TY - JOUR
T1 - On the Erdélyi-Magnus-Nevai conjecture for Jacobi polynomials
AU - Krasikov, Ilia
PY - 2008/8
Y1 - 2008/8
N2 - 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.
AB - 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.
KW - Jacobi polynomials
UR - http://www.scopus.com/inward/record.url?scp=36949037135&partnerID=8YFLogxK
U2 - 10.1007/s00365-007-0674-0
DO - 10.1007/s00365-007-0674-0
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AN - SCOPUS:36949037135
SN - 0176-4276
VL - 28
SP - 113
EP - 125
JO - Constructive Approximation
JF - Constructive Approximation
IS - 2
ER -