On the maximum value of Jacobi polynomials

Abstract

A remarkable inequality, with utterly explicit constants, established by Erdélyi, Magnus, and Nevai, states that for > − 1 2 , the orthonormal Jacobi polynomials P[Erdélyi et al., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614]. They conjectured that the real order of the maximum is O( 1/2 ).Here we will make half a way towards this conjecture by proving a new inequality which improves their result by a factor of order ( 1 + 1 k ) −1/3 . We also confirm the conjecture, even in a stronger form, in some limiting cases.

Inequalities for orthonormal Laguerre polynomials

“…with some absolute constants c 1 , c 2 [7]. Using a similar method in the Hermite case, the inequalities (with two rather poor constants)…”

Compressed sensing with preconditioning for sparse recovery with subsampled matrices of Slepian prolate functions

“…This second construction could be the most straightforward one for extending the preconditioned framework proposed in §4 of the present paper, even at the price of heavier computations with bounds on Gegenbauer polynomials taken from [38,31]. Finally, the "wavelet prolate functions" studied in [33,25] may also be convenient, at least for the techniques developed in our §3.…”

On the Erdelyi-Magnus-Nevai Conjecture for Jacobi Polynomials

“…We deduce this result from the following two theorems. The first, which has been established in [7], gives a sharp inequality for the interval containing all the local maxima of the function M α,β k (x). The second one will be proven here and in fact demonstrates equioscillatory behaviour of M α k (x, d) under an appropriate choice of d.…”