Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 816363, 29 pages doi:10.1155/2010/816363 Research Article Derivatives of Orthonormal Polynomials and Coefficients of Hermite-Fej ´ er Interpolation Polynomials with Exponential-Type Weights H. S. Jung 1 and R. Sakai 2 1 Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, South Korea 2 Department of Mathematics, Meijo University, Nagoya 468-8502, Japan Correspondence should be addressed to H. S. Jung, hsun90@skku.edu Received 10 November 2009; Accepted 14 January 2010 Academic Editor: Vijay Gupta Copyright q 2010 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 R −∞, ∞,andletQ ∈ C 2 : R → 0, ∞ be an even function. In this paper, we consider the exponential-type weights w ρ x|x| ρ exp−Qx,ρ>−1/2,x∈ R, and the orthonormal polynomials p n w 2 ρ ; x of degree n with respect to w ρ x. So, we obtain a certain differential equation of higher order with respect to p n w 2 ρ ; x and we estimate the higher-order derivatives of p n w 2 ρ ; x and the coefficients of the higher-order Hermite-Fej ´ er interpolation polynomial based at the zeros of p n w 2 ρ ; x. 1. Introduction Let R −∞, ∞ and R  0, ∞.LetQ ∈ C 2 : R → R  be an even function and let wxexp−Qx be such that  ∞ 0 x n w 2 xdx < ∞ for all n  0, 1, 2, For ρ>−1/2, we set w ρ  x  : | x | ρ w  x  ,x∈ R. 1.1 Then we can construct the orthonormal polynomials p n,ρ xp n w 2 ρ ; x of degree n with respect to w 2 ρ x.Thatis,  ∞ −∞ p n,ρ  x  p m,ρ  x  w 2 ρ  x  dx  δ mn  Kronecker  sdelta  , p n,ρ  x   γ n x n  ··· ,γ n  γ n,ρ > 0. 1.2 2 Journal of Inequalities and Applications We denote the zeros of p n,ρ x by −∞ <x n,n,ρ <x n−1,n,ρ < ···<x 2,n,ρ <x 1,n,ρ < ∞. 1.3 A function f : R  → R  is said to be quasi-increasing if there exists C>0 such that fx ≤ Cfy for 0 <x<y. For any two sequences {b n } ∞ n1 and {c n } ∞ n1 of nonzero real numbers or functions, we write b n  c n if there exists a constant C>0 independent of n or x such that b n ≤ Cc n for n being large enough. We write b n ∼ c n if b n  c n and c n  b n .We denote the class of polynomials of degree at most n by P n . Throughout C, C 1 ,C 2 , denote positive constants independent of n, x, t,and polynomials of degree at most n. The same symbol does not necessarily denote the same constant in different occurrences. We shall be interested in the following subclass of weights from 1. Definition 1.1. Let Q : R → R  be even and satisfy the following properties. a Q  x is continuous in R,withQ00. b Q  x exists and is positive in R \{0}. c One has lim x →∞ Q  x   ∞. 1.4 d The function T  x  : xQ   x  Q  x  ,x /  0 1.5 is quasi-increasing in 0, ∞ with T  x  ≥ Λ > 1,x∈ R  \ { 0 } . 1.6 e There exists C 1 > 0 such that Q   x  | Q   x  | ≤ C 1 | Q   x  | Q  x  , a.e.x∈ R \ { 0 } . 1.7 Then we write w ∈FC 2 . If there also exist a compact subinterval J 0 of R and C 2 > 0 such that Q   x  | Q   x  | ≥ C 2 | Q   x  | Q  x  , a.e.x∈ R \ J, 1.8 then we write w ∈FC 2 . Journal of Inequalities and Applications 3 In the following we introduce useful notations. a Mhaskar-Rahmanov-SaffMRS numbers a x is defined as the positive roots of the following equations: x  2 π  1 0 a x uQ   a x u   1 − u 2  1/2 du, x > 0. 1.9 b Let η x   xTa x   −2/3 ,x>0. 1.10 c The function ϕ u x is defined as the following: ϕ u  x   ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a 2 2u − x 2 u  a u  x  a u η u  a u − x  a u η u  1/2 , | x | ≤ a u , ϕ u  a u  ,a u < | x | . 1.11 In 2, 3 we estimated the orthonormal polynomials p n,ρ xp n w 2 ρ ; x associated with the weight w 2 ρ  |x| 2ρ exp−2Qx,ρ>−1/2 and obtained some results with respect to the derivatives of orthonormal polynomials p n,ρ x. In this paper, we will obtain the higher derivatives of p n,ρ x. To estimate of the higher derivatives of the orthonormal polynomials sequence, we need further assumptions for Qx as follows. Definition 1.2. Let wxexp−Qx ∈FC 2  and let ν be a positive integer. Assume that Qx is ν-times continuously differentiable on R and satisfies the followings. a Q ν1 x exists and Q i x,0≤ i ≤ ν  1arepositiveforx>0. b There exist positive constants C i > 0 such that for x ∈ R \{0}    Q i1  x     ≤ C i    Q i  x     | Q   x  | Q  x  ,i 1, ,ν. 1.12 c There exist constants 0 ≤ δ<1andc 1 > 0 such that on 0,c 1  Q ν1  x  ≤ C  1 x  δ . 1.13 Then we write wx ∈F ν C 2 . Furthermore, wx ∈F ν C 2  and Qx satisfies one of the following. a Q  x/Qx is quasi-increasing on a certain positive interval c 2 , ∞. b Q ν1 x is nondecreasing on a certain positive interval c 2 , ∞. c There exists a constant 0 ≤ δ<1 such that Q ν1 x ≤ C1/x δ on c 2 , ∞. Then we write wx ∈  F ν C 2 . 4 Journal of Inequalities and Applications Now, consider some typical examples of FC 2 . Define for α>1andl ≥ 1, Q l,α  x  : exp l  | x | α  − exp l  0  . 1.14 More precisely, define for α  m>1, m ≥ 0, l ≥ 1andα ≥ 0, Q l,α,m  x  : | x | m  exp l  | x | α  − α ∗ exp l  0   1.15 where α ∗  0ifα  0, otherwise α ∗  1, and define Q α  x  :  1  |x|  |x| α − 1,α>1. 1.16 In the following, we consider the exponential weights with the exponents Q l,α,m x. Then we have the following examples see 4. Example 1.3. Let ν be a positive integer. Let m  α − ν>0. Then one has the following. a wxexp−Q l,α,m x belongs to F ν C 2 . b If l ≥ 2andα>0, then there exists a constant c 1 > 0 such that Q  l,α,m x/Q l,α,m x is quasi-increasing on c 1 , ∞. c When l  1, if α ≥ 1, then there exists a constant c 2 > 0 such that Q  l,α,m x/Q l,α,m x is quasi-increasing on c 2 , ∞,andif0 <α<1, then Q  l,α,m x/Q l,α,m x is quasidecreasing on c 2 , ∞. d When l  1and0<α<1, Q ν1 l,α,m x is nondecreasing on a certain positive interval c 2 , ∞. In this paper, we will consider the orthonormal polynomials p n,ρ x with respect to the weight class  F ν C 2 . Our main themes in this paper are to obtain a certain differential equation for p n,ρ x of higher-order and to estimate the higher-order derivatives of p n,ρ x at the zeros of p n,ρ x and the coefficients of the higher-order Hermite-Fej ´ er interpolation polynomials based at the zeros of p n,ρ x. More precisely, we will estimate the higher-order derivatives of p n,ρ x at the zeros of p n,ρ x for two cases of an odd order and of an even order. These estimations will play an important role in investigating convergence or divergence of higher-order Hermite-Fej ´ er interpolation polynomials see 5–16. This paper is organized as follows. In Section 2, we will obtain the differential equations for p n,ρ x of higher-order. In Section 3, we will give estimations of higher-order derivatives of p n,ρ x at the zeros of p n,ρ x in a certain finite interval for two cases of an odd order and of an even order. In addition, we estimate the higher-order derivatives of p n,ρ x at all zeros of p n,ρ x for two cases of an odd order and of an even order. Furthermore, we will estimate the coefficients of higher-order Hermite-Fej ´ er interpolation polynomials based at the zeros of p n,ρ x,inSection 4. Journal of Inequalities and Applications 5 2. Higher-Order Differential Equation for Orthonormal Polynomials In the rest of this paper we often denote p n,ρ x and x k,n,ρ simply by p n x and x kn , respectively. Let ρ n  ρ if n is odd, ρ n  0 otherwise, and define the integrating functions A n x and B n x with respect to p n x as follows: A n  x  : 2b n  ∞ −∞ p 2 n  u  Qx, uw 2 ρ  u  du, B n  x  : 2b n  ∞ −∞ p n  u  p n−1  u  Qx, uw 2 ρ  u  du, 2.1 where Qx, uQ  x − Q  u/x − u and b n γ n−1 /γ n . Then in 3, Theorem 4.1 we have a relation of the orthonormal polynomial p n x with respect to the weight w 2 ρ x: p  n  x   A n  x  p n−1  x  − B n  x  p n  x  − 2ρ n p n  x  x . 2.2 Theorem 2.1 cf. 6, Theorem 3.3. Let ρ>−1/2 and wx ∈FC 2 . Then for |x| > 0 one has the second-order differential relation as follows: a  x  p  n  x   b  x  p  n  x   c  x  p n  x   D  x   E  x   0. 2.3 Here, one knows that for any integer n  1, a  x   A n  x  ,b  x   −2Q   x  A n  x  − A  n  x  , c  x   b n A 2 n  x  A n−1  x  b n−1  A n  x  B n  x  B n−1  x  − xA n  x  A n−1  x  B n  x  b n−1  A n  x  B  n  x  − A  n  x  B n  x  − 2ρ n A n  x  A n−1  x  b n−1 : c 1  x   c 2  x   c 3  x   c 4  x   c 5  x   c 6  x  , D  x   d  x  p n  x  x ,E  x   e 1  x  p  n  x  x  e 2  x  p n  x  x 2 , 2.4 where d  x   2ρ n  A n  x  B n  x  − A  n  x    2ρ n−1 A n  x  B n  x  , e 1  x   2  ρ n  ρ n−1  A n  x  ,e 2  x   −2ρ n A n  x  . 2.5 Especially, when n is odd, one has a  x  p  n  x   b  x  p  n  x   c  x  p n  x   d  x  q n−1  x   2ρA n  x  q  n−1  x   0, 2.6 where q n−1 x is the polynomial of degree n − 1 with p n xxq n−1 x. 6 Journal of Inequalities and Applications Proof. We may similarly repeat the calculation 6, Proof of Theorem 3.3, and then we obtain the results. We stand for A n : A n x,B n : B n x simply. Applying 2.2 to p  n−1 x we also see p  n−1  x   A n−1 p n−2  x  − B n−1 p n−1  x  − 2ρ n−1 p n−1  x  x , 2.7 and so if we use the recurrence formula xp n−1  x   b n p n  x   b n−1 p n−2  x  2.8 and use 2.2 too, then we obtain the following: p  n−1  x   1 b n−1 A n   xA n−1 − b n−1 B n−1  p  n  x    xA n−1 B n − b n−1 B n B n−1 − b n A n A n−1  p n  x   2ρ n x  xA n−1 − b n−1 B n−1  p n  x  − 2ρ n−1 b n−1 x  p  n  x   B n p n  x    . 2.9 We differentiate the left and right sides of 2.2 and substitute 2.2 and 2.9. Then consequently, we have, for n ≥ 1, p  n  x   −  B n−1  B n − xA n−1 b n−1 − A  n A n  p  n  x  −  b n A n−1 A n b n−1  B n−1 B n − xA n−1 B n b n−1  B  n − A  n B n A n − 2ρ A n−1 b n−1  p n  x  − 2ρ n  B n − A  n A n  p n  x  x − 2ρ n xp  n  x  − p n  x  x 2 − 2ρ n−1 p  n  x   B n p n  x  x . 2.10 Using the recurrence formula 2.8 and u/u − x1  x/u − x, we have B n  B n−1  2  ∞ −∞ p n−1  u   b n p n  u   b n−1 p n−2  u   Qx, uw 2 ρ  u  du  2  ∞ −∞ p 2 n−1  u  Q   u  w 2 ρ  u  du − 2Q   x   2x  ∞ −∞ p 2 n−1  u  Qx, uw 2 ρ  u  du  −2Q   x   xA n−1 b n−1 , 2.11 because Q  u is an odd function. Therefore, we have b  x   −2Q   x  A n − A  n . 2.12 When n is odd, since xp  n x − p n xx 2 q  n−1 x, 2.6 is proved. Journal of Inequalities and Applications 7 For the higher-order differential equation for orthonormal polynomials, we see that for j  0, 1, 2, ,ν− 2and|x| > 0 D j  x   j  t0  j  it  −1  i−t j!  j − i  !t! d j−i  x  x −i−t1  p t n  x  , E j  x   j  t0  j  it  −1  i−t j!  j − i  !t! e j−i 1  x  x −i−t1  p t1 n  x   j  t0  j  it  −1  i−t j!  i − t  1   j − i  !t! e j−i 2  x  x −i−t2  p t n  x  . 2.13 Let  j −1   0 for nonnegative integer j. In the following theorem, we show the higher-order differential equation for orthonormal polynomials. Theorem 2.2. Let ρ>−1/2 and wx ∈FC 2 .Letν  2 and j  0, 1, ,ν− 2. Then one has the following equation for |x| > 0: B j j2  x  p j2 n  x   B j j1  x  p j1 n  x   j  s0 B j s  x  p s n  x   0, 2.14 where B j j2  x   a  x  ,B j j1  x   ja   x   b  x   e 1  x  x , 2.15 and for j ≥ 1 and 1 ≤ s ≤ j B j s  x    j s − 2  a j−s2  x    j s − 1  b j−s1  x    j s  c j−s  x   j  is  −1  i−s j!  j − i  !s! d j−i  x  x −i−s1  j  is−1  −1  i−s1 j!  j − i  !  s − 1  ! e j−i 1  x  x −i−s2  j  is  −1  i−s j!  i − s  1   j − i  !s! e j−i 2  x  x −i−s2 , 2.16 and for j ≥ 0 B j 0  x   c j  x   j  i0  −1  i j!  j − i  ! d j−i  x  x −i1  j  i0  −1  i j!  i  1   j − i  ! e j−i 2  x  x −i2 . 2.17 Proof. It comes from Theorem 2.1 and 2.13. 8 Journal of Inequalities and Applications Corollary 2.3. Under t he same assumptions as Theorem 2.1,ifn is odd, then C j j2  0  p j2 n  0   C j j1  0  p j1 n  0   j  s1 C j s  0  p s n  0   0,j≥ 1, C 0 2  0  p  n  0   C 0 1  0  p  n  0   0,j 0, 2.18 where C j j2 xA n 02ρ/j  2A n 0 and for 1 ≤ s ≤ j  1 C j s  0    j s − 2  a j−s2  0    j s − 1  b j−s1  0    j s  c j−s  0   1 s  j s − 1  d j−s1  0    j s − 2  2ρA j−s2 n  0   . 2.19 Proof. Let n be odd. Then we will consider 2.6. Since q j n−1 0p j1 n 0/j  1, we have  dxq n−1 x2ρA n  x  q  n−1  x   j    x0  2ρA n  0  p j2 n  0  j  2   d  0   2jρA  n  0   p j1 n  0  j  1  j  s2  j s − 1  d j−s1  0    j s − 2  2ρA j−s2  0   p s n  0  s  d j  0  p  n  0  , 2.20 and we have  axp  n xbxp  n xc  x  p n  x   j    x0  a  0  p j2 n  0    ja   0   b  0   p j1 n  0   j  s0  j s − 2  a j−s2  0    j s − 1  b j−s1  0    j s  c j−s  0   p s n  0  . 2.21 Therefore, we have the result from 2.6. In the rest of this paper, we let ρ>−1/2andwxexp−Qx ∈  F ν C 2  for positive integer ν ≥ 1 and assume that 1  2ρ − δ ≥ 0forρ<0and a n  n 1/1ν−δ , 2.22 where 0 ≤ δ<1 is defined in 1.13. Journal of Inequalities and Applications 9 In Section 3, we will estimate the higher-order derivatives of orthonormal polynomials at the zeros of orthonormal polynomials with respect to exponential-type weights. 3. Estimation of Higher-Order Derivatives of Orthonormal Polynomials From 3, Theorem 4.2 we know that there exist C and n 0 > 0 such that for n ≥ n 0 and |x|≤a n 1  η n , A n  x  2b n ∼ ϕ n  x  −1  a 2 n  1  2η n  2 − x 2  −1/2 , | B n  x  |  A n  x  . 3.1 If Tx is unbounded, then 2.22 is trivially satisfied. Additionally we have, from 17, Theorem 1.3, that if we assume that Q  x is nondecreasing, then for |x|≤εa n with 0 <ε<1/2 | B n  x  | <λ  ε, n  A n  x  , 3.2 where there exists a constant C>0 such that λ  ε, n   C · max  1 nθ  1  θ Λ−1 ,ε 1−1/ΛΛ−1 ,ε 1/Λ ,λ  n   , 3.3 lim ε → 0 lim n →∞ λ  ε, n   0. 3.4 Here, θ  ε Λ−1/2Λ and λnOe −n C  for some C>0. For the higher derivatives of A n x and B n x, we have the following results in 17, Theorem 1.8. Theorem 3.1 see17, Theorem 1.4. For |x|≤a n 1  η n  and j  0, ,ν− 1    A j n  x      A n  x   T  a n  a n  j ,    B j n  x      A n  x   T  a n  a n  j . 3.5 Moreover, there exists εn > 0 such that for |x|≤a n /2 and j  1, ,ν− 1,    A j n  x     ≤ ε  n  A n  x   n a n  j ,    B j n  x     ≤ ε  n  A n  x   n a n  j , 3.6 with εn → 0 as n →∞. Corollary 3.2. Let 0 <β 1 < 1/2. Then there exists a positive constant C /  Cn such that one has for |x|≤β 1 a n and j  1, ,ν− 1,    A j n  x     ≤ CA n  x   n a n  j ,    B j n  x     ≤ CA n  x   n a n  j . 3.7 10 Journal of Inequalities and Applications In the following, we have the estimation of the higher-order derivatives of orthonor- mal polynomials. Theorem 3.3. Let 1 ≤ 2s  1 ≤ ν and 0 <α<1/2. Then for a n /αn ≤|x kn |≤αa n the following equality holds for n large enough: p 2s1 n  x kn    −1  s β s  x kn ,n   n a n  2s  1  ρ 2s1  α, x kn ,n   p  n  x kn  , 3.8 where β  x, n  : b n b n−1  a n n  2 A n  x  A n−1  x  , 3.9 and | ρ 2s1 α, x kn ,n|≤Cμ 1 α, nμ 2 α, nμ 3 α, n. Moreover, for 1 ≤ 2s ≤ ν    p 2s n  x kn      Cμ 1  α, n   n a n  2s−1   p  n  x kn    . 3.10 Here, μ 1  α, n  :  ε  n   α Λ−1  α  ,μ 2  α, n  : log n n  ε  n   αλ  α, n   α 2 , μ 3  α, n  : λ  α, n  λ  α, n − 1   αλ  α, n   ε  n   ε  n  λ  α, n   1 n . 3.11 Corollary 3.4. Suppose the same assumptions as Theorem 3.3. Given any δ>0, there exists a small fixed positive constant 0 <α 0 δ < 1/2 such that 3.8 holds satisfying | ρ 2s1 α 0 ,x kn ,n|≤δ and    p 2s n  x kn     ≤ δ  n a n  2s−1   p  n  x kn    3.12 for a n /α 0 n ≤|x kn |≤α 0 a n . Corollary 3.5. For |x kn |≤a n /2 and 1 ≤ j ≤ ν    p j n  x kn       n a n  j−1   p  n  x kn    . 3.13 Theorem 3.6. Let 0 < |x kn |≤a n 1  η n  and let ν  2, 3, , j  1, 2, ,ν− 2.Then    p j2 n  x kn       A n x kn  Ta n  a n  j1   p  n  x kn    , 3.14 [...]... Kanjin and R Sakai, “Convergence of the derivatives of Hermite-Fej´ r interpolation polynomials e of higher order based at the zeros of Freud polynomials, ” Journal of Approximation Theory, vol 80, no 3, pp 378–389, 1995 12 R Sakai, “Hermite-Fej´ r interpolation, ” in Approximation Theory (Kecskem´ t, 1990), vol 58 of Colloquia e e Mathematica Societatis Janos Bolyai´ , pp 591–601, North-Holland, Amsterdam,... 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Orthonormal polynomials with exponential-type weights,” Journal of Approximation Theory, vol 152, no 2, pp 215–238, 2008 4 H S Jung and R Sakai, “Specific examples of exponential weights,” Communications of the Korean Mathematical Society, vol 24, no 2, pp 303–319, 2009 5 T Kasuga and R Sakai, “Uniform or mean convergence of Hermite-Fej´ r interpolation of higher order e for Freud weights,” Journal of Approximation... Journal of Inequalities and Applications 29 References 1 E Levin and D S Lubinsky, Orthogonal Polynomials for Exponential Weights, CMS Books in Mathematics/Ouvrages de Math´ matiques de la SMC, 4, Springer, New York, NY, USA, 2001 e 2 H S Jung and R Sakai, “Inequalities with exponential weights,” Journal of Computational and Applied Mathematics, vol 212, no 2, pp 359–373, 2008 3 H S Jung and R Sakai, Orthonormal. .. Netherlands, 1991 a 13 R Sakai, “Hermite-Fej´ r interpolation prescribing higher order derivatives, ” in Progress in Approximae tion Theory, P Nevai and A Pinkus, Eds., pp 731–759, Academic Press, Boston, Mass, USA, 1991 14 R Sakai, “Certain unbounded Hermite-Fej´ r interpolatory polynomial operators,” Acta Mathematica e Hungarica, vol 59, no 1-2, pp 111–114, 1992 15 R Sakai and P V´ rtesi, “Hermite-Fej´... reason as the proof of Corollary 3.4 Journal of Inequalities and Applications 25 Proof of Theorem 4.4 To prove the result, we proceed by induction on i From 4.2 and 4.4 1/s! and the following recurrence relation; for s 1 ≤ i ≤ m − 1 we know that es,s l, m, k, n − es,i l, m, k, n i−1 p s 1 m es,p l, m, k, n lk,n,ρ i−p ! i−p xk,n,ρ 4.28 When i s, es,s l, ν, k, n 1/s! so that 4.14 and 4.15 are satisfied... other hand, one has for |xkn | ≤ an 1 |ei m, k, n | n an i 4.19 ηn An xkn T an an i 4.20 Especially, if i is odd, then one has |ei m, k, n | T an an Q xkn 1 |xkn | An xkn T an an i−1 4.21 24 Journal of Inequalities and Applications Proof of Theorem 4.1 Theorem 4.1 is shown by induction with respect to m The case of m 1 follows from 4.6 , Corollary 3.5, and Theorem 3.6 Suppose that for the case of m... 3.1, and the definitions of μi α, n i 1, 2, 3 in Theorem 3.3, if for any δ > 0 we choose a fixed constant α0 δ > 0 small enough, then there exists an integer N N α0 such that we can make μ1 α0 , n , μ2 α0 , n , and μ3 α0 , n small enough for an /α0 n ≤ |x| ≤ α0 an with n > N 20 Journal of Inequalities and Applications Proof of Corollary 3.5 Since we have from Lemma 3.8 that |Cj 2 n/an for j ≥ 0 and induction . of Inequalities and Applications Volume 2010, Article ID 816363, 29 pages doi:10.1155/2010/816363 Research Article Derivatives of Orthonormal Polynomials and Coefficients of Hermite-Fej ´ er Interpolation Polynomials. estimate the higher-order derivatives of p n,ρ x at the zeros of p n,ρ x and the coefficients of the higher-order Hermite-Fej ´ er interpolation polynomials based at the zeros of p n,ρ x. More precisely,. derivatives of p n,ρ x at all zeros of p n,ρ x for two cases of an odd order and of an even order. Furthermore, we will estimate the coefficients of higher-order Hermite-Fej ´ er interpolation polynomials