Asymptotic expansions of the Lebesgue constants for Jacobi series

C. L. Frenzen and Roderick Sue-Chuen Wong

Vol. 122 (1986), No. 2, 391–415

DOI: 10.2140/pjm.1986.122.391

Abstract

Explicit expressions are obtained for the implied constants in the two O-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 Szegő 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.

Mathematical Subject Classification 2000

Primary: 33A65, 33A65

Secondary: 42C05

Milestones

Received: 4 September 1984

Revised: 31 May 1985

Published: 1 April 1986