Carathéodory–Fejér type extremal problems on locally compact Abelian groups

Krenedits, Sándor and Révész, Szilárd (2015) Carathéodory–Fejér type extremal problems on locally compact Abelian groups. JOURNAL OF APPROXIMATION THEORY, 194. pp. 108-131. ISSN 0021-9045

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Abstract

We consider the extremal problem of maximizing a point value ∣f(z)∣∣f(z)∣ at a given point z∈Gz∈G by some positive definite and continuous function ff on a locally compact Abelian group (LCA group) GG, where for a given symmetric open set Ω∋zΩ∋z, ff vanishes outside ΩΩ and is normalized by f(0)=1f(0)=1. This extremal problem was investigated in RR and RdRd and for ΩΩ a 0-symmetric convex body in a paper of Boas and Kac in 1945. Arestov, Berdysheva and Berens extended the investigation to TdTd, where T:=R/ZT:=R/Z. Kolountzakis and Révész gave a more general setting, considering arbitrary open sets, in all the classical groups above. Also they observed, that such extremal problems occurred in certain special cases and in a different, but equivalent formulation already a century ago in the work of Carathéodory and Fejér. Moreover, following observations of Boas and Kac, Kolountzakis and Révész showed how the general problem can be reduced to equivalent discrete problems of “Carathéodory–Fejér type” on ZZ or Zm:=Z/mZZm:=Z/mZ. We extend their results to arbitrary LCA groups.

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