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On the convergence of Fourier series of computable Lebesgue integrable functions
Author(s) -
Moser Philippe
Publication year - 2010
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200910101
Subject(s) - mathematics , fourier series , computable function , lebesgue integration , almost everywhere , computable analysis , dominated convergence theorem , baire space , lp space , baire measure , baire category theorem , pure mathematics , discrete mathematics , banach space , mathematical analysis , convergence tests , computer science , rate of convergence , channel (broadcasting) , computer network
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 p > 1, 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 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)