On Certain Theorems Regarding Summable Series and Their Application to the Double and Triple Fourier's Series

Abstract

It has been frequently pointed out f that, in maiay of the applications of ordinary and multiple Fourier's series to physical problems, the theorems regarding their summability afford results of greater generality than those regarding their convergence. Also, in the case of the ordinary Fourier's series, it has been shown . that some of the classical results regarding criteria for its convergence appear as corollaries of Fejer's theorem concerning its summability (Cl), if we make use of certain theorems in the general theory of summable series. The purpose of the present paper is to obtain, in analogous fashion, criteria regarding the convergence of double and triple Fourier's series from known results regarding their summability. In order to do this, it is necessary in the first place to prove two theorems which enable us to infer the convergence of certain types of double and triple series, when we know them to be summable (Cl). These theorems are generalizations to the cases of double and triple series of a theorem proved by Pollard in the article cited above. The conditions obtained for the convergence of the double and triple Fourier's series are analogues of those given by Dini for the convergence of the simple series. These conditions are, we believe, new.?