On the integrability of double cosine and sine series, II

Abstract

We consider double cosine and sine series whose coefficients form a null sequence of bounded variation. In this case, a double cosine (or sine) series converges to a function ƒ(x, y) ( or g(x, y) , respectively). The convergence can be understood even in the regular sense introduced by Hardy, and this implies convergence in Pringsheim's sense, too. We give sufficient conditions under which ƒ(x, y) xy and g ( x , y ) are integrable on [0, π] × [0, π] in the sense of improper Riemann integral. We conjecture that these conditions are essentially necessary. Our results are the extensions of those by R. P. Boas from one-dimensional to two-dimensional trigonometric series.