Pointwise convergence of cone-like restricted two-dimensional (C, 1) means of trigonometric Fourier series

The aim of this work is to generalize the more than 60 year old celebrated result of Marcinkiewicz and Zygmund on the convergence of the two-dimensional restricted (C, 1) means of trigonometric Fourier series. They proved for any integrable function f is an element of L-1(T-2) the a.e. convergence sigma(n(1).n(2)) f -> f provided n1/beta <= n2 <=beta n1, where beta > 1 is fixed constant. That is, the set of indices (n1, n2) remains in some positive cone around the identical function. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets (of the set of indices) in order to preserve this convergence property. (c) 2007 Elsevier Inc. All rights reserved.