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

Abstract

The paper is concerned with the convergence of double Fourier- Haar series with partial sums taken over homothetic copies of a given bounded set containing the intersection of some neighbourhood of the origin with . It is proved that for a set from a fairly broad class (in particular, for convex ) there are two alternatives: either the Fourier-Haar series of an arbitrary function converges almost everywhere or 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 . Bibliography: 12 titles.