ON CONVOLUTABLE FRECHET SPACES OF DISTRIBUTIONS

NISHU GUPTA Associate Professor, Department of Mathematics, Maharaja Agrasen Institute of Technology, Delhi, India

Abstract - In this paper, many results of Fourier analysis which are known for convolutable Banach spaces of distributions (BCD-spaces) andFrechetspacesofdistributions (FD-spaces) have been generalized to convolutable Frechet spaces of distributions (CFD-spaces). Also, we discuss the dual space of a CFD-space and obtain some useful results about homogeneous CFD-spaces.

Key Words: Fourier analysis, Banach spaces, Frechet spaces, circle group, dual space and homogeneous space.

1. INTRODUCTION

In[7]someresultsofFourieranalysis,whichareknownfor (1), p LpC and M etc., were obtained for convolutableBanachspacesofdistributions(BCD-spaces). But those results cannot be applied to some important spaces like  C (the space of all infinitely differentiable functions). Also,in[6]someresultsofFourieranalysiswere obtainedforFrechetspacesofdistributions(FD-spaces).But those results cannot be applied to some important spaces like Hardy’s spaces (1) p Hp To overcome these deficiencies,inthispaperwedefinetheconvolutableFrechet spacesofdistributions(CFD-spaces).

In Section 2, we define CFD-spaces and state some preliminaryresultsdealingwith CFD-spaces InSection3,we definehomogeneous CFD-spaces andobtainsomeimportant results about homogeneous CFD-spaces In Section 4, we discussthedualspaceofa CFD-space andobtainsomeuseful results

2. DEFINITIONS, NOTATIONS EXAMPLES AND PRELIMINARY RESULTS

Wereferto[1],[5]and[8]forallthestandarddefinitions, notations and assumptions. In particular, all our distributions are assumed to be defined on the circle groupGR/2Z, and the space of all distributions is denotedby D.

2.1 Definition

AFrechetspace E iscalledaconvolutableFrechetspaceof distributions,brieflya CFD-space,ifitcanbecontinuously embeddedin(D, strong*),andif,regardedasasubsetof D;it satisfiesthefollowingproperties:

(2.1) E E M      f f   , ,whereM denotestheset ofall(Radon)measures.

(2.2) CE isaclosedsubspaceof  C

It is obvious that every BCD-space is a CFD-space (see the definitionof BCD-spacein[7]).But  C isa CFD-space which isnota BCD-space asitisnotaBanachspace.

Throughout the paper, E, if not specified, will denote a CFD-space and E* willdenoteits strong* dual(see[8],Ch.10).

2.2. Wenowgiveanexampleofanon-emptyFrechetspace E continuously embedded in D which satisfies the assumption(2.1)butnot(2.2).

Let E bethesetofall  C f suchthat   E f where

Then E is the required space as every Banach space is a Frechetspace(see[7]).

2.3. Now,wegiveanexampleofanon-emptyFrechetspace E continuouslyembeddedin D,suchthat(2.2)issatisfiedbut not(2.1).

Let (G) dM denotethesetofpurelydiscontinuousmeasures on G i.e.,thesetofallthosemeasures μ on G forwhichthere existsacountablesubset A of G suchthat ()0 CA .Then (G) dM istherequiredspace(see[7]).

The above examples show the independence of both the assumptionstakeninthedefinitionofa CFD-space.

2.4. Theorem. Let E be a CFD-space. Then the transformation E E M   : S definedby f f S     ) , ( foreach M  and E  f , is continuous on E M  . Further, for each continuous seminorm p on E ,thereexistsacontinuousseminorm q on E suchthat ) ( || || ) ( 1 f q f p     foreach M  andfor each E  f .

Proof: Fixing μ in M, consider the mapping E E  : T1 defined by f f T    1 for all E  f . Now, E is continuouslyembeddedin D and 1T iscontinuouson D.So, usingtheclosedgraphtheorem,wecanshowthat 1T isa continuouslinearoperatoron E.Similarly,ifwedefinethe mapping E M  : T2 by f T     2 for all E   , after fixing f in E, then 2T turns out to be linear and continuous. Thus the transformation E E M   : S , defined by f f S     ) , ( is bilinear and separately continuous;andhenceby([5],p.52), S is(jointly)continuous on E M 

Further, by ([2], Exercise1, p.363), for each continuous seminorm p on E,thereexistsacontinuousseminorm q on E suchthat

M

andforeach E  f .

Taking asthenthFejer'skernelintheabovetheorem,we obtainthefollowing.

2.5. Corollary. If E isa CFD-space,thenforeachcontinuous seminorm p on E,thereexistsacontinuousseminorm q on E suchthat ) ( )) ( ( f q f p n   where f F f n n    and n F denotes the nth Fejer's kernel.

Taking μ as the Dirac measure at the point x in the above theorem,weobtainthefollowing.

2.6. Corollary. If E is a CFD-space, then E is translation invariant and for each continuous seminorm p on E there exists a continuous seminorm q on E such that ) ( ) ( f q f T p x  foreach G x andforeach E  f . where f T x denotesthetranslationof f by x Also,itshowsthat {:} G x Tx  isanequicontinuousfamily oftranslationoperatorson E

Usingtheclosedgraphtheorem,andthefactthat CE isclosedin  C ,wecaneasilyprovethefollowing.

2.7. Lemma. The inclusion map : iCEE is continuous,where CE hastherelativetopologyof  C .

2.8. Theorem. A necessary and sufficient conditions for PE to be dense in E is that CE is dense in E, where P isthesetofalltrigonometricpolynomials.

Proof: Onepartisobviousas   C P

Conversely,suppose CE isdensein E.Let d bethe metricon E inducedby

1 k kp is a countable family of seminorms on E which defines the locally convex topology of E Given f in E and 0   ,thereexists uCE suchthat /2 ) , (

Since : iCEE is continuous by above lemma, u u σ n  in CE with relative topology of E. So correspondingto 0   ,thereexists 0  N suchthat /2 ) , ( 

d n forall N n 

Hence,

nn d σufdfuduu forall N n 

Therefore E P  isdensein E

3. HOMOGENOUS CFD-SPACES

HomogenousBanachsubspacesof 1L definedonthecircle group G are discussed in [3] and many results of Fourier analysis on these spaces are generalized to homogeneous BCD-spaces in[7]andtohomogeneous FD-spaces in[6].We definehomogeneous CFD-space asfollows.

3.1. Definition. ACFD-space E issaidtobehomogenousif 0x x  in G implies f T f T x x 0  in E foreach E  f

3.2. Theorem. Every homogeneous FD-space is a homogeneous CFD-space

Proof: Let E be a homogeneous FD-space From the definitionofa FD-space andTheorem5.5(iii)of[6], E E M      f f   , and CE=C is closed in  C as CE .Thus E isa CFD-space

Butconverseoftheabovetheoremisnottrueasshownin thefollowingexample

3.3. CFD-spaces which are not FD-spaces For 1 p ,considertheHardyspace p H definedby ˆ {:()0for0}pp HfLfnn .

Clearly p H is a homogeneous BCD-space for 1 p ([7],2.2)andweknowthatevery BCD-space isa CFD-space, therefore p H isahomogeneous CFD-space for 1 p . But p H isnota FD-space as  C isnotcontainedin p H

3.4. Exampleofanon-homogeneous CFD-spacesis M

Theproofofthefollowinglemmaandtheoremarejustlike theonesgivenin5.2and5.3of[6].

3.5. Lemma. Let E beaFrechetspace,  acontinuous Evalued function on G and  1 } { n n K a summability kernel (approximateidentity),then

Now g T g T x x 0  in  C as 0x x  and therefore by Lemma2.7, g T g T x x 0  in E as 0x x  .So 0   such that

(3.2) /3 ) , ( 0 0      g T f T d x x x x

Also, 0 (,) xx dTfTf 000 (,)(,)(,). xxxxxx dTfTgdTgTgdTfTg Thusby(3.1)and(3.2),   ) , ( 0 f T f T d x x for  0x x

Hence, f T f T x x 0

UsingTheorem2.8,Theorem3.6andtheabovetheorem,we getthefollowing.

3.8. Corollary. Let E be a CFD-space. Then the following fourresultsareequivalent:

(i) E ishomogeneous;

(ii) E P  isdensein E;

(iii) CE isdensein E;

) ( ) ( 2 1 lim  

dt t t Kn n wheretheintegralistakenover G.

 

3.6. Theorem. If E isahomogeneous CFD-space,thenfor each E  f , f f n   in E as   n , where f n  denotesthe n-thCesarosumoftheFourierseriesof f

3.7. Theorem. A CFD-space E ishomogeneousifandonlyif E P  isdensein E

Proof: Let E behomogeneous.Thenbytheabovetheorem, for each E  f , f f n   in E as   n . Hence E P  isdensein E

Conversely,suppose E P  isdensein E.Let E  f and 0   .Since {:} G x Tx isanequicontinuousfamilyof translationoperatorson E,byCorollary2.6.,thereexistsa 0   suchthat

(3.1) /3. ) , ( ) , (      g T f T d g f d x x

SinceCE isdense in E (seeTheorem2.8),wemayfix afunction g in CE suchthat   ) , ( g f d

(iv)

4. DUAL SPACES

 n .

If E isaFrechetspace,then E* neednotbeaFrechetspace. So,if E isa CFD-space,wecannotsaythat E* isalsoa CFDspace. However we can still embed E* in D and treat the elementsof E* asdistributions.Forthiswehavetomakea onetoonecorrespondencebetweentheelementsof E* and theelementsof D.Thetaskwouldhavebeeneasierif  C had been contained in E, but it is not. So, we define the followingsets

(4.1) } : { E    ne Z n S , } | :| { N n S n S N    and

The proof of the following four results is similar to the correspondingresultsin[7].

4.1. Lemma. E  ne if and only if 0 ) ( ˆ  n f for some E  f

Using the definition of S and the above lemma we immediatelygetthefollowing.

Fromnowonwards,by

weshallmean

ka weshallmean

Then P isacontinuousprojectionof

k ka and,by

C onto CE .

Letusdefine jED :   by jFuFPu ()()()  forall u in  C andforall F in E* where P istheprojectiondefinedintheabovelemma.

Now,wecanclaimthefollowing(See[7],Theorem4.1)

4.4. Theorem. Let E be a CFD-space. Then E* can be continuouslyembedded into D throughthemapping j and as a subset of D, it satisfy the following properties: (4.2) , ff   MEE

(4.3) *   CE isaclosedsubspaceof  C

4.5. Theorem. Let E be a homogeneous CFD-space and   E F .Thenfor each E  f ,thefunction ()gx ,defined by ()() x gxFTf  forall G x ,iscontinuouson G and generatesthedistribution Ff 

 .Moreover,thereexistsa continuousseminorm q on E suchthat ||*||() Ffqff    E .

Proof: As E ishomogeneous, 0 0 inGT xx xxfTf  in E .

Also, F iscontinuouson E.Hence ()gx iscontinuouson G for each E  f . Now, by the Corollary 2.6., the family

{:} G x Tx of translation operators on E is equicontinuous, therefore {:} G x FoTx  is also equicontinuous.So,usingTheorem9.5.3([4],p.203),there exists a continuous seminorm q on E such that |()||()| x gxFTf  () qff  E and G x  .Hence, (4.4) ||||() E gqff  

Now our aim is to prove that ()gx generates the

distribution Ff   .Define ()() nnx gxFTf   forevery x in G.Since E ishomogeneous,byTheorem3.6, nxx TfTf   in E

Consequently, (4.5) ()()()() nnxx gxFTfFTfgx   as n  ,for every x Now,forevery f in E, lim()lim()() nn nn FfFfFf

as E is homogeneous. By([8],Theorem9.3.4.), {} n F  isanequicontinuousfamily of continuous linear functional on E which converges pointwiseto F in E* .Thereforearguing just asabove,we can find a continuous seminorm q  in E such that for all positiveintegers n ,

Hence ()gx generatesthedistribution Ff

, i.e., gFf

 inthesenseofdistribution.

Now,from(4.4), ||||()

forall f in E.

4.6. Theorem. A CFD-space E ishomogenousifandonlyif forevery E  f and   E F ,theseries

(4.6)  

) ( ˆ ) ( ˆ n f n F is(C,1)-summableto ) ( f F

Proof: OnepartisclearfromTheorem3.6andthefactthat ) ( f F n is the n-th Cesaro sum of (4.6). Conversely, supposethattheseries(4.6)is(C,1)-summableto ) ( f F forevery f in E and F in E* .Then ) ( ) ( lim f F f F n n

Thisshowsthat f n convergesweaklyto f in E forevery f in E. Thus E P  is weakly dense in E. Since E P  is

convex, E P  isstronglydensein E by([5],p.66).Hence byTheorem3.7, E ishomogenous.

5. CONCLUSIONS

In this paper we define Convolutable Frechet Spaces of Distributions(brieflywrittenas CFD-spaces)andgeneralize thepreviouslyknownresultsto CFD-spaces.Wehaveused various results and techniques of Functional analysis to obtaintheseresults.Theresultsobtainedwillbeusefulfor further analysis in the field of Fourier analysis and Functionalanalysis.

ACKNOWLEDGEMENT

My deep and profound gratitude to (Late) Dr. R.P. Sinha, Associate Professor, Department of Mathematics, I.I.T. Roorkeeforhis valuableguidance.Ido nothave words to thankhim.Hewillbeaconstantsourceofinspirationforme throughoutthelife.

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