## , the orthonormal Jacobi polynomials P(α,β)

### BibTeX

@MISC{Krasikov_,the,

author = {Ilia Krasikov and Abstract T. Erdélyi and A. P. Magnus and P (α Β},

title = {, the orthonormal Jacobi polynomials P(α,β)},

year = {}

}

### OpenURL

### Abstract

k (x) = O max 1, (α 2 + β 2) 1/4}) max x∈[−1,1] (1 − x)α+ 1 2 (1 + x) β+1 2 [Erdélyi 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, even in a stronger 4 form by giving very explicit upper bounds. We also show that δ2 − x2 2 α (1 − x) ( P (α,α)

### Citations

2054 |
Handbook of Mathematical Functions
- Abramowitz, Stegun
- 1965
(Show Context)
Citation Context ...one δ = d3. To prove the inequality (14) M α 2 k (δ) < π ) ( , d4 − d3 = O ( 1 1 + 8(k + α) 2 ) , α 2 k 3/2 (k + α) 5/2 we have to find Mα (α,α) k (0, δ). The value of P k (0) for even k is (see e.g. =-=[1]-=-), (15) P (α,α) k (0) = (−1) k/2 Γ(k + α + 1) ) . k !Γ( + α + 1) This yields P (α,α) 2k ( k 2 k (0) = (−1) k/2 2r/2 ( ) k r−k+1 2 ! Γ( 2 ) . To simplify this expression we use the following inequality... |

47 | Orthogonal Polynomials for Exponential Weights
- Levin, Lubinsky
- 2001
(Show Context)
Citation Context ...1991 Mathematics Subject Classification. 33C45. 12 I. KRASIKOV where the constant C is independent on i and a±i are Mhaskar-Rahmanov-Saff numbers for Q, was developed by A.L. Levin and D.S. Lubinsky =-=[11]-=-. Recently it has been extended to the Laguerre-type exponential weights x 2ρ e −2Q(x) [6, 12]. In the case of classical orthogonal Hermite and Laguerre polynomials explicit bounds confirming such a n... |

18 |
On gamma function inequalities
- BUSTOZ, ISMAIL
- 1986
(Show Context)
Citation Context ...α,α) k (0) = (−1) k/2 Γ(k + α + 1) ) . k !Γ( + α + 1) This yields P (α,α) 2k ( k 2 k (0) = (−1) k/2 2r/2 ( ) k r−k+1 2 ! Γ( 2 ) . To simplify this expression we use the following inequality (see e.g. =-=[2]-=-), (16) Γ(x + 1) < + 1) Γ 2 ( x 2 what yields for k + 2α ≥ 0, ( P (α,α) ) 2 k (0) < Hence, for |x| ≤ δ, we have 2 x+1 2 √ π(x + 1 2 ) M α k (δ) = Mα ( k (0, δ) = δ P (α,α) ) 2 k (0) < 2 √ r k! Γ(r − k... |

13 | Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials
- Erdélyi, Magnus, et al.
- 1994
(Show Context)
Citation Context ...in the ultraspherical case was obtained earlier by L. Lorch [13]. A remarkable result covering almost all possible range of the parameters has been established by T. Erdélyi, A.P. Magnus and P.Nevai, =-=[5]-=-, (2) M α,β k ≤ ( 2e 2 + √ α2 + β2 ) provided k ≥ 0, α, β ≥ − 1 2 . Moreover, they suggested the following conjecture: Conjecture 1. provided α ≥ β ≥ − 1 2 . M α,β k π ( { = O max 1, |α| 1/2}) , The b... |

10 |
Inequalities for ultraspherical polynomials and the gamma function
- Lorch
- 1984
(Show Context)
Citation Context ...β k ≤ 22α+1Γ(k + α + β + 1)Γ(k + α + 1) πk! (2k + α + β + 1) 2α 2 = + O Γ(k + β + 1) π where k = 0, 1, ... . A slightly stronger inequality in the ultraspherical case was obtained earlier by L. Lorch =-=[13]-=-. A remarkable result covering almost all possible range of the parameters has been established by T. Erdélyi, A.P. Magnus and P.Nevai, [5], (2) M α,β k ≤ ( 2e 2 + √ α2 + β2 ) provided k ≥ 0, α, β ≥ −... |

5 |
Monotonicity properties of the zeros of ultraspherical polynomials
- Elbert, Siafarikas
- 1999
(Show Context)
Citation Context .... < xi, be the nonnegative zeros of P (α,α) k (x). Obviously, 0 < ξ < x1, so we can use an upper bound on x1 instead. An appropriate estimate for zeros of ultraspherical polynomials has been given in =-=[4]-=-, in particular ( )−1/2 2 2k + 1 x1 < + α hk, 4k + 2 where hk is the least positive zero of the Hermite polynomial Hk(x). Since hk ≤ (18) ξ ≤ Using the formula √ 21 4k+2 , [14, sec. 6.3], we obtain √ ... |

4 |
Inequalities for orthonormal Laguerre polynomials
- Krasikov
(Show Context)
Citation Context ... In the case of classical orthogonal Hermite and Laguerre polynomials explicit bounds confirming such a nearly equioscillatory behaviour independently on the parameters involved were given in [8] and =-=[9]-=- respectively. The case of Jacobi polynomials P (α,β) k (x), w(x) = (1 − x) α (1 + x) β , is much more difficult. Let us introduce some necessary notation. We define M α,β k (x, dm, dM) = √ (x − dm)(d... |

3 |
A Bernstein-Type Inequality for the Jacobi Polynomial
- Chow, Gatteschi, et al.
- 1994
(Show Context)
Citation Context ...α,α k (x), and shorten Mα k (x, −d, d), Mα k (−d, d) to Mα k (x, d), Mα k (d) respectively. As P (α,β) k For − 1 2 (x) = (−1) kP (β,α) k (−x) we may safely assume that α ≥ β. , the following is known =-=[3]-=-: ( ) 1 , k < β ≤ α < 1 2 (1) M α,β k ≤ 22α+1Γ(k + α + β + 1)Γ(k + α + 1) πk! (2k + α + β + 1) 2α 2 = + O Γ(k + β + 1) π where k = 0, 1, ... . A slightly stronger inequality in the ultraspherical case... |

3 |
R.Sakai, Orthonormal polynomials with generalized Freud-type weights
- Kasuga
(Show Context)
Citation Context ...ependent on i and a±i are Mhaskar-Rahmanov-Saff numbers for Q, was developed by A.L. Levin and D.S. Lubinsky [11]. Recently it has been extended to the Laguerre-type exponential weights x 2ρ e −2Q(x) =-=[6, 12]-=-. In the case of classical orthogonal Hermite and Laguerre polynomials explicit bounds confirming such a nearly equioscillatory behaviour independently on the parameters involved were given in [8] and... |

3 |
On extreme zeros of classical orthogonal polynomials
- Krasikov
(Show Context)
Citation Context ... intervals [η−1, x1], [xk, η1], where η±1 are given by (5). Rather accurate bounds χ−1 and χ1 on x1 and xk, such that x1 < χ−1 < χ1 < xk, and |ηj − χj| = O ( (k + α + β) −2/3) , j = ±1, were given in =-=[10]-=-. 3. Proof of Theorem 3, even case In this section we prove Theorem 3 for ultraspherical polynomials of even degree. Without loss of generality we will assume x ≥ 0. To simplify some expressions it wi... |

3 |
UB8 3PH United Kingdom E-mail address: mastiik@brunel.ac.uk for
- Publ, 23, et al.
- 1975
(Show Context)
Citation Context ...ree. Keywords: Jacobi polynomials 1. Introduction In this paper we will use bold letters for orthonormal polynomials versus regular characters for orthogonal polynomials in the standard normalization =-=[14]-=-. Given a family {pi(x)} of orthonormal polynomials orthogonal on a finite or infinite interval I with respect to a weight √ function w(x) ≥ 0, it is an important and difficult problem to estimate sup... |

2 |
On the maximum of Jacobi Polynomials
- Krasikov
(Show Context)
Citation Context ...0, α, β ≥ − 1 2 . Moreover, they suggested the following conjecture: Conjecture 1. provided α ≥ β ≥ − 1 2 . M α,β k π ( { = O max 1, |α| 1/2}) , The best currently known bound was given by the author =-=[7]-=-, (3) M α,β k ≤ 11 ( )1/3 2 2 (α + β + 1) (2k + α + β + 1) 4k(k + α + β + 1) , = O k ( ( 2/3 α 1 + α k ) 1/3 ) , provided k ≥ 6, α ≥ β ≥ 1+√ 2 4 . We also brought some evidences in support of the foll... |

1 |
Sharp inequalities for Hermite polynomials, proceeding CTF-2005
- Krasikov
(Show Context)
Citation Context ...[6, 12]. In the case of classical orthogonal Hermite and Laguerre polynomials explicit bounds confirming such a nearly equioscillatory behaviour independently on the parameters involved were given in =-=[8]-=- and [9] respectively. The case of Jacobi polynomials P (α,β) k (x), w(x) = (1 − x) α (1 + x) β , is much more difficult. Let us introduce some necessary notation. We define M α,β k (x, dm, dM) = √ (x... |