Báo cáo hóa học: " Higher order Hermite-Fejér interpolation polynomials with Laguerre-type weights" potx

Keywords: Laguerre-type weights, orthonormal polynomials, higher order Fejér interpolation polynomials Hermite-1.. Hence, in this cle, we will investigate the higher order Hermite-Fejér

R E S E A R C H Open Access

Higher order Hermite-Fejér interpolation

polynomials with Laguerre-type weights

Heesun Jung1* and Ryozi Sakai2

Full list of author information is

available at the end of the article

AbstractLetℝ+

k=1, where 0≤ l ≤ m - 1 are positive integers

2010 Mathematics Subject Classification: 41A10

Keywords: Laguerre-type weights, orthonormal polynomials, higher order Fejér interpolation polynomials

Hermite-1 Introduction and main resultsLetℝ = [-∞, ∞) and ℝ+

= [0,∞) Let R : ℝ+® ℝ+

be a continuous, non-negative, andincreasing function Consider the exponential weights wr(x) = xrexp(-R(x)), r > -1/2,and then we construct the orthonormal polynomials {p n, ρ (x)}∞

n=0 with the weight wr

(x) Then, for the zeros {x k,n,ρ}n

estima-tions with respect to p (j) n, ρ (x k,n, ρ , k = 1, 2, , n, j = 1, 2, ,ν, in [1] Hence, in this cle, we will investigate the higher order Hermite-Fejér interpolation polynomial Ln(l,

arti-m, f; x) based at the zeros {x k,n,ρ}n

k=1, using the results from [1], and we will give adivergent theorem This article is organized as follows In Section 1, we introducesome notations, the weight classes L2, L˜ν with L(C2), L(C2+), and main results InSection 2, we will introduce the classes F(C2) and F(C2+), and then, we will obtainsome 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

of polynomials with degree n by P n

© 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

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: LetQ(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)

We consider the case d =∞, that is, the space ℝ+

= [0,∞), and we strengthen tion 1.1 slightly

Defini-Definition 1.2 We assume that R : ℝ+ ® ℝ+

has the following properties:

(a) R(x), R’(x) are continuous, positive in ℝ+

There exist a compact subinterval J∋ 0 of ℝ+

and C2> 0 such that

Definition 1.3 Letw = exp( −R) ∈ L2and letν ≥ 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)

(1) If a >ν, w(x) = e −R l,α (x)∈ ˜L ν.(2) If a≤ ν and a is an integer, we define

Let us denote the zeros of pn,r(x) by

0< x n,n, ρ < · · · < x 2,n, ρ < x 1,n, ρ < ∞.

The Mhaskar-Rahmanov-Saff numbers avis defined as follows:

v = 1π

Theorem 1.5 Letw(x) = exp( −R(x)) ∈ L(C2+)and r> -1/2.

(a) For each m≥ 1 and j = 0, 1, , we have

| (l m k,n)(j) (x k,n)| ≤ C

where 0≤ δ < 1 is defined in (1.1) Then we have the following:

(a) If j is odd, then we have for m≥ 1 and j = 0, 1, , ν - 1,

| (l m k,n)(j) (x k,n)| ≤C

k,n

(1:8)

(b) If j - s is odd, then we have for m≥ 1 and 0 ≤ s ≤ j ≤ m - 1,

Theorem 1.7 Let 0 <ε < 1/4 Let 1

ε a n n2 ≤ x k,n ≤ εa n Let s be a positive integer with 2

≤ 2s ≤ ν Then under the same conditions as the assumptions of Theorem 1.6, there

exists μ1(ε, n) > 0 such that

Theorem 1.8 [4, Lemma 10] Let 0 <ε < 1/4 Let 1

ε n a n2 ≤ x k,n ≤ εa n Let s be a positiveinteger with 2≤ 2s ≤ ν - 1 Suppose the same conditions as the assumptions of Theorem

1.6 Then

(a) for 1≤ 2s - 1 ≤ ν - 1,



(l m k,n)(2s−1)(x k,n) ≤ Cδ(ε, n) n

whereδ(ε, n) is defined in Theorem 1.7

(b) there exists b(n, k) with 0 <D1 ≤ b(n, k) ≤ D2for absolute constants D1, D2suchthat the following holds:

Theorem 1.9 [4, (4.16)], [9]Let 0 <ε < 1/4 Let 1

ε a n n2 ≤ x k,n ≤ εa n Let s be a positiveinteger with 2≤ 2s ≤ m - 1 Suppose the same conditions as the assumptions of Theo-

rem 1.6 Then for j= 0, 1, 2, , there is a polynomialΨj(x) of degree j such that (-1)jψj

(-m) > 0 for m = 1, 3, 5, and the following relation holds:

interval[a, b], a >0,

lim sup

n→∞ amax≤x≤b |L n (m, f ; x)| = ∞

2 Preliminaries

Levin and Lubinsky introduced the classes L(C2) and L(C2+) as analogies of the

classes F(C2) and F(C2+) defined on I = (−√d,√

d) They defined the following:

Definition 2.1 [10] We assume that Q : I* ® [0, ∞) has the following properties:

(a) Q(t) is continuous in I*, with Q(0) = 0;

(b) Q″(t) exists and is positive in I*\{0};

Lemma 2.2 [1]Let Q(t) = R(x), x = t2

Then we have

where W(t) = w(x); x = t2

Onℝ, we can consider the corresponding class to L˜ν as follows:

Definition 2.3 [11] LetW = exp( −Q) ∈ F(C2+)and ν ≥ 2 be an integer Let Q be acontinuous and even function onℝ For the exponent Q, we assume the following:

(a) Q(j)(x) > 0, for 0≤ j ≤ ν and t Î ℝ+

/{0}

(b) There exist positive constants Ci> 0 such that for i = 1, 2, ,ν - 1,

There are many properties of Pn, r*(t) = Pn(Wr* ; t) with respect to Wr*(t),

W∈ ˜F ν,ν = 2, 3, of Definition 2.3 in [2,3,7,11-13] They were obtained by

transfor-mations from the results in [5,6] Jung and Sakai [2, Theorem 3.3 and 3.6] estimate

P (j) n, ρ∗ (t k,n), k = 1, 2, , n, j = 1, 2, , ν and Jung and Sakai [1, Theorem 3.2 and 3.3]

obtained analogous estimations with respect to p (j) n,ρ (x k,n), k = 1, 2, , n, j = 1, 2, , ν

In this article, we consider w = exp( −R) ∈ ˜ L νand pn, r(x) = pn(wr; x) In the

follow-ing, we give the transformation theorems

Theorem 2.4 [13, Theorem 2.1] Let W(t) = W(x) with x = t2

Then the orthonormalpolynomials Pn, r*(t) on ℝ can be entirely reduced to the orthonormal polynomials pn, r

Theorem 2.5 [1, Theorem 2.5] Let Q(t) = R(x), x = t2

t

δ,

In the following, we introduce useful notations:

(a) The Mhaskar-Rahmanov-Saff numbers av and a∗u are defined as the positiveroots of the following equations, that is,

v = 1π

To prove main results, we need some lemmas as follows:

Lemma 2.7 [13, Theorem 2.2, Lemma 3.7] For the minimum positive zero, t[n/2],n([n/2]

is the largest integer≤ n/2), we have

Moreover, for some constant0 <ε ≤ 2 we have

T∗(a∗n)≤ Cn2−ε.

Remark 2.8 (a) Let W(t)∈F(C2+) Then

(a-1) T(x) is bounded⇔ T*(t) is bounded

(a-2) T(x) is unbounded⇒ an≤ C(h)nh

for any h > 0

(a-3) T(an)≤ Cn2-ε for some constant 0 <ε ≤ 2

(b) Let w(x)∈ ˜L ν Then(b-1) r > -1/2⇒ r* > -1/2

In addition, since T(x) = T*(t)/2 and a n = a∗2n2, we know that (a-2)

(b) Since w(x)∈ ˜L ν, we know that W(t)∈ ˜F ν andδ* = δ by Theorem 2.5 Thenfrom (2.3) and Lemma 2.6, we have (b-1), (b-2), and (b-3) □

Lemma 2.9 [1, Lemma 3.6] For j = 1, 2, 3, , we have

j,i P 2n (i) (t)t −2j+i,

where cj, i> 0(1≤ i ≤ j, j = 1, 2, ) satisfy the following relations: for k = 1, 2, ,

3 Proofs of main results

Our main purpose is to obtain estimations of the coefficients es, i(l, m, k, n), k = 1, 2, , 0

k,n | p

n, ρ (x k,n) |

Proof of Theorem 1.5 (a) From Theorem 3.1 we know that

Therefore, the result is proved by induction with respect to m

(b) From (2) and (3), we know es, s(l, m, k, n) = 1/s! and the following recurrencerelation: for s + 1≤ i ≤ m - 1,

Therefore, we have the result by induction on i and (3.5)

Theorem 3.2 [1, Theorem 1.6] Letw(x) = exp( −R(x)) ∈ ˜ L νand let r> -1/2 Suppose

the same conditions as the assumptions of Theorem 1.6 For each k = 1, 2, , n and j =

k,n | p

n,ρ (x k,n)|

Proof of Theorem 1.6 We use the induction method on m

(a) For m = 1, we have the result because of

(b) To prove the result, we proceed by induction on i From (1.2) and (1.3) we know

es, s(l, m, k, n) = 1/s! and the following recurrence relation: for s + 1≤ i ≤ m - 1,

Then, we have (1.9) from (1.5), (1.8), (3.6), and the assumption of induction on i □Theorem 3.3 [1, Theorem 1.7] Let 0 <ε < 1/4 Let 1

ε n a n2 ≤ x k,n ≤ εa nand let s be apositive integer with 2≤ 2s ≤ ν - 1 Suppose the same conditions as the assumptions of

Theorem 1.6 Then there exists b(n, k), 0 <D1≤ b(n, k) ≤ D2 for absolute constants D1,

D2such that the following equality holds:

Lemma 3.5 [2, Theorem 3.3] Let r* > -1/2 and W(x) = exp( −Q(x)) ∈ ˜ F ν, ν ≥ 2

Assume that1 + 2r* - δ* ≥ 0 for r* < 0 and if T*(t) is bounded, then assume

For j = 0, 1, , define jj(1): = (2j + 1)-1and for k≥ 2,

k,n

From Theorem 1.5, we know that for xk, n≤ an/4,



(l m k,n)(j) (x k,n) ≤ C n

For the second term, we have from (1.10),

φ r(ν − 1).

Now, for every j we will introduce an auxiliary polynomial determined by { j (y)}∞

j=1

as the following lemma:

Lemma 3.6 [4, Lemma 11] (i) For j = 0, 1, 2 , there exists a unique polynomial Ψj

(y) of degree j such that

j(ν) = φ j(ν), ν = 1, 2, 3,

(ii)Ψ0(y) = 1 andΨj(0) = 0, j = 1, 2,

SinceΨj(y) is a polynomial of degree j, we can replace jj(ν) in (3.7) with Ψj(y), thatis,



for an arbitrary y and j = 0, 1, 2, We use the notation Fkn(x, y) = (lk, n(x))ywhichcoincides with l y k,n (x) if y is an integer Since lk, n(xk, n) = 1, we have Fkn(x, t) > 0 for x

in a neighborhood of xk, n and an arbitrary real number t

We can show that (∂/∂x)j

Fkn(xk, n, y) is a polynomial of degree at most j withrespect to y for j = 0, 1, 2, , where (∂/∂x)j

Fkn(xk, n, y) is the jth partial derivative of

Fkn (x, y) with respect to x at (xk, n, y) [14, p 199] We prove these facts by induction

on j For j = 0 it is trivial Suppose that it holds for j ≥ 0 To simplify the notation, let

F(x) = Fkn(x, y) and l(x) = lk, n(x) for a fixed y Then F’(x)l(x) = yl’(x)F(x) By Leibniz’s

rule, we easily see that

l (s+1) (x k,n )F (j−s) (x k,n),

which shows that F(j+1)(xk, n) is a polynomial of degree at most j + 1 with respect to y

Let P kn [j] (y), j = 0, 1, 2, be defined by

Then P [j] kn (y) is a polynomial of degree at most 2j

By Theorem 1.8 (1.11), we have the following:

Lemma 3.7 [4, Lemma 12] Let j = 0, 1, 2, , and M be a positive constant Let 0 <ε <

Lemma 3.8 [4, Lemma 13] If y < 0, then for j = 0, 1, 2 ,

(∂/∂x) 2r F kn (x k,n,−m)(l m

k,n)(2s −2r) (x k,n)+

(∂/∂x) 2r+1 F kn (x k,n,−m)(l m

k,n)(2s −2r−1) (x k,n)

By (1.11), (3.12) and (3.13), we see that the first sum s

r=0has the form of



r(−m)φs −r (m) + ˜η s(−m, ε, xk,n , n)





Theorem 1.9 is important to show a divergence theorem with respect to Ln(m, f; x),where m is an odd integer

Proof of Theorem 1.9 We prove (1.12) by induction on s Since e0(m, k, n) = 1 and

Ψ0(y) = 1, (1.12) holds for s = 0 From (3.6) we write e2s(m, k, n) in the form of



r(−m)φs −r (m)(1 + η r)(1 +ξ s −r),

where ξs-r : = ξs-r(m, ε, xk, n, n) and hr : = hr(m, ε, xk, n, n) which are defined in(1.11) and (1.12) Then, using Lemma 3.9 and j0 (m) = 1, we have the following form:

we see that |h s(m,ε, xk, n, n)|® 0 as n ® ∞ and ε ® 0 (recall above estimation of

|II|) Therefore, we proved the result

Lemma 3.10 [5, Theorem 1.3] Let ρ > −1

2and w(x) (C2+) There exists n0 such thatuniformly for n≥ n0, we have the following:

(f) LetΛ be defined in Definition 1.2 (d) There exists C > 0 such that for n ≥ 1,

a n ≤ Cn1/.

Proof (a) and (b) follow from [5, Theorem 1.3] (e) follows from [5, Theorem 1.4]

We need to prove (c), (d), and (f)

(c) For 0 <a≤ xk, n≤ b < ∞, we have (2.11);

ϕ n (x k,n)∼

√

a n

so applying (a), we have the result

(d) Let 0 <a≤ xk+1, n<xk, n ≤ b < ∞ We take a constant δ > 0 as

Hence, we have the result

Lemma 3.11 Let the function hkn (m; x) be defined by (1.4) and let 0 <c <a <b <d <

Now, choose a, b > 0 satisfying for all xk+1,n, xk, n Î [c, d],

Let xÎ [a, b] and |x -xj(x),n| = min {|x - xk, n}|; xk, n Î [a, b]},xj(c)+1,n<c≤xj(c),n, and

xj(d),n≤ d <xj(d)-1,n Moreover, we take a non-negative integer jksatisfying for each xk, n

Î [a, b] and k ≠ j(x),



j k+12

Here, by (3.16) and (3.17) we see

Therefore, we have for 0 ≤ i ≤ m - 2,



c ≤x k,n ≤d xk,n=xj(x),n

1

It follows from Lemma 3.11 that maxa ≤ x ≤ b Gn(x)≤ C with C independent of n

Therefore, it is enough to show that maxa ≤ x ≤ bFn(x)≥ C log (1 + n) We consider a,

b and jkdefined in (3.22) and (3.23) Let K (x; [c, d]) be the set of numbers defined as

where jkis a non-negative integer Then, there exist g > 0 and C > 0 such that

Consequently, the theorem is complete □

Acknowledgements

The authors thank the referees for many valuable suggestions and comments Hee Sun Jung was supported by SEOK

CHUN Research Fund, Sungkyunkwan University, 2010.

Author details

1 Department of Mathematics Education, Sungkyunkwan University Seoul 110-745, Republic of Korea 2 Department of

Mathematics, Meijo University Nagoya 468-8502, Japan

Authors ’ contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in

the sequence alignment All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 28 July 2011 Accepted: 25 November 2011 Published: 25 November 2011

References

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with exponential-type weights J Inequal Appl 2010, 29 (2010) (Article ID 816363)

3 Jung, HS, Sakai, R: Markov-Bernstein inequality and Hermite-Fejér interpolation for exponential-type weights J Approx

8 Freud, G: Orthogonal Polynomials Pergamon Press, Oxford (1971)

9 Sakai, R, Vértesi, P: Hermite-Fejér interpolations of higher order III Studia Sci Math Hungarica 28, 87 –97 (1993)

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Inequal Appl 2009, 22 (2009) (Article ID 528454)

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rule, we easily see that

l (s+1) (x k,n )F (j−s)... −2r−1) (x k,n)

By (1.11), (3.12) and (3.13), we see that the first sum... s −r),

where ξs-r : = ξs-r(m,