The summability of the triple Fourier series at points of discontinuity of the function developed

Abstract

In a previous papert the author has studied the summability of the development of a functioin of three variables in a triple Fourier series at points of continuity of the function developed, using the method originated by Fejer and applied by him to problems involving simple series. For the latter series the proof of the summability at points of discontinuity of the first kind (finite jumps) is quite similar to that for points of continuity, and the two cases can be treated by means of a single discussion. In passing to the case of the double Fourier series, it is found that the study of the behavior of the series at points of discontinuity of an analogous type presents difficulties and complications that do not arise in connection with points of continuity.+ When we go on to the case of triple series, we find that the difficulties and complications of the corresponding problem are still further increased. In the present paper a study is made of the summability of the triple Foulier series at points of discontinuity of the type that would be apt to arise in physical applications. This includes discontinuities lying on plane or curved surfaces, and such that the function to be developed approaches the same value as we approach the point along any path lying entirely within the region of continuity. The definition of summability used in the discussion of the triple series is analogous to that of Cesaro for the simple series. Designating by Slmn the sum of the I n n terms of the triple series I a,,, lying in a rectangular parallelepiped I terms high, m terms broad, and n terms deep, in the upper, left-hand, forward corner of this series, and forming