Absolute convergence of double Walsh-Fourier series and related results (Q653824)

The authors give estimates for \[ \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{mn}| \hat f(m,n)| ^r, \] where \(\{a_{mn}\}\) is a given double sequence of nonnegative real numbers and \(\hat f(m,n)\) are the Fourier coefficients in the double Walsh orthonormal system on the unit square. The results are formulated in terms of the dyadic \(L^p\)-modulus or the local dyadic \(L^p\)-modulus of continuity. The class of functions of bounded fluctuation is studied, too.