ON CONVOLUTABLE LOCALLY CONVEX SPACES OF DISTRIBUTIONS

Abstract

Many results in Fourier analysis which are known for LP 5_ oo) , C and M etc., have been obtained for Banach spaces of distributions, convolutable Banach spaces of distributions, Frechet spaces of distributions and locally convex spaces of distributions by Vishnu Kant, J.K.Nath, M.P.Singh and A.N.Mohammed in their Ph.D.theses "On the Banach Spaces of Distributions", "On Convolutable Banach Spaces of Distributions", "On Frechet Spaces of Distributions and Multiplier Operators" and "On Locally Convex Spaces of Distributions" respectively. In the present thesis we study convolutable locally convex spaces of distributions (briefly called as CD-spaces), generalize some known results to these spaces and state some new results. We have used various results and techniques of Functional Analysis to obtain these results. The thesis consists of seven chapters and is summarized as below. In Chapter 1, we explain the assumptions and terminology used in the thesis, give various definitions and examples and then state some useful results already known. In Chapter 2, first we define a CD-space E as follows. Let D denote the space of all distributions on T=R/2n-Z. A locally convex space E will be called a CD-space, if it can be continuously embedded into D (D having the weak* topology); and if regarded as a subset of D it satisfies the following properties: (i) The inclusion map i:C"' n E -÷ E is continuous, where C°'nE has the relative topology of ; (ii) pEA1,fEEp*feE and (p,f)-->p*f is a continuous bilinear mapping from M x E to E, where M denotes the set of all (Radon) measures. We give various examples to show that the properties (i) and (ii) in the above definition are independent and there exist spaces which are CD-spaces but not the previously known spaces as stated earlier. Also, we obtain some preliminary results for CD-spaces. Chapter 3 is devoted to the study of homogenous CD-spaces. A CD-space E is said to be homogenous if, for every f in E, the function x —+ Tx f is continuous from T to E. For a sequentially complete homogenous CD-space E, we show that C' n E is dense in E. We also show that E* (the dual space of E) with strong* topology is a CD-space if E is a sequentially complete homogenous CD-space. Further, we show that if E is a sequentially complete homogenous CD-space, then E 0. is homogenous if and only if for every f E E and F E E* , the series E F (n)i (—n) is (C, 1 )-summable to F(f) . In Chapter 4, we define the (C,1)- complementary space E' of a CD-space E as E' = {F E E* : lim F(o- nf) = F(f) for all f € EI, n—)pco and show that if E is a sequentially complete CD-space, then (E, fi(E* ,E)) is also a CD-space. If E is a Banach-Mackey CD-space, P is a continuous multiplier projection from C' onto C' n E such that for every integer n,P(en)= en V enE E , P(en)= 0 V en e E and F E E' , then we show that U(f , F) is a bounded Borel function that ii generates the distribution F 4.1 where U(f,F)(t). F(T,f) Vf E E and Vt E T. Moreover, there exists a continuous seminorm p on E such that F*f ILD P(f) VfEE• We also show that E' is a closed subspace of (E* ,fl(E* ,E)) if E is a sequentially complete CD-space. We say that a CD-space E is perfect if E = (E')' . We give a sufficient condition for a CD-space to be perfect. In Chapter 5, we show that if (E,Y) is a sequentially complete homogenous CD-space, then (E*,..7°) is a homogenous CD-space, where Y° is the topology of precompact convergence on E* .We also show that if E is a sequentially complete homogenous CD-space, then E is N-complete. Chapter 6 is devoted to the CD-spaces which admit conjugation. A CD-space E is said to admit conjugation if f , the conjugate distribution off is in E for every f in E. In this chapter, a necessary and sufficient condition for a distribution to have its conjugate in the same space has been established. A CD-space E is said to admit convergence if for every/ in E, Sflf f in E as n co ( Snf is the n-th partial sum of the Fourier series off ). We show that a barrelled CD-space E, having C' n E as a dense subset, admits convergence if and only if {S : n E N } is an equicontinuous family of operators on E. A CD-space E is said to be a character CD-space iff in E implies that e„ f is in E for every integer n, where e„ f (u) = f (e nu) for all u E C. If E is a barrelled sequentially complete homogenous character CD-space and {en : n E z} is an equicontinuous family of operators on E, then we show that E admits conjugation if and only if E admits convergence. In this chapter, we also find a iii relationship between the convolution and translation operators for sequentially complete homogenous CD-spaces. Chapter 7 deals with the theory of multiplier operators. Suppose F and G are two spaces of distributions, a complex valued function 0 on Z is said to be a multiplier of type (F,G) if f E F implies 0..? E G , where G denotes the set of transforms k of elements g of G. By (F,G) we denote the set of all 0 of type(F,G).A linear operator U from F to G is said to be a multiplier of type(F,G) if there exists a unique 0 on Z, of type (F,G), such that f E F (Uf)^ = .We denote by U E Mc (F ,G) the set of all continuous multiplier operators of type (F,G). If E is a sequentially complete homogenous CD-space, then we show that U E mc(E,C) if and only if there exists Fin E* such that Uf = F* f for each f E E .If E is a weakly sequentially complete homogenous CD-space and U E mc(E* ,C), then there exists a distribution f E E such that OF = F * f for each F in E" . If E is a sequentially complete homogenous CD-space and U is a multiplier of type (E, E), then we show that each closed invariant subspace of E is stable under U i