Computable analysis: an introduction
Computable analysis: an introduction
Baire categories on small complexity classes and meager--comeager laws
Information and Computation
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This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of L^p-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing L^p-computable Baire categories. We show that L^p-computable Baire categories satisfy the following three basic properties. Singleton sets {f} (where f is L^p-computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of L^p-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^p-computable functions and show that whereas for every p1, the Fourier series of every L^p-computable function f converges to f in the L^p norm, the set of L^1-computable functions whose Fourier series does not diverge almost everywhere is meager.