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Fourier coefficients of functions in power-weighted L2-spaces and conditionality constants of bases in Banach spaces

Part of: Approximations and expansions Normed linear spaces and Banach spaces; Banach lattices Harmonic analysis in one variable

Published online by Cambridge University Press:  30 March 2022

J. L. Ansorena Open the ORCID record for J. L. Ansorena [Opens in a new window]
J. L. Ansorena*
Affiliation:
Department of Mathematics and Computer Sciences, Universidad de La Rioja, Logroño 26004, Spain (joseluis.ansorena@unirioja.es)
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Abstract

We prove that, given $2< p<\infty$, the Fourier coefficients of functions in $L_2(\mathbb {T}, |t|^{1-2/p}\,{\rm d}t)$ belong to $\ell _p$, and that, given $1< p<2$, the Fourier series of sequences in $\ell _p$ belong to $L_2(\mathbb {T}, \vert {t}\vert ^{2/p-1}\,{\rm d}t)$. Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every $1< p<\infty$ and every $0\le \alpha <1$, there is a Schauder basis of $\ell _p$ whose conditionality constants grow as $(m^{\alpha })_{m=1}^{\infty }$, and there is an almost greedy basis of $\ell _p$ whose conditionality constants grow as $((\log m)^{\alpha })_{m=2}^{\infty }$.

Keywords

Fourier coefficientsFourier series, conditionality constants, Hilbert spaces, ℓp-spaces, almost greedy bases

MSC classification

Primary: 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series
Secondary: 46B15: Summability and bases 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 3 , June 2023 , pp. 784 - 810
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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