Logan's problem for Jacobi transforms

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) vertical bar 2 rho-i lambda Gamma(alpha+1)Gamma(i lambda)\Gamma()rho+i lambda)/2 Gamma((rho+i lambda)/2-beta vertical bar(-2) d lambda, respectively. We solve the following generalized Logan problem: to find the infimum inf Lambda((-1)(m-1) f), m is an element of N, where Delta(f) = sup {lambda > 0: 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 >= 2, then we additionally assume that integral(infinity)(0) lambda(2k) f (lambda) d sigma(lambda) = 0 for k = 0,center dot center dot center dot, 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 H-d is also solved. Using the 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.