## Stieltjes polynomials and Lagrange interpolation (1997)

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Venue: | Math. Comp |

Citations: | 11 - 5 self |

### BibTeX

@ARTICLE{Ehrich97stieltjespolynomials,

author = {Sven Ehrich and Giuseppe Mastroianni},

title = {Stieltjes polynomials and Lagrange interpolation},

journal = {Math. Comp},

year = {1997},

volume = {66},

pages = {311--331}

}

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### Abstract

Abstract. Bounds are proved for the Stieltjes polynomial En+1, andlower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials Pn. This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials Gn. Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of En+1, and for the extended Lagrange interpolation process with respect to the zeros of PnEn+1 in the uniform and weighted L p norms. The corresponding Lebesgue constants

### Citations

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Orthogonal Polynomials
- Szegö
- 1939
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Citation Context ...OLYNOMIALS AND LAGRANGE INTERPOLATION 325 We argue analogously for µ = � � � n+1 2 +1,...,n+ 1, and, observing 1 − x2 µ ∼ � 1 − ξ2 µ, we obtain (31). Using (34), (35) and a standard bound for Pn (cf. =-=[37]-=-), we also obtain (32). We now proceed with the proofs of the results in §3. In the following, we use some properties of the Hilbert transform H(f), defined by � f(x) H(f,t) = lim ɛ→0 |x−t|>ɛ x − t dx... |

26 |
Stieltjes polynomials and related quadrature rules
- Monegato
- 1982
(Show Context)
Citation Context ...rfer [34] and of the first author of this paper. For a more complete history of the problem under consideration, the interested reader may consult the exhaustive surveys of Gautschi [15] and Monegato =-=[28]-=-. Nevertheless, the interpolation process based on the zeros of Stieltjes polynomials and/or the extended interpolation process that uses the zeros of the polynomials K2n+1 = PnEn+1 have received litt... |

17 |
Mean convergence of Lagrange interpolation
- NEVAI
- 1976
(Show Context)
Citation Context ...mma 4.5 to obtain � � I1 ≤ � E � ′ n+1(ξ) E ′ n+1 (ξd) � � � � �f�∞ � � � 4 � 1 − ξ2 d ≤ C � � n √ n E′ � � � n+1(ξ) � � �f�∞, where ξd <ξ<ξd+1. Applying the weighted Bernstein inequality (cf., e.g., =-=[30]-=-), observing 4� 1 − ξ2 d ∼ 4� 1 − ξ2 , and using (9) we obtain I1 ≤ C �f�∞ for a positive constant C. Proof of Theorem 3.2. Let q ± ∈ Pn such that q− ≤ f ≤ q + . Using [30, (25)], we have �[f − Ln+1f]... |

16 |
Gauss–Kronrod quadrature — a survey
- Gautschi
- 1988
(Show Context)
Citation Context ...papers of Peherstorfer [34] and of the first author of this paper. For a more complete history of the problem under consideration, the interested reader may consult the exhaustive surveys of Gautschi =-=[15]-=- and Monegato [28]. Nevertheless, the interpolation process based on the zeros of Stieltjes polynomials and/or the extended interpolation process that uses the zeros of the polynomials K2n+1 = PnEn+1 ... |

14 |
A Note on Extended Gaussian Quadrature Rules
- Monegato
- 1976
(Show Context)
Citation Context ...ic polynomials of degree at most k). Some years later Barrucand [1] observed that ξµ,n+1 are precisely the zeros of the Stieltjes polynomials En+1. In the second half of the seventies, G. Monegato in =-=[25]-=-, proved that the interlacing property of the zeros of En+1 with those of Pn is equivalent to the positivity of the coefficients B GK µ,n+1, and then proved that the Gauss–Kronrod formula has positive... |

12 |
On a set of polynomials
- Geronimus
- 1930
(Show Context)
Citation Context ...) = α0,n sin(n +1)θ+α1,n sin(n − 1)θ + ···+ � αn 2 −1,n sin3θ + α n 2 ,n sin θ, n even, α n−1 2 ,n sin2θ, n odd, is important in connection with a class of polynomials Gn considered by Geronimus (cf. =-=[18]-=-; cf. also [28, 34, 36]). The connection is (cf. [28, 36]) (11) sin θGn(cos θ) = en(θ). As a byproduct of the previous theorem, we also obtain bounds for the Geronimus polynomial Gn. Theorem 2.2. For ... |

10 |
Asymptotic Properties of Stieltjes Polynomials and Gauss–Kronrod Quadrature Formulae
- Ehrich
- 1995
(Show Context)
Citation Context ...ds in Theorem 2.1 are smaller by a factor ∼ n−1/2 at the endpoints if compared to the interior of the interval. This conforms to recent results about the asymptotic behaviour of Stieltjes polynomials =-=[10]-=-, namely that the formula � �� 2nsin θ En+1(cos θ)=2 cos n + π 1 � θ + 2 π � + o( 4 √ (10) n) holds uniformly for ɛ ≤ θ ≤ π − ɛ. On the other hand, a comparison of (5) and (10) shows that in fixed clo... |

10 |
Quadrature sums involving pth powers of polynomials
- Lubinsky, Máté, et al.
- 1987
(Show Context)
Citation Context ... sequence (αν,n) be defined as in (3). Then 1 3 √ n ≤ m� ′ αν,n ≤ 5 3 √ n . (18) ν=0 Here the prime means that αm,n has to be replaced by 1 2 αm,n if n is odd. Proof of Lemma 4.1. Szegö[36]provedthat =-=(19)-=- α0,n =1, αν,n < 0, ν =1,2,..., In [10], it was proved that for the product ∞� αν,n =0. (α0,n + ···+αk,n)(f0,n + ···+fk,n) =1+Rk,n, where fν,n is defined in (3), we have (20) Rk,n < 0. Usingalowerboun... |

9 |
On the asymptotic behaviour of functions of the second kind and Stieltjes polynomials and on Gauss-Kronrod quadrature formulas
- Peherstorfer
- 1992
(Show Context)
Citation Context ...acing properties of the zeros and to construct extended positive quadrature formulas. Among them, we mention Gautschi and Notaris [16], Gautschi and Rivlin [17], and the recent papers of Peherstorfer =-=[34]-=- and of the first author of this paper. For a more complete history of the problem under consideration, the interested reader may consult the exhaustive surveys of Gautschi [15] and Monegato [28]. Nev... |

9 |
Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören
- Szegő
- 1935
(Show Context)
Citation Context ...n(1) = 1. The polynomials En+1 defined (up to a multiplicative constant) by � 1 −1 En+1(x)Pn(x) x k dx =0, k=0,1,...,n, n≥1, were introduced by Stieltjes more than one hundred years ago. In 1934 Szegö=-=[36]-=-, following Stieltjes idea, introduced the wider class of polynomials E (λ) , defined by � 1 −1 n+1 wλ(x)E (λ) (λ) n+1 (x)P n (x) x k dx =0, k=0,1,...,n, n≥1, where wλ(x) =(1−x2 ) λ−1/2 ,λ>−1 2 ,andP(... |

8 |
Nicholson-type integrals for products of Gegenbauer functions and related topics, Theory and
- Durand
- 1975
(Show Context)
Citation Context ...n(x−iɛ)) = 2 Qn(x), x ∈(−1,1), where, for complex z �∈ [−1, 1], Qn(z)= 1 � 1 Pn(t) 2 −1 z−t dt is the Legendre function of the second kind.sSTIELTJES POLYNOMIALS AND LAGRANGE INTERPOLATION 321 Durand =-=[9]-=- proved that the symmetric function �� 2 sin θ π Qn(cos �2 θ) +[Pn(cos θ)] 2 � is monotonically increasing for 0 <θ≤ π 2 ,andthat �� 2 sin θ π Qn(cos �2 θ) +[Pn(cos θ)] 2 � � � � n 1 Γ 2 + 2 ≤ √ � n π... |

8 |
Convergence of generalized Jacobi series and interpolating polynomials
- Xu, Mean
- 1993
(Show Context)
Citation Context ...ort K,thenwehave � � (36) GH(F) = − FH(G), K K see, for instance, [31]. Moreover, let u ∈ V and v ∈ V be two GJ weights with u ≤ v, u ∈ Lp and v−1 ∈ Lq ,1<p<∞,p−1 +q−1 =1,then (37) see, for instance, =-=[29, 38]-=-. �H(f)u�p≤C�fv�p, Proof of Theorem 3.1. It is sufficient to prove that |Ln+1(f,x)| ≤ Clog n �f�∞, −1 ≤ x ≤ 1. Let d be chosen such that ξd ≤ x<ξd+1. Letalso|x−ξd|≤|ξd+1 − x| (the other case can be tr... |

7 |
Positivity of Weights of Extended Gauss–Legendre Quadrature Rules
- Monegato
- 1978
(Show Context)
Citation Context ...f Pn is equivalent to the positivity of the coefficients B GK µ,n+1, and then proved that the Gauss–Kronrod formula has positive weights even if it is constructed with respect to the weight wλ, 0<λ≤1 =-=[26]-=-. Kronrod’s idea together with the results of Barrucand and Monegato urged a lot of mathematicians to consider Stieltjes polynomials for more general weight functions, to study the interlacing propert... |

6 |
Intégration Numérique, Abscisses de Kronrod-Patterson et Polynômes de
- Barrucand
- 1970
(Show Context)
Citation Context ...997 American Mathematical Societys312 SVEN EHRICH AND GIUSEPPE MASTROIANNI i.e. R GK 2n+1 (p)=0ifp∈P3n+1 (Pk is the space of all algebraic polynomials of degree at most k). Some years later Barrucand =-=[1]-=- observed that ξµ,n+1 are precisely the zeros of the Stieltjes polynomials En+1. In the second half of the seventies, G. Monegato in [25], proved that the interlacing property of the zeros of En+1 wit... |

6 |
A Family of Gauss–Kronrod Quadrature Formulae
- Gautschi, Rivlin
- 1988
(Show Context)
Citation Context ...eneral weight functions, to study the interlacing properties of the zeros and to construct extended positive quadrature formulas. Among them, we mention Gautschi and Notaris [16], Gautschi and Rivlin =-=[17]-=-, and the recent papers of Peherstorfer [34] and of the first author of this paper. For a more complete history of the problem under consideration, the interested reader may consult the exhaustive sur... |

6 |
Two-weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform.StudiaMath.55(1976
- Muckenhoupt, Wheeden
(Show Context)
Citation Context ...ort K,thenwehave � � (36) GH(F) = − FH(G), K K see, for instance, [31]. Moreover, let u ∈ V and v ∈ V be two GJ weights with u ≤ v, u ∈ Lp and v−1 ∈ Lq ,1<p<∞,p−1 +q−1 =1,then (37) see, for instance, =-=[29, 38]-=-. �H(f)u�p≤C�fv�p, Proof of Theorem 3.1. It is sufficient to prove that |Ln+1(f,x)| ≤ Clog n �f�∞, −1 ≤ x ≤ 1. Let d be chosen such that ξd ≤ x<ξd+1. Letalso|x−ξd|≤|ξd+1 − x| (the other case can be tr... |

5 |
An overview of results and questions related to Kronrod schemes
- Monegato
- 1979
(Show Context)
Citation Context ... ≤ x ≤1, |En+1(x)| ≤ C √ (9) � �1 �1−x 2 1 n 2 + , n −1 ≤ x ≤1, where C is a positive constant.s314 SVEN EHRICH AND GIUSEPPE MASTROIANNI The best bound for Stieltjes polynomials in literature is (cf. =-=[27]-=-) |En+1(x)| ≤ 4 , −1 ≤ x ≤1. γn While this bound behaves uniformly on the whole interval [−1, 1], the bounds in Theorem 2.1 are smaller by a factor ∼ n−1/2 at the endpoints if compared to the interior... |

4 |
Pointwise simultaneous convergence of extended Lagrange interpolation with additional knots
- Criscuolo, Mastroianni, et al.
- 1992
(Show Context)
Citation Context ... of two or three orthogonal polynomials with respect to different weights. By using the method of additional nodes they proved convergence theorems in uniform and weighted L p norms (see for instance =-=[4, 5, 6, 20, 21, 32]-=-). The reasons for the absence of results on interpolation processes based on the zeros of En+1 and/or PnEn+1 are first of all the fact that in literature there are no accurate bounds available for th... |

4 |
On estimates for the weights
- Förster, Petras
- 1990
(Show Context)
Citation Context ...ch functions, we cannot use (12), but the following theorem is useful. Theorem 3.3. Assume u √ ϕ ∈ Lp and (u √ ϕ) −1 ∈ Lq , 1 <p<∞,andp−1 + q−1 =1.Iff∈ACloc and f ′ ϕ2/pu ∈ L1 ,then �[f−Ln+1f]u�p ≤ C =-=(14)-=- n �f ′ ϕu� L p [(ξ1,ξn+1)] + c �f ′ ϕ 2/p u� L 1 (I ′ n ), where I ′ n =[−1,1]\(ξ1,ξn+1) and the constants are independent of n and f. In particular if f ′ ϕu ∈ Lp ,then �[f−Ln+1f]u�p ≤ C n En−1(f ′ ... |

4 |
Uniform Convergence of derivatives of Lagrange interpolation
- Mastroianni
- 1992
(Show Context)
Citation Context ... of two or three orthogonal polynomials with respect to different weights. By using the method of additional nodes they proved convergence theorems in uniform and weighted L p norms (see for instance =-=[4, 5, 6, 20, 21, 32]-=-). The reasons for the absence of results on interpolation processes based on the zeros of En+1 and/or PnEn+1 are first of all the fact that in literature there are no accurate bounds available for th... |

3 |
Mean convergence of the derivatives of extended Lagrange interpolation with additional nodes
- Criscuolo, Mastroianni, et al.
- 1993
(Show Context)
Citation Context ... of two or three orthogonal polynomials with respect to different weights. By using the method of additional nodes they proved convergence theorems in uniform and weighted L p norms (see for instance =-=[4, 5, 6, 20, 21, 32]-=-). The reasons for the absence of results on interpolation processes based on the zeros of En+1 and/or PnEn+1 are first of all the fact that in literature there are no accurate bounds available for th... |

3 |
Convergence of extended Lagrange interpolation
- Criscuolo, Mastroianni, et al.
- 1990
(Show Context)
Citation Context |

2 |
Fourier and Lagrange Operators in some weighted Sobolevtype spaces
- Criscuolo, Mastroianni
- 1995
(Show Context)
Citation Context ...d f. For example, if p =2,u(x)= 4√ 1−x2and f(x) = log(1 + x), from (14) we obtain �[f − Ln+1(f)]u�2 = O(n−1 ). The interested reader may find estimates of En(g)u,p with a GJ weight u and g ∈ ACloc in =-=[3]-=-. The case p = 1 is interesting in the applications, because it is connected with the error of the product quadrature rule. Estimates of �[f − Ln+1]u�1 in the L1 norm and the same weight u are only po... |

2 |
An Algebraic and Numerical Study of Gauss–Kronrod Quadrature Formulae for Jacobi Weight Functions
- Gautschi, Notaris
- 1988
(Show Context)
Citation Context ...jes polynomials for more general weight functions, to study the interlacing properties of the zeros and to construct extended positive quadrature formulas. Among them, we mention Gautschi and Notaris =-=[16]-=-, Gautschi and Rivlin [17], and the recent papers of Peherstorfer [34] and of the first author of this paper. For a more complete history of the problem under consideration, the interested reader may ... |

2 |
Approximation of functions by extended Lagrange interpolation
- Mastroianni
- 1995
(Show Context)
Citation Context |

2 |
Nevai; Mean convergence of Lagrange interpolation
- P
- 1976
(Show Context)
Citation Context ...k,n)(f0,n + ···+fρk,n) = k� ν� fν−µ,nαµ,n + ρk� k� fν−µ,nαµ,n + ν=0 µ=0 ν=k+1 µ=0 k� ν� αν,n fρk+1−µ,n. ν=1 µ=1 By (3), the first term is equal to 1, and the second term is greater than 0. Therefore, =-=(22)-=- (α0,n + ···+αk,n)(f0,n + ···+fρk,n) > 1+ k� ν� αν,n fρk+1−µ,n ν=1 µ=1 > 1 − f (ρ−1)k+1,n |α1,n +2α2,n + ···+kαk,n|. Now let k = m = ⌊(n+1)/2⌋. We again obtain, by elementary estimates, from [10, (40)... |

2 |
Weighted L p error of Lagrange interpolation
- Mastroianni, Vértesi
(Show Context)
Citation Context ...of of Theorem 3.3. Let q ± be defined as in the previous proof. Let ⎧ ⎪⎨f(ξ1), x < ξ1, fn(x)= f(x), ξ1 ≤ x ≤ ξn+1, ⎪⎩ � q f(ξn+1), ξn+1 <x. We have �[f − Ln+1f]u�p ≤ �[f −fn]u�p + �[fn−Ln+1fn]u�p. By =-=[23]-=-, if f ∈ ACloc and f ′ ϕ2/pu ∈ L1 ,weobtain �[f−fn]u�p ≤ C�f ′ ϕ 2/p u� L 1 (I ′ n ).s328 SVEN EHRICH AND GIUSEPPE MASTROIANNI We observe that fn is a bounded and measurable function, and, using (13),... |

2 |
Hilbert Transform and Lagrange Interpolation
- Nevai
- 1990
(Show Context)
Citation Context ...fined by � f(x) H(f,t) = lim ɛ→0 |x−t|>ɛ x − t dx, f ∈ L1 . We recall that if G ∈ L∞ and F log + F ∈ L1 ,whereFand G have compact support K,thenwehave � � (36) GH(F) = − FH(G), K K see, for instance, =-=[31]-=-. Moreover, let u ∈ V and v ∈ V be two GJ weights with u ≤ v, u ∈ Lp and v−1 ∈ Lq ,1<p<∞,p−1 +q−1 =1,then (37) see, for instance, [29, 38]. �H(f)u�p≤C�fv�p, Proof of Theorem 3.1. It is sufficient to p... |

1 |
Error bounds for product quadrature rules in L1 weighted norm
- Criscuolo, Scuderi
- 1994
(Show Context)
Citation Context ...s connected with the error of the product quadrature rule. Estimates of �[f − Ln+1]u�1 in the L1 norm and the same weight u are only possible under strong conditions on the weight u (see for instance =-=[7, 24]-=-). From the previous theorems, we can derive better estimates than (12) when p = 1 by some assumption on the weight u. For instance since �[f − Ln+1f]u�1 ≤ � �u�1 �[f − Ln+1f] √ u�2, if (uϕ) ±1 ∈ L1 ,... |

1 |
Boundedness of Lagrange and Hermite operators
- Vecchia, Mastroianni, et al.
(Show Context)
Citation Context ....Forx∈[−1,x1]∪[xn,1], there holds (6) |En+1(x)| ≤25 + ɛ(n), limn→∞ ɛ(n) ≤ 0, ɛ(n) < 30. Furthermore, En+1(1) ≥ 2 3 √ , π n ≥1. (7) According to Theorem 2.1, rough bounds are |En+1(x)| ≤2C ∗ � 2n+1� 4 =-=(8)-=- 1−x2 +57, π and −1 ≤ x ≤1, |En+1(x)| ≤ C √ (9) � �1 �1−x 2 1 n 2 + , n −1 ≤ x ≤1, where C is a positive constant.s314 SVEN EHRICH AND GIUSEPPE MASTROIANNI The best bound for Stieltjes polynomials in ... |

1 |
Förster: A Comparison Theorem for Linear Functionals and its application in Quadrature
- –J
- 1982
(Show Context)
Citation Context ... α n 2 ,n sin θ, n even, α n−1 2 ,n sin2θ, n odd, is important in connection with a class of polynomials Gn considered by Geronimus (cf. [18]; cf. also [28, 34, 36]). The connection is (cf. [28, 36]) =-=(11)-=- sin θGn(cos θ) = en(θ). As a byproduct of the previous theorem, we also obtain bounds for the Geronimus polynomial Gn. Theorem 2.2. For n ≥ 1, |Gn(x)| ≤2C ∗ � 2n+1 1 n 4√ 1−x2 + 2 √ 1−x2 , x1 where C... |

1 |
Schranken für die Varianz und die Gewichte von Quadraturformeln
- Förster
- 1987
(Show Context)
Citation Context ..., −1=t0 <t1 <···<tr−1 <tr=1, |x|≤1. k=0 We state some convergence theorems of Ln+1f to f in the Lp norm with weight u. Theorem 3.2. Let u ∈ Lp with 1 ≤ p<∞. Then for any continuous function f we have =-=(12)-=- �[f − Ln+1f]u�p ≤ C En(f)∞, where C is independent of n and f. Furthermore, if u √ ϕ ∈ Lp and (u √ ϕ) −1 ∈ Lq , ϕ(x)= √ 1−x2 ,p−1 +q−1 =1,1<p<∞, then, for any function f :[−1,1] → R which is bounded ... |

1 |
Förster: Inequalities for Legendre polynomials and application to quadrature
- –J
- 1981
(Show Context)
Citation Context ...ndent of n and f. Furthermore, if u √ ϕ ∈ Lp and (u √ ϕ) −1 ∈ Lq , ϕ(x)= √ 1−x2 ,p−1 +q−1 =1,1<p<∞, then, for any function f :[−1,1] → R which is bounded and measurable, we have �[f − Ln+1f]u�p ≤ C ˜ =-=(13)-=- En(f)u,p, where C is independent of n and f.s316 SVEN EHRICH AND GIUSEPPE MASTROIANNI The assumptions about u which were made in Theorem 3.2 to obtain (13) are stronger than those to obtain (12). But... |

1 |
Vértesi: Error estimates of product quadrature rules
- Mastroianni, P
- 1993
(Show Context)
Citation Context ...s connected with the error of the product quadrature rule. Estimates of �[f − Ln+1]u�1 in the L1 norm and the same weight u are only possible under strong conditions on the weight u (see for instance =-=[7, 24]-=-). From the previous theorems, we can derive better estimates than (12) when p = 1 by some assumption on the weight u. For instance since �[f − Ln+1f]u�1 ≤ � �u�1 �[f − Ln+1f] √ u�2, if (uϕ) ±1 ∈ L1 ,... |

1 |
Una buona matrice di nodi, Calcolo 30
- Occorsio
- 1993
(Show Context)
Citation Context |

1 |
Convergence of extended Lagrange interpolation in weighted Lp
- Occorsio
- 1994
(Show Context)
Citation Context .... n Proof of Lemma 4.5. We recall that the zeros of En+1 are used as additional nodes for the Gauss-Kronrod formulas. For their weights A GK ν,n and B GK µ,n+1 in (1), we obtain from [10, (93), (94)] =-=(33)-=- (34) A GK ν,n B GK µ,n+1 = a G ν,n + P ′ n(xν)En+1(xν) = 2 Pn(ξµ)E ′ n+1 2 , ν =1,...,n, (ξµ) , µ=1,...,n+1, where aG ν,n , ν =1,...,n, are the Gaussian quadrature weights. Now, we use the positivity... |