Convergence of Almost Periodic Fourier Series
Andrew Bailey (Birmingham)
Tuesday 2nd February, 2010 16:00-17:00 214
Almost periodic functions are well studied for their applications in differential equations but surprisingly little work has been done in considering some of the classical questions in Fourier analysis in this broader context. In particular, many natural questions relating to convergence of almost periodic Fourier series have not yet been answered. The principal difficulty working with almost periodic functions is that many of the "standard tools" of Fourier analysis have either not been adapted or fail to work completely. This talk will mainly be focussed on showing that a certain maximal operator bound in the Stepanov almost periodic function spaces implies almost everywhere convergence of dyadic partial sums of almost periodic Fourier series. Along the way, some of the adaptations of "standard tools" will be highlighted and attention will be drawn to some of places where they seemingly cannot be adapted. Time permitting, the talk will conclude with some discussion of ongoing research into an almost periodic version of Carleson's famous theorem asserting almost everywhere convergence of (regular partial sums of) Fourier series.