Higher order Hermite-Fejér interpolation polynomials with Laguerre-type weights

Jung Heesun, Sakai, Sakai Ryozi

Description

Let ℝ + = [0, ∞) and R : ℝ + → ℝ + be a continuous function which is the Laguerre-type exponent, and p n , ρ ( x ), ρ > - 1 2 be the orthonormal polynomials with the weight w ρ ( x ) = x ρ e - R ( x ) . For the zeros { x k , n , ρ } k = 1 n of p n , ρ ( x ) = p n ( w ρ 2 ; x ) , we consider the higher order Hermite-Fejér interpolation polynomial L n ( l , m , f ; x ) based at the zeros { x k , n , ρ } k = 1 n , where 0 ≤ l ≤ m - 1 are positive integers. 2010 Mathematics Subject Classification : 41A10.

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 5
• Langue: English

Extrait

Jung and Sakai Journal of Inequalities and Applications 2011, 2011:122
http://www.journalofinequalitiesandapplications.com/content/2011/1/122
RESEARCH Open Access
Higher order Hermite-Fejér interpolation
polynomials with Laguerre-type weights
1* 2Heesun Jung and Ryozi Sakai
* Correspondence: hsun90@skku. Abstract
edu
1 + + +Department of Mathematics Letℝ = [0, ∞) and R :ℝ ®ℝ be a continuous function which is the Laguerre-
Education, Sungkyunkwan 1type exponent, and p (x), be the orthonormal polynomials with theρ> −n, rUniversity Seoul 110-745, Republic 2
of Korea r -R(x) n 2weight w (x)= x e . For the zeros {x } of p (x)= p (w ;x), we considerk,n,ρ n,ρ nr ρk=1Full list of author information is
available at the end of the article the higher order Hermite-Fejér interpolation polynomial L (l, m, f; x) based at then
nzeros {x } , where 0 ≤ l ≤ m - 1 are positive integers.k,n,ρ k=1
2010 Mathematics Subject Classification: 41A10.
Keywords: Laguerre-type weights, orthonormal polynomials, higher order Hermite-
Fejér interpolation polynomials
1. Introduction and main results
+ + +
Letℝ=[-∞, ∞)andℝ =[0, ∞). Let R :ℝ ®ℝ be a continuous, non-negative, and
r
increasing function. Consider the exponential weights w (x)= x exp(-R(x)), r > -1/2,r
∞{p (x)}and then we construct the orthonormal polynomials with the weight wn,ρ rn=0
n 2(x). Then, for the zeros {x } of p (x)= p (w ;x), we obtained various estima-k,n,ρ n,ρ nk=1 ρ
(j)tions with respect to , k = 1, 2, ..., n, j = 1, 2, ..., ν, in [1]. Hence, in this arti-p (x )n,ρ k,n,ρ
cle, we will investigate the higher order Hermite-Fejér interpolation polynomial L (l,n
n{x }m, f; x) based at the zeros k,n,ρ , using the results from [1], and we will give ak=1
divergent theorem. This article is organized as follows. In Section 1, we introduce
2 2some notations, the weight classes , ˜ with , , and main results. InL L(C ) L(C +)2 Lν
22Section 2, we will introduce the classes F(C ) and , and then, we will obtainF(C +)
2 2some relations of the factors derived from the classes , and the classesF(C ) F(C +)
2 2L(C +), L(C +). Finally, we will prove the main theorems using known results in
[1-5], in Section 3.
+We say that f :ℝ®ℝ is quasi-increasing if there exists C > 0 such that f(x) ≤ Cf(y)
for 0 <x <y. The notation f(x)~ g(x) means that there are positive constants C , C1 2
such that for the relevant range of x, C ≤ f(x)/g(x) ≤ C . The similar notation is used1 2
for sequences, and sequences of functions. Throughout this article, C, C , C , ... denote1 2
positive constants independent of n, x, t or polynomials P (x). The same symbol doesn
not necessarily denote the same constant in different occurrences. We denote the class
of polynomials with degree n by .Pn
© 2011 Jung and Sakai; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.Jung and Sakai Journal of Inequalities and Applications 2011, 2011:122 Page 2 of 24
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First, we introduce classes of weights. Levin and Lubinsky [5,6] introduced the class
+of weights onℝ as follows. Let I = [0, d), where 0 <d≤∞.
Definition 1.1. [5,6] We assume that R : I® [0, ∞) has the following properties: Let
2Q(t)= R(x) and x = t .
√
(a) is continuous in I, with limit 0 at 0 and R(0) = 0;xR(x)
√
(b) R″(x) exists in (0, d), while Q″ is positive in ;(0, d)
(c)
lim R(x)= ∞;
x→d−
(d) The function
xR (x)
T(x):=
R(x)
is quasi-increasing in (0, d), with
1
T(x) ≥ > , x ∈ (0,d);
2
(e) There exists C > 0 such that1
| R (x) | R (x)
≤ C ,a.e. x ∈ (0,d).1
R(x) R(x)
2Then, we write w ∈L(C ). If there also exist a compact subinterval J* ∋0of
√ √
∗ and C > 0 such thatI =(− d, d) 2
Q (t) | Q (t) | ∗ ∗≥ C ,a.e. t ∈ I \J ,2| Q (t) | Q(t)
2then we write w ∈L(C +).
+
We consider the case d = ∞, that is, the spaceℝ = [0, ∞), and we strengthen Defini-
tion 1.1 slightly.
+ +
Definition 1.2. We assume that R :ℝ ®ℝ has the following properties:
+(a) R(x), R’(x) are continuous, positive inℝ , with R(0) = 0, R’(0) = 0;
+(b) R″(x) > 0 exists inℝ \{0};
(c)
lim R(x)= ∞;
x→∞
(d) The function
xR (x)
T(x):=
R(x)Jung and Sakai Journal of Inequalities and Applications 2011, 2011:122 Page 3 of 24
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+is quasi-increasing inℝ \{0}, with
1 +T(x) ≥ > , x ∈ \{0};
2
(e) There exists C > 0 such that1
R (x) R (x) +≤ C ,a.e. x ∈ \{0}.1R (x) R(x)
+There exist a compact subinterval J ∋0ofℝ and C > 0 such that2
R (x) R (x) +≥ C ,a.e. t ∈ \J,2R (x) R(x)
then we write w ∈L .2
To obtain estimations of the coefficients of higher order Hermite-Fejér interpolation
npolynomial based at the zeros {x } , we need to focus on a smaller class ofk,n,ρ k=1
weights.
Definition 1.3. Let w = exp(−R) ∈L and let ν ≥ 2 be an integer. For the exponent2
R, we assume the following:
(j) (j)
(a) R (x) > 0, for 0 ≤ j ≤ ν and x > 0, and R (0) = 0, 0 ≤ j ≤ ν-1.
(b) There exist positive constants C >0, i=1,2,..., ν - 1 such that for i=1,2,...,i
ν-1
R (x)(i+1) (i) +
R (x) ≤ C R (x) ,a.e. x ∈ \{0}.i
R(x)
(c) There exist positive constants C, c > 0 and 0 ≤ δ < 1 such that on xÎ (0, c )1 1
δ
1(ν) (1:1)R (x) ≤ C .
x
(d) There exists c > 0 such that we have one among the following2
√
(d1) T(x)/ x is quasi-increasing on (c , ∞),2
(ν)(d2) R (x) is nondecreasing on (c , ∞).2
−R(x)Then we write ˜ .w(x)= e ∈Lν
Example 1.4. [6,7] Let ν ≥ 2 be a fixed integer. There are some typical examples
satisfying all conditions of Definition 1.3 as follows: Let a>1, l ≥ 1, where l is an inte-
ger. Then we define
αR (x) = exp (x ) −exp (0),l,α l l
where exp (x) = exp(exp(exp ... exp(x)) ...) is the l-th iterated exponential.lJung and Sakai Journal of Inequalities and Applications 2011, 2011:122 Page 4 of 24
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−R (x)l,α ˜(1) If a >ν, .w(x)= e ∈Lν
(2) If a ≤ ν and a is an integer, we define
(j)r R (0)
l,α∗ α jR (x) = exp (x ) − exp (0) − x .l,α l l j!
j=1
∗−R (x) ˜Then l,α .w(x)= e ∈Lν
˜In the remainder of this article, we consider the classes L and ;Let w ∈L or2 L 2ν
1 r˜ . For ρ> − ,weset w (x): = x w(x). Then we can construct the ortho-rw ∈L ν ≥ 2ν 2
2 2normal polynomials p (x)= p (w ;x) of degree n with respect to w (x). That is,n,ρ n ρρ
∞
2p (u)p (u)w (u)du = δ (Kronecker’s delta) n,m=0,1,2,....n,ρ m,ρ nmρ
0
Let us denote the zeros of p (x)byn,r
0 < x < ··· < x < x < ∞.n,n,ρ 2,n,ρ 1,n,ρ
The Mhaskar-Rahmanov-Saff numbers a is defined as follows:v
1 1 a tR (a t)v v
v = dt, v > 0.
π 0 t(1 −t)
(l)
Let l, m be non-negative integers with 0 ≤ l <m ≤ ν. For fÎ C (ℝ), we define the (l,
m)-order Hermite-Fejér interpolation polynomials L (l,m,f;x) ∈P as follows: Forn mn−1
each k = 1, 2, ..., n,
(j) (j)L (l,m,f;x )= f (x ), j=0,1,2,...,l,n k,n,ρ k,n,ρ
(j)
L (l,m,f;x )=0, j = l+1,l+2,...,m −1.n k,n,ρ
For each P ∈P ,wesee L (m-1, m, P; x)= P(x). The fundamental polynomialsmn−1 n
h (m;x) ∈Ps,k,n,ρ mn−1, k = 1, 2, ..., n,of L (l, m, f; x) are defined byn
m−1
m ih (l,m;x)= l (x) e (l,m,k,n)(x −x ) . (1:2)s,k,n,ρ s,i k,n,ρk,n,ρ
i=s
Here, l (x) is a fundamental Lagrange interpolation polynomial of degree n-1k, n, r
[[8], p. 23] given by
2p (w ;x)n ρ
l (x)=k,n,ρ 2(x −x )p (w ;x )k,n,ρ k,n,ρn ρ
and h , (l, m; x) satisfiess k, n, r
(j) (1:3)h (l,m;x )= δ δ j,s=0,1,...,m −1, p=1,2,...,n.p,n,ρ s,j k,ps,k,n,ρJung and Sakai Journal of Inequalities and Applications 2011, 2011:122 Page 5 of 24
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Then
n l
(s)L (l,m,f;x)= f (x )h (l,m;x).n k,n,ρ s,k,n,ρ
k=1 s=0
In particular, for fÎ C(ℝ), we define the m-order Hermite-Fejér interpolation poly-
L (m,f;x) ∈Pnomials as the (0, m)-order Hermite-Fej&#