Beurling Weighted Spaces, Product-Convolution Operators, and the Tensor Product of Frames

Abstract

G. Gaudry solved the multiplier problem for Beurling algebras, i.e., he identified the space of all multipliers of a Beurling algebra with a weighted space of bounded measures. In the first part of this thesis, we solve multiplier problems for some Beurling weighted spaces. We identify the space of all multipliers of some Beurling weighted spaces with the dual of spaces of Figa-Talamanca type. A paper by R.C Busby and H.A.Smith gives necessary and sufficient conditions for the compactness of product-convolution operators. In the second part of this thesis, we present some applications of the result of R.C Busby and H.A. Smith; and we prove that the eigenfunctions of certain product-convolution operators can be obtained as solutions of some differential equations. Incidentally, we obtain classical special functions as eigenfunctions of these product-convolution operators. In the third part of this thesis, we prove that the tensor product of two sequences is a frame(Riesz basis) if and only if each part of this tensor product is a frame (Riesz basis). We use this result to extend the Lyubarskii and Seip Wallsten theorem, characterizing Gabor frames generated by the Gaussian function, to higher dimensions.