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Almost everywhere convergence of dyadic partial sums of Fourier series for almost periodic functions

Bailey, Andrew David (2009)
M.Phil. thesis, University of Birmingham.

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Bailey09MPhil.pdf
Bailey09MPhil_addendum.pdf

Abstract

It is a classical result that for a function \(f\) \(\in\) L\(^p\)(\(\char{bbold10}{0x54}\)), dyadic partial sums of the Fourier series of \(f\) converge almost everywhere for \(p\) \(\in\) (1, \(\infty\)). In 1968, E. A. Bredihina established an analogous result for the Stepanov spaces of almost periodic functions in the case \(p\) = 2. Here, a new proof of the almost everywhere convergence result for Stepanov spaces is presented by way of a bound on an appropriate maximal operator for \(p\) = 2\(^k\), \(k\) \(\in\) \(\char{bbold10}{0x4E}\). In the process of establishing this, a number of general results are obtained that will facilitate further work pertaining to operator bounds and convergence issues in Stepanov spaces.

Type of Work:M.Phil. thesis.
Supervisor(s):Bennett, Jonathan
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:Mathematics
Additional Information:

The addendum file dates from October 2010.

Keywords:Almost Periodic Functions, Fourier Series, Convergence, Maximal Operator, Stepanov Space, Fourier Summation, Littlewood-Paley Theory, Hilbert Transform
Subjects:QA Mathematics
Institution:University of Birmingham
Library Catalogue:Check for printed version of this thesis
ID Code:268
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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