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

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 Heesun Jung1* and Ryozi Sakai2 * Correspondence: hsun90@skku. edu 1 Department of Mathematics Education, Sungkyunkwan University Seoul 110-745, Republic of Korea Full list of author information is available at the end of the article Abstract Let ℝ+ = [0, ∞) and R : ℝ+ ® ℝ+ be a continuous function which is the Laguerretype exponent, and pn, r(x), ρ > − 12 be the orthonormal polynomials with the weight wr(x) = xr e-R(x). For the zeros {xk,n,ρ }nk=1 of pn,ρ (x) = pn (w2ρ ; x) , we consider the higher order Hermite-Fejér interpolation polynomial Ln(l, m, f; x) based at the zeros {xk,n,ρ }nk=1 , where 0 ≤ l ≤ m - 1 are positive integers. 2010 Mathematics Subject Classification: 41A10. Keywords: Laguerre-type weights, orthonormal polynomials, higher order HermiteFejér interpolation polynomials 1. Introduction and main results Let ℝ = [-∞, ∞) and ℝ+ = [0, ∞). Let R : ℝ+ ® ℝ+ be a continuous, non-negative, and increasing function. Consider the exponential weights wr(x) = xr exp(-R(x)), r > -1/2, and then we construct the orthonormal polynomials {pn,ρ (x)}∞ n=0 with the weight wr (x). Then, for the zeros {xk,n,ρ }nk=1 of pn,ρ (x) = pn (w2ρ ; x) , we obtained various estimations with respect to p(j) n,ρ (xk,n,ρ ) , k = 1, 2, ..., n, j = 1, 2, ..., ν, in [1]. Hence, in this article, we will investigate the higher order Hermite-Fejér interpolation polynomial Ln (l, m, f; x) based at the zeros {xk,n,ρ }nk=1 , using the results from [1], and we will give a divergent theorem. This article is organized as follows. In Section 1, we introduce some notations, the weight classes L2 , L̃ν with L(C2 ) , L(C2 +) , and main results. In Section 2, we will introduce the classes F (C2 ) and F (C2 +) , and then, we will obtain some relations of the factors derived from the classes F (C2 ) , F (C2 +) and the classes L(C2 +) , L(C2 +) . 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 C1, C2 such that for the relevant range of x, C1 ≤ f(x)/g(x) ≤ C2. The similar notation is used for sequences, and sequences of functions. Throughout this article, C, C1, C2, ... denote positive constants independent of n, x, t or polynomials Pn(x). The same symbol does 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 http://www.journalofinequalitiesandapplications.com/content/2011/1/122 First, we introduce classes of weights. Levin and Lubinsky [5,6] introduced the class of weights on ℝ+ 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 Q(t) = R(x) and x = t2. (a) √ xR(x) is continuous in I, with limit 0 at 0 and R(0) = 0; √ (b) R″(x) exists in (0, d), while Q″ is positive in (0, d) ; (c) lim R(x) = ∞; x→d− (d) The function T(x) := xR (x) R(x) is quasi-increasing in (0, d), with T(x) ≥  > 1 , 2 x ∈ (0, d); (e) There exists C1 > 0 such that | R (x) | R (x) ≤ C1 , R(x) R(x) a.e. x ∈ (0, d). Then, we write w ∈ L(C2 ) . If there also exist a compact subinterval J* ∋ 0 of √ √ I∗ = (− d, d) and C2 > 0 such that Q (t) | Q (t) | ≥ C2 ,  | Q (t) | Q(t) a.e. t ∈ I∗ \J∗ , then we write w ∈ L(C2 +) . We consider the case d = ∞, that is, the space ℝ+ = [0, ∞), and we strengthen Definition 1.1 slightly. Definition 1.2. We assume that R : ℝ+ ® ℝ+ has the following properties: (a) R(x), R’(x) are continuous, positive in ℝ+, with R(0) = 0, R’(0) = 0; (b) R″(x) > 0 exists in ℝ+\{0}; (c) lim R(x) = ∞; x→∞ (d) The function T(x) := xR (x) R(x) Page 2 of 24 Jung and Sakai Journal of Inequalities and Applications 2011, 2011:122 http://www.journalofinequalitiesandapplications.com/content/2011/1/122 Page 3 of 24 is quasi-increasing in ℝ+\{0}, with T(x) ≥  > 1 , 2 x ∈ Ê+ \{0}; (e) There exists C1 > 0 such that R (x) R (x) ≤ C , 1 R (x) R(x) a.e. x ∈ Ê+ \{0}. There exist a compact subinterval J ∋ 0 of ℝ+ and C2 > 0 such that R (x) R (x) ≥ C , 2 R (x) R(x) a.e. t ∈ Ê+ \J, then we write w ∈ L2 . To obtain estimations of the coefficients of higher order Hermite-Fejér interpolation polynomial based at the zeros {xk,n,ρ }nk=1 , we need to focus on a smaller class of weights. Definition 1.3. Let w = exp(−R) ∈ L2 and let ν ≥ 2 be an integer. For the exponent R, we assume the following: (a) R(j) (x) > 0, for 0 ≤ j ≤ ν and x > 0, and R(j) (0) = 0, 0 ≤ j ≤ ν - 1. (b) There exist positive constants Ci > 0, i = 1, 2, ..., ν - 1 such that for i = 1, 2, ..., ν-1 R(i+1) (x) ≤ Ci R(i) (x) R (x) , R(x) a.e. x ∈ Ê+ \{0}. (c) There exist positive constants C, c1 > 0 and 0 ≤ δ < 1 such that on x Î (0, c1) (ν) R  δ 1 (x) ≤ C . x (1:1) (d) There exists c2 > 0 such that we have one among the following √ (d1) T(x)/ x is quasi-increasing on (c2, ∞), (d2) R(ν)(x) is nondecreasing on (c2, ∞). Then we write w(x) = e−R(x) ∈ L̃ν . 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 integer. Then we define Rl,α (x) = expl (xα ) − expl (0), where expl (x) = exp(exp(exp ... exp(x)) ...) is the l-th iterated exponential. Jung and Sakai Journal of Inequalities and Applications 2011, 2011:122 http://www.journalofinequalitiesandapplications.com/content/2011/1/122 Page 4 of 24 (1) If a >ν, w(x) = e−Rl,α (x) ∈ L̃ν . (2) If a ≤ ν and a is an integer, we define R∗l,α (x) = expl (xα ) − expl (0) − (j) r  Rl,α (0) j=1 j! xj . ∗ Then w(x) = e−Rl,α (x) ∈ L̃ν . In the remainder of this article, we consider the classes L2 and L̃ν ; Let w ∈ L2 or 1 r w ∈ L̃ν ν ≥ 2 . For ρ > − , we set wr(x): = x w(x). Then we can construct the ortho2 normal polynomials pn,ρ (x) = pn (w2ρ ; x) of degree n with respect to w2ρ (x) . That is,  ∞ 0 pn,ρ (u)pm,ρ (u)w2ρ (u)du = δnm (Kronecker’s delta) n, m = 0, 1, 2, . . . . Let us denote the zeros of pn,r(x) by 0 < xn,n,ρ < · · · < x2,n,ρ < x1,n,ρ < ∞. The Mhaskar-Rahmanov-Saff numbers av is defined as follows: v= 1 π  1 0 av tR (av t)  dt, t(1 − t) v > 0. Let l, m be non-negative integers with 0 ≤ l <m ≤ ν. For f Î C(l) (ℝ), we define the (l, m)-order Hermite-Fejér interpolation po (...truncated)