  This paper studies how well computable functions can be approximated by their
Fourier series. To this end, we equip the space of Lp-computable functions
(computable Lebesgue integrable functions) with a size notion, by introducing
Lp-computable Baire categories.
  We show that Lp-computable Baire categories satisfy the following three basic
properties. Singleton sets {f} (where f is Lp-computable) are meager, suitable
infinite unions of meager sets are meager, and the whole space of Lp-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 Lp-computable functions and
show that whereas for every p>1, the Fourier series of every Lp-computable
function f converges to f in the Lp norm, the set of L1-computable functions
whose Fourier series does not diverge almost everywhere is meager.
