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On almost everywhere convergence of strong arithmetic means of Fourier series


Author: Bobby Wilson
Journal: Trans. Amer. Math. Soc. 367 (2015), 1467-1500
MSC (2010): Primary 42A20, 42A24
Published electronically: September 5, 2014
MathSciNet review: 3280051
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Abstract | References | Similar Articles | Additional Information

Abstract: This article establishes a real-variable argument for Zygmund's theorem on almost everywhere convergence of strong arithmetic means of partial sums of Fourier series on $ \mathbb{T}$, up to passing to a subsequence. Our approach extends to, among other cases, functions that are defined on $ \mathbb{T}^d$, which allows us to establish an analogue of Zygmund's theorem in higher dimensions.


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Additional Information

Bobby Wilson
Affiliation: Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, Illinois 60615

DOI: https://doi.org/10.1090/S0002-9947-2014-06297-1
Keywords: Fourier series on $L^1$, $\ell^2$ averages, Calder\'on-Zygmund decomposition, partial sums, weak $L^1$, strong arithmetic means
Received by editor(s): May 8, 2013
Received by editor(s) in revised form: October 1, 2013
Published electronically: September 5, 2014
Additional Notes: This is in partial fulfillment of the author’s requirements for the Doctor of Philosophy degree in Mathematics at the University of Chicago
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.