Lp Computable Functions and Fourier Series

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of L-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing L-computable Baire categories. We show that L-computable Baire categories satisfy the following three basic properties. Singleton sets {f} (where f is L-computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of L-computable functions is not meager. We give an alternative characterization of meager sets via Banach Mazur games. We study the convergence of Fourier series for L-computable functions and show that whereas for every p > 1, the Fourier series of every L-computable function f converges to f in the L norm, the set of L-computable functions whose Fourier series does not diverge almost everywhere is meager.