On the Erdélyi-Magnus-Nevai conjecture for Jacobi polynomials

Ilia Krasikov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


T. Erdelyi, A.P. Magnus and P. Nevai conjectured that for α, β ≥ - 1/2, the orthonormal Jacobi polynomials Pk (α, β) (x) satisfy the inequality max x∈[-1,1] (1-x)α+1/2(1+x)β+1/2 (P k(α, β) (x))2 =O (max{1,(α2 + β2)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 α = β ≥ (1+ √2)/4, even in a stronger form by giving very explicit upper bounds. We also show that √δ2-x 2(1-x2)α (P2k (α , α) (x))2 < 2/π (1+1/8(2k + α)2 for a certain choice of δ, such that the interval (-δ, δ) contains all the zeros of P2k (α, α) (x). Slightly weaker bounds are given for polynomials of odd degree.

Original languageEnglish
Pages (from-to)113-125
Number of pages13
JournalConstructive Approximation
Issue number2
StatePublished - Aug 2008
Externally publishedYes


  • Jacobi polynomials


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