Absolute convergence of double Walsh–Fourier series and related results - Móricz, Ferenc; Veres, Antal (Acta mathematica Hungarica, 2011)
We consider the double Walsh orthonormal system on the unit square , where { w m ( x )} is the ordinary Walsh system on the unit interval in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh–Fourier series of a function for some 1< p ≦2. More generally, we give best possible sufficient conditions for the finiteness of the double series where { a mn } is a given double sequence of nonnegative real numbers satisfying a mild assumption and 0< r <2. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity of f .