Carathéodory–Fejér type extremal problems on locally compact Abelian groups

Abstract

Abstract. We consider the extremal problem of maximizing a point value |f (z)| at a given point z ∈ G by some positive definite and continuous function f on a locally compact Abelian group (LCA group) G, where for a given symmetric open set Ω ∋ z, f vanishes outside Ω and is normalized by f (0) = 1.This extremal problem was investigated in R and R d and for Ω a 0-symmetric convex body in a paper of Boas and Kac in 1943. Arestov and Berdysheva extended the investigation to T d , where T := 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 Z or Z m := Z/mZ. We extend their results to arbitrary LCA groups.Mathematics Subject Classification (2000): Primary 43A35, 43A70. Secondary 42A05, 42A82.

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

“…Until that work the equivalence remained unclear in spite of the fact that, e.g., Boas and Kac found ways to deduce the solution of the trigonometric extremal problem (2) from their results on Problem 1. The recent results of [20] capitalized on this observation when proving analogous equivalences even in LCA groups. Theorem 7 (Krenedits and Révész).…”

“…Condition 2, or, in Sections 7 and 8, under another assumption defined later as roundness, see Definition 17. As we will explain in the sequel, this latter extension is still general enough to cover all the known extensions, including the unconditional results of [20] for LCA groups.…”

“…Let us admit right here that although [18] extended the problem even to sets Ω in classical groups not satisfying Condition 2, in the generality of non-Abelian groups we cannot describe the problem without any extra assumption yet. Still, the unconditional extension can be handled in locally compact Abelian groups, to which we refer the reader to [20].…”

“…In view of (17) below, giving that for f " 0 }f } 8 " f peq, the trivial estimate or trivial (upper) bound for the Carathéodory-Fejér constants is simply C c G pΩ, zq ď C # G pΩ, zq ď 1. Note that also the lower estimation C # G pΩ, zq ě C c G pΩ, zq ě 1{2 can be worked out analogously to [20,Proposition 3.2].…”

“…Also, the information that the degree of the polynomial T is at most N means that supp c k ⊂ [−N, N ], and with z := 1 ∈ Z this is easily seen to match the condition formulated in (1.1) with Ω := [−N, N ]. For more details about this interpretation, we refer the reader to [20].…”