## Generalized Jacobi Weights, Christoffel Functions, And Jacobi Polynomials (1994)

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Venue: | SIAM J. Math. Anal |

Citations: | 14 - 2 self |

### BibTeX

@ARTICLE{Erdélyi94generalizedjacobi,

author = {Thomas Erdélyi and Alphonse P. Magnus and Paul Nevai},

title = {Generalized Jacobi Weights, Christoffel Functions, And Jacobi Polynomials},

journal = {SIAM J. Math. Anal},

year = {1994},

volume = {25},

pages = {602--614}

}

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### Abstract

We obtain such upper bounds for Jacobi polynomials which are uniform in all the parameters involved and which contain explicit constants. This is done by a combination of some results on generalized Christo#el functions and some estimates of Jacobi polynomials in terms of Christo#el functions. 1.

### Citations

605 |
An Introduction to Orthogonal Polynomials
- Chihara
- 1978
(Show Context)
Citation Context ...eneralized Christoffel Functions and Generalized Polynomials Generalized Christoffel Functions. Given w(≥ 0) ∈ L1 (R) andp∈(0, ∞), λ ∗ n(w, p, z) def � f = inf f∈| GCAP |n−1 p (t) f p=-= w(t) dt, z ∈ C, (3) (z) where n-=- ≥ 1isreal, thatis,nis not necessarily an integer. Remark 2. Of course, λ∗ n(w, p) ≡ λ∗ np−p+1(w, 1). As a matter of fact, this is one of the underlying reasons for the usefulness of the c... |

62 |
Orthogonal Polynomials," Pergamon
- FREUD
- 1971
(Show Context)
Citation Context ...ty it remains valid for all g ∈|GCAP |N. Hence, for every N ≥ 0. Thus, �G(cos ·)� L ∞ (R) ≤ � � � � 1 − (·) 2 � � F � ≤ L∞ ([−1,1]) (1 + N) e 4π (2 + N) e 2π � =-=π −π � 1 −1 G(cos θ) dθ, ∀ G ∈|GCAP |N, (8) F (t) dt, ∀ � 1-=- − (·) 2 F ∈|GCAP |N+1, N ≥ 0. Applying this inequality with F = f p w, Theorem 3 follows immediately. � §3. Christoffel Functions and Jacobi Polynomials I’ve tried A! I’ve tried B! I’... |

57 | Géza Freud, orthogonal polynomials and Christoffel functions: A case study - Nevai - 1986 |

56 | of Equations and System of Equations - Ostrowski, Solutions - 1960 |

56 | Orthogonal Polynomials - Freud - 1971 |

41 | Orthogonal Polynomials - Nevai - 1979 |

26 |
On Asymptotics of Jacobi Polynomials
- Chen, Ismail
- 1991
(Show Context)
Citation Context ...)p2 n (w, x) suggest that small values of n give relevant information. For instance, with α =10andβ=2, n 0 1 2 3 4 5 6 7 8 9 10 max 1.478 1.251 1.191 1.161 0.845 0.747 1.123 0.727 1.112 0.703 0.685 =-=π (2)s4 THOMAS ERDÉLYI, ALPHO-=-NSE P. MAGNUS, AND PAUL NEVAI For n = 0, explicit calculation yields � max 1 − x2 2 w(x)p0 (w, x) = x∈[−1,1] that is, 4 Γ(α + β +2) 2 α+β+1 Γ(α +1)Γ(β+1) max x∈[−1,1] (1 − x)α+... |

18 |
Bounds on the extreme zeros of orthogonal polynomials
- Ismail, Li
- 1992
(Show Context)
Citation Context ...ed from the other one by the three–term recurrence formula. Combination of (i) and (ii) yields (iii) The differential equation: (1 − x 2 )p ′′ n(w, x)+[β−α−(α+β+2)x]p ′ n(w, x)+n(n+�=-=�+β+1)pn(w, x) = 0 (12) (cf. [-=-22, formula (4.2.1), p. 60] or [3, formula (2.20), p. 149]). In order to compute r ′ (±1), we proceed as follows. From (11), γn(w) pn+1(w, ±1) γn+1(w) pn(w, ±1) and from (12), so that we have r... |

17 | Mean convergence of Lagrange interpolation - NEVAI - 1976 |

12 |
Inequalities for ultraspherical polynomials and the gamma function
- LORCH
- 1984
(Show Context)
Citation Context ...sharper version of it). Bernstein’s results can be extended to Jacobi (i.e., ultraspherical or Gegenbauer) polynomials with parameters − 1 2 <α=β<1 2 (cf. [22, (7.33.4) and (7.33.5), p. 171] and=-= also [15] for a-=- refinement). In addition, for a wider range of the parameters, similar inequalities have been proved in [13] (α = β>−1 2 )and[7](α=β>1 2 ). For instance, L. Lorch [15, formula (10), p. 115] pro... |

12 | On the zeros of Jacobi polynomials P (αn,βn) n (x - Moak, Saff, et al. - 1979 |

10 |
New inequalities for the zeros of Jacobi polynomials
- Gatteschi
- 1987
(Show Context)
Citation Context ...6.626 of the 1980 edition of Gradshteyn and Ryzhik’s book [11], coming from Theorem 1.82.1 of Szegő [22 Section 1.82 p.19], known as “Sturm’s theorem for open intervals”, see also the introdu=-=ction of [9]. 5 Here i-=-s a self–contained proof. Suppose that Y (x1) =0forsomex1 ∈(a, b). Since the equations are homogeneous, we may assume that Z(x) > 0on(a, b), and (x0 − x1)Y ′ (x1) > 0. 6 Therefore, Y (x) = �... |

10 | Bernstein’s inequality in Lp,0 < p < 1 - Nevai - 1979 |

5 |
Alternative proof of a sharpened form of Bernstein’s inequality for Legendre polynomials
- LORCH
- 1993
(Show Context)
Citation Context ...le trigonometric functions. For Legendre polynomials this is somewhat more complicated, and the appropriate inequality was proved by S. Bernstein (cf. [22, (7.3.8), p. 165] for Bernstein’s result an=-=d [1, 14] for a-=- sharper version of it). Bernstein’s results can be extended to Jacobi (i.e., ultraspherical or Gegenbauer) polynomials with parameters − 1 2 <α=β<1 2 (cf. [22, (7.33.4) and (7.33.5), p. 171] an... |

5 |
Bernstein’s inequality in Lp for 0 and (C, 1) bounds for orthogonal polynomials
- MÁTÉ, NEVAI
- 1980
(Show Context)
Citation Context ...0.sGENERALIZED JACOBI WEIGHTS AND JACOBI POLYNOMIALS 13 the Newton–Raphson iteration method (see, for instance, [21, Chapter 9, p. 55]). Hence, by (13), 2(β +1) −1+ n(n+α+β+1) ≤xkn(w) 2(α +1=-=) ≤ 1 − , (16) n(n+α+β+1) k -=-=1,2,...,n, is valid for every α>−1, β>−1, and n ≥ 1(andisexact if n =1). However, considering only the upper bound, it behaves like 1 − 2(α +1)/n 2 for large n, instead of 1 − j 2 α /(2... |

5 | Remez-type inequalities on the size of generalized polynomials - Erdlyi |

4 | Inequalities for generalized non–negative polynomials
- Erdélyi, Máté, et al.
- 1992
(Show Context)
Citation Context ...neralized polynomials is (one of) the perfect setting for studying Jacobi polynomials. As a matter of fact, this was the primary reason for introducing generalized polynomials in the first place (cf. =-=[6, 5]-=-). This paper is a modest attempt to demonstrate the applicability of generalized polynomials to problems which have not yet been settled in a satisfactory way despite more than a hundred years of und... |

3 | Nikol’skiLtype inequalities for generalized polynomials and zeros of orthogonal polynomials - ERDLYI - 1991 |

2 |
Inequalities for ultraspherical polynomials and application to quadrature, manuscript
- Förster
- 1993
(Show Context)
Citation Context ... 1 2 <α=β<1 2 (cf. [22, (7.33.4) and (7.33.5), p. 171] and also [15] for a refinement). In addition, for a wider range of the parameters, similar inequalities have been proved in [13] (α = β>−1 =-=2 )and[7](α=β>1 2 ). For instance, L. Lor-=-ch [15, formula (10), p. 115] proved1 max x∈[−1,1] � � �(1 − x 2 ) λ 2 P (λ) n (x) � � � ≤ 21−λ (n + λ) λ−1 Γ(λ) for n =0,1,... and 0 <λ<1, which, in terms of the ortho... |

2 | Orthogonal polynomials on the real line associated with the weight |x| αe−|x|β - Nevai - 1973 |

2 | Weighted Markov and Bernstein type inequalities for generalized polynomials - Erdélyi - 1992 |

2 | Inequalities for generalized non-negative polynomials - ERDLYI, MATS, et al. - 1992 |

1 |
An estimate of the remainder in the expansion of the generating function for the Legendre polynomials (Generalization and improvement of Bernstein’s inequality), Vestnik Leningrad Univ
- Antonov, Hol˘sevnikov
- 1981
(Show Context)
Citation Context ...le trigonometric functions. For Legendre polynomials this is somewhat more complicated, and the appropriate inequality was proved by S. Bernstein (cf. [22, (7.3.8), p. 165] for Bernstein’s result an=-=d [1, 14] for a-=- sharper version of it). Bernstein’s results can be extended to Jacobi (i.e., ultraspherical or Gegenbauer) polynomials with parameters − 1 2 <α=β<1 2 (cf. [22, (7.33.4) and (7.33.5), p. 171] an... |

1 |
Generalized polynomial weights, Christoffel functions and zeros of orthogonal polynomials
- Erdélyi, Nevai
- 1992
(Show Context)
Citation Context ...neralized polynomials is (one of) the perfect setting for studying Jacobi polynomials. As a matter of fact, this was the primary reason for introducing generalized polynomials in the first place (cf. =-=[6, 5]-=-). This paper is a modest attempt to demonstrate the applicability of generalized polynomials to problems which have not yet been settled in a satisfactory way despite more than a hundred years of und... |

1 |
Bernstein type inequalities for Jacobi polynomials
- Gatteschi
- 1988
(Show Context)
Citation Context ...x∈[−1,1] � − x2w(x) pn(w, x) � � ≤ � � 2Γ(n+1) n+α+ πΓ(n +2α+1) 1 �α 2 for n =0,1,...,and−1 1 <α< 2 2 where w(x) =(1−x2 ) α . For nonsymmetric Jacobi weights much less =-=is known. In 1988, L. Gatteschi [10] extended Bernstein’s resul-=-ts to Jacobi polynomials with − 1 1 <α,β< . For instance, he 2 2 proved that if − 1 1 2 <α,β< 2 and α + β>0then2 max θ∈[0, π � � 1 α+ 2 β+ �(sin θ/2) (cos θ/2) 2 ] 1 � 2 (α,... |

1 |
Inequalities for Legendre functions and Gegenbauer functions
- Lohöfer
- 1991
(Show Context)
Citation Context ...als with parameters − 1 2 <α=β<1 2 (cf. [22, (7.33.4) and (7.33.5), p. 171] and also [15] for a refinement). In addition, for a wider range of the parameters, similar inequalities have been proved=-= in [13] (α = β>−1 2 )and[7](α=β>1 2 ). -=-For instance, L. Lorch [15, formula (10), p. 115] proved1 max x∈[−1,1] � � �(1 − x 2 ) λ 2 P (λ) n (x) � � � ≤ 21−λ (n + λ) λ−1 Γ(λ) for n =0,1,... and 0 <λ<1, which, i... |

1 | Department of Mathematics, The Ohio State University, 231 West 18th Avenue - Soc, Providence - 1967 |

1 | Weighted Markov and Bernstein type inequalities for generalized non-negative polynomials - ERDLYI - 1992 |