Convergence of double Fourier series and $W$-classes

Abstract

The double Fourier series of functions of the generalized bounded variation class {n/ln(n + 1)}*BV are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class A* BV, defined here, have quadrant limits at every point and, for f E A*BV, there exist at most countable sets P and Q such that, for x ¬∈ P and y ¬∈ Q, f is continuous at (x, y). It is shown that the previously studied class ABV contains essentially discontinuous functions unless the sequence A satisfies a strong condition.