New Exceptional Sets and Convergence of the Square Partial Sums of Walsh–Fourier Series

For double Walsh–Fourier series and with f ∈ L~2(0, 1) × 0, 1)) we prove two almost orthogonality results relative to the linearized maximal square partial sums operator S_(N(x,y))f(x, y).Assumptions are N(x, y) non-decreasing as a function of x and of y and, roughly speaking, partial derivatives with approximately constant ratio ■≌2~(n_0) for all x and y, where n_0 is any fixed non-negative integer. Estimates, independent of N(x, y) and n_0, are then extended to L~r, 1 r 2.We give an application to the family N(x, y) = λxy on 0, 1) × 0, 1), any λ 10.

New Exceptional Sets and Convergence of the Square Partial Sums of Walsh-Fourier Series

For double Walsh-Fourier series and with f ∈ L2(0, 1) × 0, 1)) we prove two almost orthogonality results relative to the linearized maximal sq