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arXiv · 1101.2461

On the Convergence of Lacunary Walsh-Fourier Series

Yen Do·Michael T. Lacey
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Abstract

We study the Walsh-Fourier series of S_{n_j}f, along a lacunary subsequence of integers {n_j}. Under a suitable integrability condition, we show that the sequence converges to f a.e. Integral condition is only slightly larger than what the sharp integrability condition would be, by a result of Konyagin. The condition is: f is in L loglog L (logloglog L). The method of proof uses four ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency Calderon-Zygmund Decomposition of Nazarov-Oberlin-Thiele, (3) a classical inequality of Zygmund, giving an improvement in the Hausdorff-Young inequality for lacunary subsequences of integers, and (4) the extrapolation method of Carro-Martin, which generalizes the work of Antonov and Arias-de-Reyna.

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Yen Do, Michael T. Lacey. 2011-07-02. On the Convergence of Lacunary Walsh-Fourier Series. https://doi.org/10.1112/blms%2Fbdr088

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