Logan's problem for Jacobi transform

TL;DR

This paper solves a generalized Logan problem for Jacobi transforms, identifying extremizers and their properties, and applies these results to Fourier transforms on hyperboloids, with implications for zero distribution of positive definite functions.

Contribution

It extends the solution of Logan's problem to Jacobi transforms with general parameters, characterizes extremizers, and connects Jacobi functions to Chebyshev systems.

Findings

Identifies extremizers for the generalized Logan problem.

Proves Jacobi functions form Chebyshev systems.

Provides bounds on zeros of positive definite functions.

Abstract

We consider direct and inverse Jacobi transforms with measures $d\mu(t)=2^{2\rho}(\sinh t)^{2\alpha+1}(\cosh t)^{2\beta+1}\,dt$ and $d\sigma(\lambda)=(2\pi)^{-1}\bigl|\frac{2^{\rho-i\lambda}\Gamma(\alpha+1)\Gamma(i\lambda)} {\Gamma((\rho+i\lambda)/2)\Gamma((\rho+i\lambda)/2-\beta)}\bigr|^{-2}\,d\lambda$, respectively. We solve the following generalized Logan problem: to find \[ \inf\Lambda((-1)^{m-1}f), \quad m\in \mathbb{N}, \] where $\Lambda(f)=\sup\,\{\lambda>0\colon f(\lambda)>0\}$ 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 $m\ge 2$, then we additionally assume that $\int_{0}^{\infty}\lambda^{2k}f(\lambda)\,d\sigma(\lambda)=0$ for $k=0,\dots,m-2$. 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 $\alpha=\beta=-1/2$. We find a unique (up to multiplication by a positive constant) extremizer $f_m$. The corresponding Logan problem for the Fourier transform on the hyperboloid $\mathbb{H}^{d}$ is also solved. Using properties of the extremizer $f_m$ 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.