Divergent Fourier series in function spaces near L1[0;1

Abstract

Bochkariev's theorem states that for any uniformly bounded orthonormal system Φ, there is a Lebesgue integrable function such that the Fourier series of it with respect to the system Φ diverges on the set of positive measure. In this paper, we extended Bochkariev's theorem for some class of variable exponent Lebesgue spaces. We characterized the class of variable exponent Lebesgue spaces Lp(⋅)[0;1], 1<p(x)<∞ a.e. on [0;1], such that above mentioned Bochkarev's theorem is valid.