### 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 language | English |
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Pages (from-to) | 859-874 |

Journal | Mathematische Zeitschrift |

Volume | 289 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 1 Aug 2018 |

### Keywords

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

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## Cite this

Gorbachev, D. V., & Tikhonov, S. Y. (2018). Wiener’s problem for positive definite functions.

*Mathematische Zeitschrift*,*289*(3-4), 859-874. https://doi.org/10.1007/s00209-017-1978-9