On the Convergence of Fourier Series of Computable Lebesgue Integrable Functions

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.