THE GENERALIZATION OF TITCHMARSH’S CONVOLUTION THEOREM TO FUNCTIONS OF SEVERAL VARIABLES

Abstract

Аs known, among the widely known facts of mathematical analysis the so-called Tichmarshao convolution theorem that the equality of zeros of functions of one variable on a finite segment entails their vanishing on segments whose sum of lengths equals the length of the end segment of the convolution definition occupies a certain place. In this article, this theorem is generalized  to functions of several variables. Also the definite analogue theorems of Gilbert- Schmidt, occupying function into a Fourier series in terms of the eigenfunctions of the named operator is installed. In our case, the convolution is shown to be more accurate, and namely, the uniform convergence of the series to the convolution function against the known convergence in the mean [1],  in a Hilbert space L₂. When establishing the main results of the article, along with the known facts, recalled above, in the theory of operator equations with completely continuous symmetric linear operators, the method proposed by the author in [1], the so-called transition method for the equations convolutions is used.