Lp Computable Functions and Fourier Series
<|reference_start|>Lp Computable Functions and Fourier Series: 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.<|reference_end|>
@article{moser2006lp,
    title={Lp Computable Functions and Fourier Series},
    author={Philippe Moser},
    journal={arXiv preprint arXiv:cs/0608106},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0608106},
    primaryClass={cs.CC}
}