Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials
Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 542653, 15 pages
http://dx.doi.org/10.1155/2013/542653
Research Article
Higher-Order Hermite-Fejér Interpolation for
Stieltjes Polynomials
Hee Sun Jung1 and Ryozi Sakai2
1
2
Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea
Department of Mathematics, Meijo University, Nagoya 468-8502, Japan
Correspondence should be addressed to Hee Sun Jung;
Received 5 July 2013; Accepted 22 August 2013
Academic Editor: Qiankun Song
Copyright © 2013 H. S. Jung and R. Sakai. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Let 𝑤𝜆 (𝑥) := (1−𝑥2 )𝜆−1/2 and 𝑃𝜆,𝑛 be the ultraspherical polynomials with respect to 𝑤𝜆 (𝑥). Then, we denote the Stieltjes polynomials
1
𝐸𝜆,𝑛+1 with respect to 𝑤𝜆 (𝑥) satisfying ∫−1 𝑤𝜆 (𝑥)𝑃𝜆,𝑛 (𝑥)𝐸𝜆,𝑛+1 (𝑥)𝑥𝑚 𝑑𝑥 (= 0, 0 ≤ 𝑚 < 𝑛 + 1; ≠ 0, 𝑚 = 𝑛 + 1). In this paper, we
consider the higher-order Hermite-Fejér interpolation operator 𝐻𝑛+1,𝑚 based on the zeros of 𝐸𝜆,𝑛+1 and the higher order extended
Hermite-Fejér interpolation operator H2𝑛+1,𝑚 based on the zeros of 𝐸𝜆,𝑛+1 𝑃𝜆,𝑛 . When m is even, we show that Lebesgue constants
of these interpolation operators are 𝑂(𝑛max{(1−𝜆)𝑚−2,0} )(0 < 𝜆 < 1) and 𝑂(𝑛max{(1−2𝜆)𝑚−2,0} )(0 < 𝜆 < 1/2), respectively; that is,
‖H2𝑛+1,𝑚 ‖ = 𝑂(𝑛max{(1−2𝜆)𝑚−2,0} )(0 < 𝜆 < 1) and ‖𝐻𝑛+1,𝑚 ‖ = 𝑂(𝑛max{(1−𝜆)𝑚−2,0} )(0 < 𝜆 < 1/2). In the case of the Hermite-Fejér
interpolation polynomials H2𝑛+1,𝑚 [⋅] for 1/2 ≤ 𝜆 < 1, we can prove the weighted uniform convergence. In addition, when m is
odd, we will show that these interpolations diverge for a certain continuous function on [−1, 1], proving that Lebesgue constants
of these interpolation operators are similar or greater than log n.
We note that, by definition, 𝐻𝑛,1 is the Lagrange, 𝐻𝑛,2 is
the Hermite-Fejér, and 𝐻𝑛,4 is the Krylov-Stayermann interpolatory polynomial. By (2), we may write
1. Introduction
Let 𝑋 := {𝑥𝑘,𝑛 } ⊂ [−1, 1] and
− 1 < 𝑥1,𝑛 < 𝑥2,𝑛 < ⋅ ⋅ ⋅ < 𝑥𝑛−1,𝑛 < 𝑥𝑛,𝑛 < 1,
𝑛 = 1, 2, . . . .
(1)
𝑘=1
For any real-valued function 𝑓 on [−1, 1] and an integer
𝑚 ≥ 1, we recall that there exist unique Hermite and HermiteFejér interpolatory polynomials of higher order denoted by
𝐻𝑛,𝑚 (𝑓, 𝑋), and of degree ≤ 𝑛𝑚 − 1, defined as follows:
𝐻𝑛,𝑚 (𝑓, 𝑋, 𝑥𝑘,𝑛 ) = 𝑓 (𝑥𝑘,𝑛 ) ,
(𝑡)
𝐻𝑛,𝑚
(𝑓, 𝑋, 𝑥𝑘,𝑛 ) = 0,
1 ≤ 𝑘 ≤ 𝑛;
1 ≤ 𝑡 ≤ 𝑚 − 1,
1 ≤ 𝑘 ≤ 𝑛.
𝑛
𝐻𝑛,𝑚 (𝑓, 𝑋, 𝑥) = ∑ 𝑓 (𝑥𝑘,𝑛 ) ℎ𝑘,𝑛,𝑚 (𝑋, 𝑥) ,
(3)
𝑛 = 1, 2, . . . .
The polynomials
𝑚−1
(2)
𝑖
𝑚
ℎ𝑘,𝑛,𝑚 (𝑋, 𝑥) = 𝑙𝑘,𝑛
(𝑋, 𝑥) ∑ 𝑒𝑖,𝑘,𝑛,𝑚 (𝑥 − 𝑥𝑘,𝑛 ) ,
𝑖=0
(4)
1≤𝑘≤𝑛
2
Journal of Applied Mathematics
are unique, of degree exactly 𝑛𝑚 − 1 and satisfy the relations
(𝑡)
ℎ𝑘,𝑛,𝑚
(𝑋, 𝑥𝑙,𝑛 ) = 𝛿0,𝑡 𝛿𝑙,𝑘 ,
1 ≤ 𝑘, 𝑙 ≤ 𝑛,
0 ≤ 𝑡 ≤ 𝑚 − 1,
(5)
= 𝐻𝑛,𝑚 (𝑃, 𝑋, 𝑥) + ∑ ∑ 𝑃(𝑡) (𝑥𝑘,𝑛 ) ℎ𝑡,𝑘,𝑛,𝑚 (𝑋, 𝑥) .
𝑡=1 𝑘=1
1, 𝑢 = V;
:= {
0, 𝑢 ≠ V.
𝑤𝑛 (𝑥)
,
𝑤𝑛 (𝑥𝑘,𝑛 ) (𝑥 − 𝑥𝑘,𝑛 )
𝑛
(7)
𝑤𝑛 (𝑥) := ∏ (𝑥 − 𝑥𝑘,𝑛 ) ,
𝑘=1
and the coefficients 𝑒𝑖,𝑘 may be obtained from the relations
ℎ𝑘,𝑛,𝑚 (𝑋, 𝑥𝑙,𝑛 ) = 𝛿𝑙,𝑘 ,
(𝑡)
(𝑋, 𝑥𝑙,𝑛 ) = 0,
ℎ𝑘,𝑛,𝑚
1 ≤ 𝑘, 𝑙 ≤ 𝑛;
1 ≤ 𝑡 ≤ 𝑚 − 1,
(8)
If 𝑓 ∈ 𝐶(𝑚−1) [−1, 1], then the Hermite interpolation polynô𝑛,𝑚 (𝑓, 𝑋, 𝑥) of degree ≤ 𝑛𝑚 − 1 with respect to 𝑋 is
mial 𝐻
defined by
̂(𝑡) (𝑓, 𝑋, 𝑥𝑘,𝑛 ) := 𝑓(𝑡) (𝑥𝑘,𝑛 ) ,
𝐻
𝑛,𝑚
1 ≤ 𝑘 ≤ 𝑛, 0 ≤ 𝑡 ≤ 𝑚 − 1.
(9)
̂𝑛,𝑚 (𝑓, 𝑋, 𝑥) as
We may express 𝐻
𝑚−1 𝑛
̂𝑛,𝑚 (𝑓, 𝑋, 𝑥) = ∑ ∑ 𝑓(𝑡) (𝑥𝑘,𝑛 ) ℎ𝑡,𝑘,𝑛,𝑚 (𝑋, 𝑥) ,
𝐻
𝑡=0 𝑘=1
(10)
𝑚 = 1, 2, . . . ,
where for 0 ≤ 𝑡 ≤ 𝑚 − 1
ℎ𝑡,𝑘,𝑛,𝑚 (𝑋, 𝑥)
𝑡
(𝑥 − 𝑥𝑘,𝑛 ) 𝑚−1−𝑡
𝑖
∑ 𝑒𝑡,𝑖,𝑘,𝑛,𝑚 (𝑥 − 𝑥𝑘,𝑛 )
𝑡!
𝑖=0
(11)
is the unique polynomial of degree 𝑛𝑚 − 1 satisfying
(𝑖)
ℎ𝑡,𝑘,𝑛,𝑚
(𝑋, 𝑥𝑗,𝑛 ) = 𝛿𝑡,𝑖 𝛿𝑘,𝑗 ,
0 ≤ 𝑖, 𝑡 ≤ 𝑚 − 1,
1 ≤ 𝑗, 𝑘 ≤ 𝑛.
In what follows, we abbreviate several notations as ℎ𝑘 (𝑥) :=
ℎ𝑘,𝑛,𝑚 (𝑥), 𝑒𝑖,𝑘 := 𝑒𝑖,𝑘,𝑛,𝑚 , and 𝑒𝑡,𝑖,𝑘 := 𝑒𝑡,𝑖,𝑘,𝑛,𝑚 if there is
no confusion. Here, we are interested in Hermite-Fejér and
Hermite interpolations with respect to 𝑋 whose elements are
the zeros of a sequence of Stieltjes polynomials and the product polynomials of Stieltjes polynomials and the ultraspherical polynomials, respectively. To be precise, we first consider
the generalized Stieltjes polynomials 𝐸𝜆,𝑛+1 (𝑥) defined (up to
a multiplicative constant) by
1
∫ 𝑤𝜆 (𝑥) 𝑃𝜆,𝑛 (𝑥) 𝐸𝜆,𝑛+1 (𝑥) 𝑥𝑘 𝑑𝑥 = 0,
−1
1 ≤ 𝑘, 𝑙 ≤ 𝑛.
𝑚
= 𝑙𝑘,𝑛
(𝑋, 𝑥)
(13)
(6)
Here, 𝑙𝑘,𝑛 (𝑋, 𝑥) are the well-known fundamental Lagrange
polynomials of degree 𝑛 − 1 given by
𝑙𝑘,𝑛 (𝑋, 𝑥) :=
̂𝑛,𝑚 (𝑃, 𝑋, 𝑥)
𝑃 (𝑥) = 𝐻
𝑚−1 𝑛
where for nonnegative integers 𝑢 and V
𝛿𝑢,V
𝑒0,𝑖,𝑘,𝑛,𝑚 for 1 ≤ 𝑘 ≤ 𝑛 and 0 ≤ 𝑖, 𝑡 ≤ 𝑚 − 1 (see [1]). Now, we
have for any polynomial 𝑃 of degree ≤ 𝑛𝑚 − 1,
(12)
Then, we easily see from the relations (5) and (12) that
ℎ0,𝑘,𝑛,𝑚 (𝑥) = ℎ𝑘,𝑛,𝑚 (𝑥), 𝑒0,𝑖,𝑘,𝑛,𝑚 = 𝑒𝑖,𝑘,𝑛,𝑚 , and 𝑒𝑡,𝑖,𝑘,𝑛,𝑚 =
(14)
𝑘 = 0, 1, 2, . . . , 𝑛, 𝑛 ≥ 1,
where 𝑤𝜆 (𝑥) = (1 − 𝑥2 )𝜆−1/2 , 𝜆 > −1/2, and 𝑃𝜆,𝑛 (𝑥) is the 𝑛th
ultraspherical polynomial for the weight function 𝑤𝜆 (𝑥). In
1935, Szegö [2] showed that the zeros of the generalized
Stieltjes polynomials 𝐸𝜆,𝑛+1 (𝑥) are real and inside [−1, 1] and
interlace with the zeros of 𝑃𝜆,𝑛 (𝑥) whenever 0 ≤ 𝜆 ≤ 2.
For the properties of interpolation operators based at
the zeros of 𝐸𝜆,𝑛+1 and the zeros of 𝑃𝜆,𝑛 𝐸𝜆,𝑛+1 , Ehrich and
Mastroianni [3, 4] proved that Lagrange interpolation
operators 𝐿 𝑛+1 based on the zeros of 𝐸𝜆,𝑛+1 and extended
Lagrange interpolation operators L2𝑛+1 based on the zeros
of 𝐸𝜆,𝑛+1 𝑃𝜆,𝑛 have Lebesgue constants ‖𝐿 𝑛+1 ‖∞ (0 < 𝜆 < 1)
and ‖L2𝑛+1 ‖∞ (0 < 𝜆 ≤ 1/2) of optimal order, that is,
𝑂(log 𝑛). For the Hermite-Fejér interpolation operator
𝐻𝑛+1 based on the zeros of 𝐸𝜆,𝑛+1 and the extended
Hermite-Fejér interpolation operator H2𝑛+1 based on the
zeros of 𝐸𝜆,𝑛+1 𝑃𝜆,𝑛 , it is proved that Lebesgue constants
‖𝐻𝑛+1 ‖∞ (0 < 𝜆 < 1) and ‖H2𝑛+1 ‖∞ (0 < 𝜆 ≤ 1/2) are of
optimal order, that is, 𝑂(1), in [5]. In this paper, we consider
the higher-order Hermite-Fejér interpolation operator
𝐻𝑛+1,𝑚 based on the zeros of 𝐸𝜆,𝑛+1 and the higher-order
extended Hermite-Fejér interpolation operator H2𝑛+1,𝑚
based on the zeros of 𝐸𝜆,𝑛+1 𝑃𝜆,𝑛 . When 𝑚 is even, we show
that Lebesgue constants of these interpolation operators are
𝑂(𝑛max{(1−𝜆)𝑚−2,0} ) and 𝑂(𝑛max{(1−2𝜆)𝑚−2,0} ), respectively; that
is, ‖𝐻𝑛+1,𝑚 ‖ = 𝑂(𝑛max{(1−𝜆)𝑚−2,0} ) (0 < 𝜆 < 1) and ‖H2𝑛+1,𝑚 ‖
= 𝑂(𝑛max{(1−2𝜆)𝑚−2,0} ) (0 < 𝜆 < 1/2). In the case of the
Hermite-Fejér interpolation polynomials H2𝑛+1,𝑚 [⋅] for
1/2 ≤ 𝜆 < 1, we can prove the weighted uniformconvergence.
In addition, when 𝑚 is odd, we will show that these
interpolations diverge for a certain continuous function
on [−1, 1], proving that Lebesgue constants of these
interpolation operators are similar or greater than log 𝑛.
Journal of Applied Mathematics
3
This paper is organized as follows. In Section 2, we will
introduce the main results. In Section 3, we will show
the auxiliary propositions and estimate the coefficients of
Hermite-Fejér interpolation polynomials in order to prove
the ma (...truncated)
