Wiener’s problem for positive definite functions

© 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.