier transforms j of

f

provedtobeequivalenttooneanother.

In [4] (Theorems

5.1,

5.2)

^we proved an analogue of Theorem 4.1 for the Dini-Lipschitz functions in

L2(R). ^In

this section weshall prove Theorem 5.2 in

[4]

for functionsin

L2(G).

THEOREM

4.2.

Let f(z)

belongto

L2(G).

^Thenthe conditions

w2(J’,k ^0(hO/(Log h)7),

^hG_k

(4.1)

areequivalent.

Here

h

m ^-1.

PROOF.

That the first implication is tree follows from Theorem 3.1 where it is proved that

I](,,)I o(,(f,))q

n=mk

Taking p q 2 and substitutingfor h

rn

^{-1 we obtain}

^(4.2).

^We also hint that anargument basedon the Parseval’s identity similar to that of Titchmarsh’s leads independently to thesame result.

To

prove theconverselet

(4.2)

^{hold. Then}

mk+

-1

](n)

²

0(m-2/(Logrn’ 0(m’2+al/(Log rn& ₊ 1) 27) (4.3)

SinceG has the boundedness property

(P);

hence every

Pk =mt ₊ 1link

^{isfinite for all}

^keN,

^the

sameistrueof

Log P

_k. Thustherighthand sides of

(4.2)

^and

(4.3)

^{are thesame. Thisappliesto}

estimatesof the form

ink+ 1-1

and

[.(n)[

²

0(m-2a/(Log(m) 7) (4.2)

612 M. S. OUNIS

To

sumup,bythe Parseval’s identityoneobtains

[[f(z+h)-f(z)[2dz= _ [(n)[2[(n,h)-l[2

G G

oo

mk

+ --01 m’2o/(Log m)27

m/r ^o

This isequivalentto

(4.1)

^uponsubstituting for h

m ^"1,

^{heGtand theproofiscomplete.}

REMARK

4.2.

We

concludeby indicating that Theorem5.1 in

[4]

^{istrueforVilenkinFourier}

series, since, it canbe deduced asa special caseofTheorem4.1. Wealso add that for 0<^c

<

^1, Theorems 3.1 and4.1 of thepresentpapercanbeprovedforhigherdifferences of

f(z)eLP(G).

^The

statementsand theproofsarealmost straightforward andwillnotbegiven.

ACKNOWLEDGEMENT:

Thisresearchwassupported

by

agrant from Yarmouk University.

REFERENCES

1.

ONNEWEER, C.W.,

Absolute

Convergence

ofFourier Serieson Certain

Groups. II, ]ke

Math,

Joal,

Vol.41

(1974),

679-688.

2.

QUEK, T.S.

and

YAP, L.Y.H.,

Absolute

Convergence

of Vilenkin-Fourier. Series,

J.

Math.

Analysis A_v_vl. Vol.74

(1980)

^1-14.

3.

TITCHMARSH, E.C.,

TheoryofFourier

Integrals,

2ndEd. OxfordUniv.

Px’ess,

1948.

4.

YOUNIS, M.S.,

Fourier Transforms ofDini-Lipschitz Functions, ]Jtt.xtAL

J.

Math.

&

Sci.Vol. 9

No.

2

(1986)

301-312.

5.

YOUNIS, M.S.,

FourierTransformsin

L

^pspaces,

M.

Phil.Thesis, Chelsea Coll., London, 1970.

6.

YOUNIS, M.S.,

FourierTransformsofLipschitzFunctionsoncompact

Groups,

Ph.D. Thesis,

McMaster

Univ., Hamilton, Ontario,

Canada,

1974.