Logan's problem for Jacobi transform

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 $$\inf \mathrm{\Lambda }((-1{)}^{m-1}f),m\in \mathbb{N},$$ where $\mathrm{\Lambda }(f)=\sup \{\lambda >0\text{\hspace{0.17em}⁣}:f(\lambda )>0\}$ 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 ${\int }_0^{\mathrm{\infty }}{\lambda }^{2k}f(\lambda )d\sigma (\lambda )=0$ 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 $\alpha =\beta =-1\mathrm{/}2$. 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.