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 Lp = Lp[−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 Lp, p ∈ (0, 1), cannot be improved.