Convergence of double Fourier series and $W$-classes

Abstract:

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