Minimal Fourier majorants in \(L^p\)

Denote the coefficients in the complex form of the Fourier series of a function \(f\) on the interval \([-\pi, \pi)\) by \(\hat f(n)\). It is known that if \(p = 2j/(2j-1)\) for some integer \(j>0\), then for each function \(f\) in \(L^p\) there exists another function \(F\) in \(L^p\) that major...

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Bibliographic Details
Published in:arXiv.org 2021-12
Main Authors: Fournier, John J F, Vrecko, Dean
Format: Article
Language:English
Subjects:
Fourier series
Online Access:Get full text
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Summary:Denote the coefficients in the complex form of the Fourier series of a function \(f\) on the interval \([-\pi, \pi)\) by \(\hat f(n)\). It is known that if \(p = 2j/(2j-1)\) for some integer \(j>0\), then for each function \(f\) in \(L^p\) there exists another function \(F\) in \(L^p\) that majorizes \(f\) in the sense that \(\hat F(n) \ge |\hat f(n)|\) for all \(n\), and for which \(\|F\|_p \le \|f\|_p\). When \(j > 1\), the existence proofs for such small majorants do not provide constructions of them, but there is a unique majorant of minimal \(L^p\) norm. We modify previous existence proofs to say more about the form of that majorant.
ISSN:2331-8422