On the integrability of double cosine and sine series, I

Abstract

We consider double cosine and sine series whose coefficients form a null sequence of bounded variation. We prove that in this case the sum of a double cosine series is integrable in the sense of improper Riemann integral and the series in question is the Fourier series of its sum in the same sense. On the other hand, the sum of a double sine series is not necessarily integrable in the sense of improper Riemann integral, but the series is always a generalized Fourier sine series of its sum. These imply the important corollary that if the sum of a double cosine or sine series, with coefficients tending to zero and of bounded variation, is Lebesgue integrable, then the series is the Fourier series of its sum. Our results are the extensions of those by R. P. Boas and N. K. Bary from one-dimensional to two-dimensional trigonometric series.