Abstract
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 p = L p[−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, p ∈ (0, 1), cannot be improved.
References
H. K. Bari, Trigonometric Series (Fizmatgiz, Moscow, 1961) [in Russian].
A. Kolmogoroff, “Sur les fonctions harmoniques conjuguées et les séries de Fourier,” Fundamenta 7, 24–29 (1925).
V. I. Filippov, “On the Kolmogorov theorems on Fourier series and conjugate functions,” Izv. Vyssh. Uchebn. Zaved.Mat., No. 7, 21–34 (2012) [Russian Math. (Iz. VUZ) 56 (7), 18–29 (2012)].
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Original Russian Text © M. R. Gabdullin, 2016, published in Matematicheskie Zametki, 2016, Vol. 99, No. 6, pp. 878–886.
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Gabdullin, M.R. On the divergence of Fourier series in the spaces ϕ(L) containing L . Math Notes 99, 861–869 (2016). https://doi.org/10.1134/S0001434616050242
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DOI: https://doi.org/10.1134/S0001434616050242