Convergence, Uniqueness, and Summability of Multiple Trigonometric Series

In this paper our primary interest is in developing further insight into convergence properties of multiple trigonometric series, with emphasis on the problem of uniqueness of trigonometric series. Let E be a subset of positive (Lebesgue) measure of the k dimensional torus. The principal result is that the convergence of a trigonometric series on E forces the boundedness of the partial sums almost everywhere on E where the system of partial sums is the one associated with the system of all rectangles situated symmetrically about the origin in the lattice plane with sides parallel to the axes. If E has a countable complement, then the partial sums are bounded at every point of E. This result implies a uniqueness theorem for double trigonometric series, namely, that if a double trigonometric series converges unrestrictedly rectangularly to zero everywhere, then all the coefficients are zero. Although uniqueness is still conjectural for dimensions greater than two, we obtain partial results and indicate possible lines of attack for this problem. We carry out an extensive comparison of various modes of convergence (e.g., square, triangular, spherical, etc.). A number of examples of pathological double trigonometric series are displayed, both to accomplish this comparison and to indicate the "best possible" nature of some of the results on the growth of partial sums. We obtain some compatibility relationships for summability methods and finally we present a result involving the (C, a, 0) summability of multiple Fourier series. Introduction. The main interest of this paper will be the theory of multiple trigonometric series. Multiple Fourier series (the most important type of multiple trigonometric series) will be discussed only in connection with the theory of uniqueness and again in the last chapter. For the definitions of any unfamiliar terms used in the introduction the reader is referred to ?1. One of the main difficulties in multiple series arises in connection with the usual consistency theorems for summation methods. In order to maintain the validity of the typical theorem "convergence implies summability," even in the case of Poisson summation one has to have the added condition that all partial sums be bounded. If one attempts to restrict himself to regular methods of forming the partial sums, it is easy to construct examples where this condition fails. However, by introducing unrestricted rectangular partial sums, convergence of a multiple trigonometric Received by the editors January 22, 1971. AMS 1970 subject classifications. Primary 42A92, 42A48, 42A20, 42A24, 40B05; Secondary 40G10, 40A05, 40D15.