Asymptotic expansions of the Lebesgue constants for Jacobi series

Abstract

Explicit expressions are obtained for the implied constants in the two 0-terms in Lorch's asymptotic expansions of the Lebesgue constants associated with Jacobi series [Amer. J. Math., 81 (1959), 875-888]. In particular, a question of Szeg concerning asymptotic monotonicity of the Lebesgue constants for Laplace series is answered. Our method differs from that of Lorch, and makes use of some recently obtained uniform asymptotic expansions for the Jacobi polynomials and their zeros.