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Volume 2010, Article ID 816363, 29 pagesdoi:10.1155/2010/816363 Research Article Derivatives of Orthonormal Polynomials and Coefficients of Hermite-Fej ´er Interpolation Polynomials with

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

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

Copyrightq 2010 H S Jung and R Sakai This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited

LetR  −∞, ∞, and let Q ∈ C2 :R → 0, ∞ be an even function In this paper, we consider the exponential-type weights w ρ x  |x| ρexp−Qx, ρ > −1/2, x ∈ R, and the orthonormal

Let R  −∞, ∞ and R  0, ∞ Let Q ∈ C2 : R → R be an even function and let

wx  exp−Qx be such that∞

0 x n w2xdx < ∞ for all n  0, 1, 2, For ρ > −1/2, we set

w ρ x : |x| ρ wx, x ∈ R. 1.1

Then we can construct the orthonormal polynomials p n,ρ x  p n w2

ρ ; x of degree n with respect to w2

,

p n,ρ x  γ n x n  · · · , γ n  γ n,ρ > 0.

1.2

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 → Ris said to be quasi-increasing if there exists C > 0 such that fx ≤ Cfy for 0 < x < y For any two sequences {b n}∞n1 and {c n}∞n1 of nonzero realnumbersor 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, C1, C2, 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 from1

Definition 1.1 Let Q : R → Rbe even and satisfy the following properties

a Qx is continuous in R, with Q0  0.

b Qx exists and is positive in R \ {0}.

In the following we introduce useful notations.

a Mhaskar-Rahmanov-Saff MRS numbers a x is defined as the positive roots of thefollowing equations:

x  2π

ρ  |x| 2ρexp−2Qx, 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 Qx as follows.

Definition 1.2 Let wx  exp−Qx ∈ FC2 and let ν be a positive integer Assume that Qx is ν-times continuously differentiable on R and satisfies the followings.

a Q ν1 x exists and Q i x, 0 ≤ i ≤ ν  1 are positive for x > 0.

b There exist positive constants C i > 0 such that for x ∈ R \ {0}

a Qx/Qx is quasi-increasing on a certain positive interval c2, ∞.

b Q ν1 x is nondecreasing on a certain positive interval c2, ∞.

c There exists a constant 0 ≤ δ < 1 such that Q ν1 x ≤ C1/x δonc2, ∞.

Then we write wx ∈ Fν C2

Now, consider some typical examples ofFC2 Define for α > 1 and l ≥ 1,

In the following, we consider the exponential weights with the exponents Q l,α,m x.

Then we have the following examplessee 4 

Example 1.3 Let ν be a positive integer Let m  α − ν > 0 Then one has the following.

a wx  exp−Q l,α,m x belongs to F ν C2

b If l ≥ 2 and α > 0, then there exists a constant c1 > 0 such that Ql,α,m x/Q l,α,m x is

quasi-increasing onc1, ∞.

c When l  1, if α ≥ 1, then there exists a constant c2 > 0 such that Q l,α,m x/Q l,α,m x

is quasi-increasing on c2, ∞, and if 0 < α < 1, then Ql,α,m x/Q l,α,m x is

quasidecreasing onc2, ∞.

d When l  1 and 0 < α < 1, Q ν1 l,α,m x is nondecreasing on a certain positive interval

c2, ∞.

In this paper, we will consider the orthonormal polynomials p n,ρ x with respect to

the weight class Fν C2 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 ofhigher-order Hermite-Fej´er interpolation polynomialssee 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 InSection 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

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:

whereQx, u  Qx − Qu/x − u and b n  γ n−1 /γ n Then in3, Theorem 4.1 we have

a relation of the orthonormal polynomial p n x with respect to the weight w2

Proof We may similarly repeat the calculation6, Proof of Theorem 3.3 , and then we obtain the results We stand for A n : An x, B n : Bn x simply Applying 2.2 to p

xp n−1 x  b n p n x  b n−1 p n−2 x 2.8and use2.2 too, then we obtain the following:

We differentiate the left and right sides of 2.2 and substitute 2.2 and 2.9 Then

consequently, we have, for n ≥ 1,

For the higher-order differential equation for orthonormal polynomials, we see that

 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 wx ∈ FC2 Let ν  2 and j  0, 1, , ν − 2 Then one has the following equation for |x| > 0:

Corollary 2.3 Under the same assumptions as Theorem 2.1 , if n is odd, then

Therefore, we have the result from2.6

In the rest of this paper, we let ρ > −1/2 and wx  exp−Qx ∈ Fν C2 for

positive integer ν ≥ 1 and assume that 1  2ρ − δ ≥ 0 for ρ < 0 and

where 0≤ δ < 1 is defined in 1.13

InSection 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

If Tx is unbounded, then 2.22 is trivially satisfied Additionally we have, from 17,

Theorem 1.3 , that if we assume that Qx is nondecreasing, then for |x| ≤ εa n with

Here, θ  ε Λ−1/2Λ and λn  Oe −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,

In the following, we have the estimation of the higher-order derivatives of mal polynomials.

orthonor-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:

and especially if j is even, then

To prove these results we need some lemmas

Lemma 3.7 a For s ≥ r > 0

Proof. a It is 1, Lemma 3.11c b It is 1, Lemma 3.8c c It comes from 3.1 d Since

j  1 ≤ ν, Q j1 x is increasing So, we obtain d by 1.12

Lemma 3.8 Let ax, bx, cx, dx, and e i x, i  1, 2, be defined in Theorem 2.1

a For |x| ≤ a n /2 and 1 ≤ k ≤ ν − 1, there exists εn satisfying εn → 0 as n → 0 such that

Moreover, for |x| ≤ a n 1  η n  and 0 ≤ k ≤ ν − 3,

Lemma 3.9 Let 0 < α < 1/2, 0 ≤ j ≤ ν − 2, and L1> 0 Let a n /αn ≤ |x| ≤ αa n Then

B

j

j1 x

B j2 j x

B j1 j x

B j2 j x

≤ C a n n 3.38

Moreover, for |x| ≤ a n 1  η n ,

... Cμ1α, n n

where μ1α, n is defined in Theorem 3.3 and for L1a n /n ≤ |x| ≤ a n /2

... ≤ a n /2 from the above easily.

Lemma 3.10 Let < α < 1/2 and ≤ j ≤ ν − Let a n /αn ≤ |x| ≤ αa n Then for a n... μ3α, n, where μ2α, n, μ3α, n, and βx, n are defined in

Theorem 3.3 For L1a n