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Asymptotic expansions of the Lebesgue constants for Jacobi series. (English) Zbl 0556.33010
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 [L. Lorch, Am. J. Math. 81, 875-888 (1959; Zbl 0095.049)]. In particular, a question of Szegö 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.

MSC:
33C45Orthogonal polynomials and functions of hypergeometric type
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
42C10Fourier series in special orthogonal functions