Szegő’s conjecture on Lebesgue constants for Legendre series

Abstract

In 1926, Szego 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.