Inequalities for orthonormal Laguerre polynomials

Abstract

For 24 we will establish the following inequalitywhere the upper bound holds for k 35, and the lower one for k 2 × 10 10 . Sharp pointwise estimates on M k and related functions for x 0 are also given.

Sub-exponentially localized kernels and frames induced by orthogonal expansions

“…We shall utilize some estimates on |L α n (t)| established in [11]. As is shown in [11,Theorem 2] for α ≥ 24 and n ≥ 1 max 0<t≤ 2(α+1) 2 2n+α+1…”

On the maximum value of Jacobi polynomials

“…This approach does work in a similar problem for the Laguerre polynomials [12]. Unfortunately, in our case this would require rather tedious calculations (or, at least, we did not find any simple way) to establish the convexity of the involved functions.…”

“…seek estimates as precise as (5) (in fact, this time it is much easier, as we have one parameter less, see [12]). We assume x 0.…”

“…An astonishing fact is that this is sometimes possible under marginal restrictions on the parameters. Yet, the only known examples, putting aside asymptotics, are the Hermite polynomials with just one parameter k involved [9] (see (12) below), and the following remarkable inequality, established by Erdélyi et al [2],…”

Equioscillatory property of the Laguerre polynomials

“…[15,4]). As a corollary of Theorem 1 we improve an upper bound given in [6] and establish the following inequality:…”