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On the convergence of Fourier series of computable Lebesgue integrable functions

✍ Scribed by Philippe Moser

Publisher: John Wiley and Sons
Year: 2010
Tongue: English
Weight: 130 KB
Volume: 56
Category: Article
ISSN: 0044-3050
DOI: 10.1002/malq.200910101

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✦ Synopsis

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.

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