A Uniform Asymptotic Expansion of the Jacobi Polynomials with Error Bounds

Abstract

In a recent investigation of the asymptotic behavior of the Lebesgue constants for Jacobi polynomials, we encountered the problem of obtaining an asymptotic expansion for the Jacobi polynomials , as n → ∞, which is uniformly valid for θ in . The leading term of such an expansion is provided by the following well-known formula of “Hilb's type” [13, p. 197]:(1.1)where α > – 1, β real and ; c and are fixed positive numbers. Note that the remainder in (1.1) is always θ1/2O(n–3/2).

Szegő’s conjecture on Lebesgue constants for Legendre series

“…Although these results are in a sense refinements of the asymptotic approximations obtained by Frenzen and Wong [5], they are of quite different nature from those given in [5]. Thus, in spite of the fact that the main strategy in this paper is similar to that employed by Frenzen and Wong [6], the detailed analysis here differs considerably from that given there.…”

Asymptotic expansions of the Lebesgue constants for Jacobi series

“…This is done in §6. Since the derivations of expansions (1.19) and (1.21) are similar to that of (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14), they will not be presented here. In terms of the function g a β(θ) defined in (2.33) (and also used in [9, (30 where R n is given in (2.32).…”

“…Our approach differs from that of Lorch. We first split the interval (0, π) at the exact zeros of the Jacobi polynomial and then apply recently obtained uniform asymptotic expansions for the Jacobi polynomials and their zeros [2]. Our method may also be extended to give higher order approximations when desired.…”

“…To evaluate I k9 we shall use the following results given in [2]; see, in particular, the main theorem, Corollary 2, and the first paragraph in §5 of that reference. …”

Uniform asymptotics for Jacobi polynomials with varying large negative parameters— a Riemann-Hilbert approach

“…In view of its importance in application, there is a tremendous amount of literature on the asymptotic behavior of these polynomials as n grows to infinity. For fixed α and β, classical results on this subject can be found in the definitive book by Szegö [13]; for more recent work, we refer to [8], [16] and the references given there. Some asymptotic results are now also available, when α and β depend on n and tend to infinity with n; see, e.g., [1], [2] and [9].…”