International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 11 Issue: 07| July 2024 www.irjet.net p-ISSN: 2395-0072
A RELATIONSHIP BETWEEN THE CONVOLUTION AND TRANSLATION OPERATORS FOR
CONVOLUTABLE FRECHET SPACES OF DISTRIBUTIONS
NISHU GUPTA
Associate Professor, Department of Mathematics, Maharaja Agrasen Institute of Technology, Delhi, India
Abstract - In this paper, a relationship between the convolution and translation operators is established for ConvolutableFrechetSpacesofDistributions(CFD-spaces).
Key Words: Fourier analysis, Banach spaces, Frechet spaces, Locally convex spaces, circle group and homogeneous space.
1. INTRODUCTION
If E is (1) p Lp orC , fE and 1gL ,thenitis shownin[1,Vol.I,3.19]that gf isthelimitin E offinite linearcombinationsoftranslatesof f .Also,arelationship between the convolution and translation operators is establishedin[2]and[5]forConvolutableBanachSpacesof DistributionsandforFrechetSpacesofDistributions(FDspaces).Inthispaper,weextendthatresulttohomogeneous ConvolutableFrechetSpacesofDistributions(CFD-spaces).
2. DEFINITIONS AND NOTATIONS
Allthenotationsandconventionsusedin[3]and[6]willbe continued in thispaper.In particular,GR/2Z will denotethecirclegroupand D willdenotethespaceofall distributionson G.Fortheconvenienceofthereader,we repeatthefollowingdefinitionsgivenin[3].
2.1 Definition
AFrechetspace E iscalledaconvolutableFrechetspaceof distributions,brieflya CFD-space,ifitcanbecontinuously embeddedin(D, strong*),andif,regardedasasubsetof D;it satisfiesthefollowingproperties:
(2.1) , MfEfE ,whereMdenotesthe setofall(Radon)measures.
(2.2) CE isaclosedsubspaceof C .
Throughoutthepaper E ,ifnotspecified,willdenotea CFDspaceand E willdenoteits strong* dual(see[7],Ch.10).
2.2. Definition
A CFD-space E issaidtobehomogenousif 0xx in G implies 0xx TfTf in E foreach fE
3. SOME USEFUL RESULTS
Using Theorems 3.25 and 3.27 of [4] we can state the following.
3.1. Theorem. Let X be a Frechet space and be a complexBorelmeasureonacompactHausdorffspace Q.If : fQX is continuous, then the integral Q fd exists(andliesin X)suchthat
() QQ FfdFfdFX
3.2.Theorem. If pisacontinuousseminormonaFrechet space X and Q yfd isdefinedasabove,thenthere exists FX suchthat ()() Fypy and |()|() Fxpx forall xX
(SeeTheorem3.3of[4]).
Inparticular,
QQ pfdFfd
()()||QQFfdpfd
Theabovetworesultswillbeusefultofindtherelationship between the convolution and translation operators for homogeneous CFD-spaces.
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 11 Issue: 07| July 2024 www.irjet.net p-ISSN: 2395-0072
4. CONVOLUTIONS AND TRANSLATIONS
4.1. Theorem. Let M and E beahomogeneous CFD-space.If gE ,then g isthelimitin E offinite linearcombinationsoftranslatesof g .
Proof: Let g V denote the closed linear subspaceof E generatedby () a TgaG ,i.e.,theclosurein E oftheset ofallfinitelinearcombinationsofelements a Tg.Weshall firstprovethetheoremfor f in 1L insteadof in M
Let 1 : gSfLfgV . Now obviously S is a linearsubspaceof 1L .Also S isclosedin 1L .Toseethis,let f bealimitpointof S .Thenthereexistsasequence nf in S suchthat n ff in 1L .ByTheorem2.4.of[3], n fgfg in E .But ng fgV forevery n,and g V isaclosedlinearsubspaceof E ,therefore gfgV andhence fS .Thus S isclosedin 1L
Now we shall show that S contains all characteristic functionsofthesubintervalsof [0,2] .Toseethis,first take I f where [,]Iab and 02 ab .
Since E is homogeneous, the vector valued integral y I Tgdy isdefinedandbelongsto E (seeTheorem3.1). Now,forevery n
Partition I byafinitenumberofsubintervals kI whose lengths kI aremajorizedbyanumber tobechosen later.Chooseandfixapoint ka ineach kI .Wethenhave
,say.
Nowgiven 0 andacontinuousseminorm p on E ,we canchoose 0 sosmallthat kkya yapTgTg
Then,byTheorem3.2,
k k kyak I phpTgTgdyI foreach k
Hence,
|| 222 k kak kk pfgITgII
Sincethisholdsforeachcontinuousseminorm p ,
lim 2 kka k k ITgfg
But 1 2 kka k ITg
is a finite linear combination of translates of g , hence gfgV , i.e., fS where I f
Asfinitelinearcombinationsofcharacteristicfunctionsof theform I aredensein 1L and S isclosedin 1L ,wecan concludethat 1SL
Now, let be in M . Then 1 n L for all n, i.e., n S forall n.Hence, ng gV forall n.But
nn pggpgg
So, by Theorem 2.4. of [3], there exists a continuous seminorm q on E suchthat 1 nn pggqgg
But n gg byTheorem3.6.of[3],therefore 0as . n pggn
Hence * ggV .
International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 11 Issue: 07| July 2024 www.irjet.net p-ISSN: 2395-0072
5. CONCLUSIONS
Inthispaper,wehaveobtainedarelationshipbetweenthe convolution and translation operators for homogeneous ConvolutableFrechetSpacesofDistributions(CFD-spaces). WehaveusedvariousresultsandtechniquesofFunctional analysis to obtain theresult. Theresult obtained willbe usefulforfurtheranalysisinthefieldofFourieranalysisand Functionalanalysis.
ACKNOWLEDGEMENT
Mydeepandprofoundgratitudeto(Late) Dr. R.P. Sinha, Associate Professor, Department of Mathematics, I.I.T. Roorkeeforhisvaluableguidance.Idonothavewordsto thankhim.Hewillbeaconstantsourceofinspirationforme throughoutthelife.
REFERENCES
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[3] Nishu Gupta, “On Convolutable Frechet spaces of distributions”, International Research Journal of EngineeringandTechnology(IRJET),Volume10Issue7 July2023.
[4] W.Rudin,FunctionalAnalysis,McGraw-Hill,Inc.,New York,1991.
[5] M.P. Singh, “On Frechet spaces of distributions”, MathematicalSciencesInternationalResearchJournal, Volume3Issue2(2014).
[6] R.P.SinhaandJ.K.Nath,“OnConvolutableBanachspaces ofdistributions”,Bull.Soc.Math.Belg.,ser.B,41(1989), No.2,229-237.
[7] A. Wilansky, Modern Methods in Topological Vector SpacesMcGraw-Hill,Inc.,NewYork,1978.