On convergence factors in double series and the double Fourier’s series

Abstract

It is generally recognized that in order to show that an actual solution of the physical problem is given by the formal series arising in many problems of mathematical physics, it is necessary to apply the theory of convergence factors to these series. Results in this direction of the greatest generalitr have been obtained by a method originated by FEJER,t and applied by him fo problems involving the ordinary Fourier's series. In a previous paper t the writer has applied FEJER'S method to problems that involve developments in Bessel's functions. The object of the present paper is to make such an application to problems that involve the development of a function of two variables in a double Fourierns series. Such a discussion naturally involves the consideration of certain general facts in the theory of the summability of double series, a theory that has as yet been scarcely touched on. We have not made, however, any attempt to found a very broad theory of this sort. Such an attempt would have led us too far from the central aim of the paper, and in any case it seems preferable that the theory of summability of double series should be developed at first from the point of view of its applications, and that the study of such a theory from the abstract point of view can be made to better advantage when the facts of widest use in the application of the theory have been brought to light. The consideration of the summability of double series naturally suggests the consideration of the summability of multiple series of any order, and the study of the summability of the double Fourier's series suggests the study of the summability of the triple Fourier's series and the problems connected with it. Generalizations of the results of the present paper to such cases may be obtained by methods similar to those employed in the subsequent discussion.