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

Bailey, Andrew David (2009). Almost everywhere convergence of dyadic partial sums of Fourier series for almost periodic functions. University of Birmingham. M.Phil.

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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.