## Abstract

T. Erdelyi, A.P. Magnus and P. Nevai conjectured that for α, β ≥ - 1/2, the orthonormal Jacobi polynomials P_{k} ^{(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,(α^{2} + β^{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 α = β ≥ (1+ √2)/4, even in a stronger form by giving very explicit upper bounds. We also show that √δ^{2}-x ^{2}(1-x^{2})^{α} (P_{2k} ^{(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 P_{2k} ^{(α, α)} (x). Slightly weaker bounds are given for polynomials of odd degree.

Original language | English |
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Pages (from-to) | 113-125 |

Number of pages | 13 |

Journal | Constructive Approximation |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2008 |

Externally published | Yes |

## Keywords

- Jacobi polynomials