ON LOCALLY CONVEX SPACES OF DISTRIBUTIONS

Abstract

Many results in Fourier Analysis which are known for L" (1 < p < oo), C, M and Orlicz spaces etc., have been obtained for Banach spaces of distributions, convolutable Banach spaces of distributions and Frechet spaces of distributions by Vishnu Kant, J.K. Nath and M.P. Singh in their Ph. D. theses "On the Banach Spaces of Distributions", "On Convolutable Banach Spaces of Distributions" and "On Frechet Spaces of Distributions and Multiplier Operators" respectively. In the present thesis we study locally convex spaces of distributions (briefly called as LCD-spaces), generalize some known results to these spaces and state some new results. The thesis consists of six 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 an LCD-space E as follows. Let D denote the space of all distributions on T = R / 2irZ . A locally convex space E is called an LCD- space if it can be continuously embedded into D (D having the weak* -topology), and if, regarded as a subspace of D, it satisfies the following properties: (i) C° c E, and the inclusion map is continuous; (ii) E is translation invariant and {Tx I x E T} , the family of all translation operators on E, is equicontinuous on E;