The convergence of double Fourier-Haar series over homothetic copies of sets

摘要

The paper is concerned with the convergence of double Fourier-Haar series with partial sums taken over homothetic copies of a given bounded set W subset of R-+(2) containing the intersection of some neighbourhood of the origin with R-+(2). It is proved that for a set W from a fairly broad class (in particular, for convex W) there are two alternatives: either the Fourier-Haar series of an arbitrary function f is an element of L([0, 1](2)) converges almost everywhere or Lln(+) L([0, 1](2)) is the best integral class in which the double Fourier-Haar series converges almost everywhere. Furthermore, a characteristic property is obtained, which distinguishes which of the two alternatives is realized for a given W.