Logan's problem for Jacobi transforms

Abstract

We consider direct and inverse Jacobi transforms with measures [equaction presented] respectively. We solve the following generalized Logan problem: to find inf Λ ((−1) m−1 f ) , m ∈ N, where Λ (f ) = sup {λ > 0: f (λ) > 0} and the infimum is taken over all nontrivial even entire functions f of exponential type that are Jacobi transforms of positive measures with supports on an interval. Here, if m ≥ 2, then we additionally assume that ∫0∞ λ2k f (λ) dσ (λ) = 0 for k = 0, . . ., m − 2. We prove that admissible functions for this problem are positive definite with respect to the inverse Jacobi transform. The solution of Logan’s problem was known only when α = β = −1/2. We find a unique (up to multiplication by a positive constant) extremizer fm. The corresponding Logan problem for the Fourier transform on the hyperboloid Hd is also solved. Using properties of the extremizer fm allows us to give an upper estimate of the length of a minimal interval containing not less than n zeros of positive definite functions. Finally, we show that the Jacobi functions form the Chebyshev systems. © 2023 Cambridge University Press. All rights reserved.