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Almost everywhere divergence of lacunary subsequences of partial sums of fourier series

  • S. V. Konyagin
Article

Abstract

If an increasing sequence {n m } of positive integers and a modulus of continuity ω satisfy the condition Σ m=1 ω(1/n m )/m < ∞, then it is known that the subsequence of partial sums \(S_{n_m } \left( {f,x} \right)\) converges almost everywhere to f(x) for any function fH 1 ω . We show that this sufficient convergence condition is close to a necessary condition for a lacunary sequence {n m }.

Keywords

Fourier series Lebesgue measure modulus of continuity 

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Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of the Russian Academy of SciencesMoscowRussia

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