Szegő’s conjecture on Lebesgue constants for Legendre series

C. K. Qu and Roderick Sue-Chuen Wong

Vol. 135 (1988), No. 1, 157–188

DOI: 10.2140/pjm.1988.135.157

Abstract

In 1926, Szegö conjectured that the Lebesgue constants for Legendre series form a monotonically increasing sequence. In this paper, we prove that his conjecture is true. Our method is based on an asymptotic expansion together with an explicit error bound, and makes use of some recent results of Baratella and Gatteschi concerning uniform asymptotic approximations of the Jacobi polynomials.

Mathematical Subject Classification 2000

Primary: 42C10

Secondary: 33A45

Milestones

Received: 11 May 1987

Published: 1 November 1988