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