Wiener’s problem for positive definite functions

D. V. Gorbachev, S. Yu Tikhonov

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    3 Citations (Scopus)

    Abstract

    © 2017, Springer-Verlag GmbH Deutschland. We study the sharp constant Wn(D) in Wiener’s inequality for positive definite functions ∫Tn|f|2dx≤Wn(D)|D|-1∫D|f|2dx,D⊂Tn.Wiener proved that W1([- δ, δ]) < ∞, δ∈ (0 , 1 / 2). Hlawka showed that Wn(D) ≤ 2 n, where D is an origin-symmetric convex body. We sharpen Hlawka’s estimates for D being the ball δBn and the cube δIn. In particular, we prove that Wn(δBn) ≤ 2 (0.401⋯+o(1))n. We also obtain a lower bound of Wn(D). Moreover, for a cube D=1qIn with q= 3 , 4 , … , we obtain that Wn(D) = 2 n. Our proofs are based on the interrelation between Wiener’s problem and the problems of Turán and Delsarte.
    Original languageEnglish
    Pages (from-to)859-874
    JournalMathematische Zeitschrift
    Volume289
    Issue number3-4
    DOIs
    Publication statusPublished - 1 Aug 2018

    Keywords

    • Hlawka’s inequality
    • Linear programming bound problem
    • Positive definite function
    • Sharp constant
    • Wiener’s problem

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