On the divergence of Fourier series in the spaces ϕ( L) containing L.

The paper deals with the question of the divergence of Fourier series in function spaces wider than L = L[−π, π], but narrower than L = L[−π, π] for all p ∈ (0, 1). It is proved that the recent results of Filippov on the generalization to the space ϕ( L) of Kolmogorov's theorem on the convergence of Fourier series in L, p ∈ (0, 1), cannot be improved. [ABSTRACT FROM AUTHOR]