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Asymptotic expansions of the Lebesgue constants for Jacobi series. (English) Zbl 0556.33010

Summary: 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 0095.04903)]. 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.
See also the next review [C. L. Frenzen and R. Wong, C. R. Math. Acad. Sci., Soc. R. Can. 6, 267–271 (1984; Zbl 0556.33011)] of further work building upon these results.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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