Turán’s and Fejér’s Extremal Problems for Jacobi Transform

Abstract

We give the solution of the Turan extremal problem for compact supported functions on the half-line with nonnegative Jacobi transform and the dual Fejer extremal problem for even nonnegative entire functions of exponential type that are Jacobi transforms. We prove the uniqueness of the extremal functions. The Markov quadrature formula on the half-line at zeros of the modified Jacobi function is used for the proof of these results.