On classical inequalities for autocorrelations and autoconvolutions

J. A. Jiménez Madrid, Jaume de Dios Pont

Publication date: 25 June 2021

Abstract: In this paper we study an autocorrelation inequality proposed by Barnard and Steinerberger. The study of these problems is motivated by a classical problem in additive combinatorics. We establish the existence of extremizers to this inequality, for a general class of weights, including Gaussian functions (as studied by the second author and Ramos) and characteristic function (as originally studied by Barnard and Steinerberger). Moreover, via a discretization argument and numerical analysis, we find some almost optimal approximation for the best constant allowed in this inequality. We also discuss some other related problem about autoconvolutions.

Has companion code repository: https://github.com/jaumededios/suprema-autocorrelations

Mathematics Subject Classification ID

Discrete version of topics in analysis (39A12) Convolution, factorization for one variable harmonic analysis (42A85) Trigonometric polynomials, inequalities, extremal problems (42A05) Lagrange's equations (70H03)

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