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 ∈L ¹(T ^m), with ¦f(x)¦≥α>0,x ∈T ^m, there exists a signum functionε(x)=±1,x ∈T ^m such that the Fourier series of the productf (x)ε(x) is divergent with respect to rectangles almost everywhere.