Fourier majorants that match norms

2021/05/25 by Fournier, John J. F.
#43A15 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 42A32 #Secondary 47B10

paper · doi:10.48550/arxiv.2105.11642

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, but that also satisfies ‖F‖_p ≤ ‖f‖_p. Rescaling F suitably then gives a majorant with the same L^p norm as f. We show how that majorant comes from a variant in L^2j of the notion of exact majorant in L².

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