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The Point Value Maximization Problem for Positive Definite Functions Supported in a Given Subset of a Locally Compact Group

Part of: Harmonic analysis in one variable Abstract harmonic analysis Locally compact groups and their algebras

Published online by Cambridge University Press:  24 January 2018

Sándor Krenedits  and
Szilárd Gy. Révész
Sándor Krenedits*
Affiliation:
Faculty of Mechanical Engineering, Institute of Mathematics and Informatics, St. Stephen University Gödöllő, Páter Károly u. 1. 2100 Hungary (krenedits@t-online.hu)
Szilárd Gy. Révész
Affiliation:
Faculty of Sciences, Institute of Mathematics and Informatics, University of Pécs, Pécs, Vasvári Pál utca 4, 7622 Hungary Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences Budapest, Reáltanoda utca 13–15. 1053 Hungary (revesz.szilard@renyi.mta.hu)
*
*Corresponding author.
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Abstract

The century-old extremal problem, solved by Carathéodory and Fejér, concerns a non-negative trigonometric polynomial $T(t) = a_0 + \sum\nolimits_{k = 1}^n {a_k} \cos (2\pi kt) + b_k\sin (2\pi kt){\ge}0$, normalized by a0=1, where the quantity to be maximized is the coefficient a1 of cos (2π t). Carathéodory and Fejér found that for any given degree n, the maximum is 2 cos(π/n+2). In the complex exponential form, the coefficient sequence (ck) ⊂ ℂ will be supported in [−n, n] and normalized by c0=1. Reformulating, non-negativity of T translates to positive definiteness of the sequence (ck), and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c: ℤ → ℂ, supported in [−n, n]. Boas and Kac, Arestov, Berdysheva and Berens, Kolountzakis and Révész and, recently, Krenedits and Révész investigated the problem in increasing generality, reaching analogous results for all locally compact abelian groups. We prove an extension to all the known results in not necessarily commutative locally compact groups.

Keywords

Carathéodory–Fejér extremal problemlocally compact topological groupsabstract harmonic analysisHaar measuremodular functionconvolution of functions and of measurespositive definite functionsBochner–Weil theoremconvolution squareFejér–Riesz theorem

MSC classification

Primary: 43A35: Positive definite functions on groups, semigroups, etc. 43A70: Analysis on specific locally compact and other abelian groups
Secondary: 42A05: Trigonometric polynomials, inequalities, extremal problems 42A82: Positive definite functions 22D05: General properties and structure of locally compact groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 179 - 200
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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