Logan's problem for Jacobi transform

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.