Minimal Fourier majorants in L^p

2021/12/27 by Fournier, John J. F., Vrecko, Dean
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2112.13799

Abstract

Denote the coefficients in the complex form of the Fourier series of a function f on the interval [-π, π) by 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 F(n) ≥ | f(n)| for all n, and for which ‖F‖_p ≤ ‖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.

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