Divergence almost everywhere of rectangular partial sums of multiple fourier series of bounded functions

Abstract

In this paper we establish the following results, which are the multidimensional generalizations of well-known theorems: 1) Suppose that a functionf ∈C(T m ) has no intervals of constancy inT m ; then there exists a homeomorphism ϕ:T m →T m such that the Fourier series of the superpositionF=f o ϕ is divergent with respect to rectangles almost everywhere; 2) for any integrable functionf ∈L1(T m ), with ¦f(x)¦≥α>0,x ∈T m , there exists a signum functione(x)=±1,x ∈T m such that the Fourier series of the productf(x)e(x) is divergent with respect to rectangles almost everywhere.