Pointwise convergence of cone-like restricted two-dimensional Fejér means of Walsh-Fourier series

Abstract

For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, β−1 ≤ n1/n2 ≤ β is provided with some fixed parameter β ≥ 1. In this paper we generalize the result of Gat and Weisz. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets in order to preserve this convergence property.