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

Abstract

The century old extremal problem, solved by Carathéodory and Fejér, concerns a nonnegative trigonometric polynomial T ptq " a 0`ř n k"1 a k cosp2πktq`b k sinp2πktq ě 0, normalized by a 0 " 1, and the quantity to be maximized is the coefficient a 1 of cosp2πtq. Carathéodory and Fejér found that for any given degree n the maximum is 2 cosp π n`2 q. In the complex exponential form, the coefficient sequence pc k q Ă C will be supported in r´n, ns and normalized by c 0 " 1. Reformulating, nonnegativity of T translates to positive definiteness of the sequence pc k q, and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c : Z Ñ C, supported in r´n, ns.Boas and Katz, 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. (2000): Primary 43A35, 43A70. Secondary 42A05, 42A82. Mathematics Subject Classification

Delsarte’s extremal problem and packing on locally compact Abelian groups

“…The statement can be found e.g. in[35, (32.8) (d)] and[35, (32.9) Theorem], see also[48, Lemma 12(v)],[47, Lemma 2.6.1 (iii)].…”

Delsarte's Extremal Problem and Packing on Locally Compact Abelian Groups

“…The statement can be found e.g. in[27, (32.8) (d)] and[27, (32.9) Theorem], see also[36, Lemma 12(v)],[35, Lemma 2.6.1 (iii)].…”