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On the Erdelyi-Magnus-Nevai Conjecture for Jacobi Polynomials

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Abstract

T. Erdelyi, A.P. Magnus and P. Nevai conjectured that for \(\alpha , \beta \ge - \frac{1}{2} ,\) the orthonormal Jacobi polynomials \(\mbox{\bf P}_k^{( \alpha , \beta )} (x)\) satisfy the inequality

$\max_{x \in [-1,1]}(1-x)^{\alpha+{1}/{2}}(1+x)^{\beta+{1}/{2}}(\mbox{\bf P}_k^{( \alpha , \beta )} (x) )^2 =O (\max\{1,(\alpha^2+\beta^2 )^{1/4} \} ),$

[Erdelyi et al., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials. SIAM J. Math. Anal., 25 (1994), 602-614.]. Here we will confirm this conjecture in the ultraspherical case \(\alpha = \beta \ge ({1+ \sqrt{2}})/{4},\) even in a stronger form by giving very explicit upper bounds. We also show that

$\sqrt{\delta^2-x^2} (1-x^2)^{\alpha}(\mbox{\bf P}_{2k}^{(\alpha , \alpha)} (x))^2 < \frac{2}{\pi} \left( 1+\frac{1}{8(2k+ \alpha)^2} \right)$

for a certain choice of \(\delta,\) such that the interval \((-\delta, \delta)\) contains all the zeros of \(\mbox{\bf P}_{2k}^{(\alpha , \alpha)} (x).\) Slightly weaker bounds are given for polynomials of odd degree.

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  1. Department of Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, United Kingdom

    Ilia Krasikov

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  1. Ilia Krasikov
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Correspondence to Ilia Krasikov.

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Krasikov, I. On the Erdelyi-Magnus-Nevai Conjecture for Jacobi Polynomials. Constr Approx 28, 113–125 (2008). https://doi.org/10.1007/s00365-007-0674-0

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  • DOI: https://doi.org/10.1007/s00365-007-0674-0

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