Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 482935, 25 pages
doi:10.1155/2012/482935
Research Article
Asymptotic Properties of Derivatives of
the Stieltjes Polynomials
Hee Sun Jung1 and Ryozi Sakai2
1
2
Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea
Department of Mathematics, Meijo University, Nagoya 468-8502, Japan
Correspondence should be addressed to Hee Sun Jung, hsun90@skku.edu
Received 16 March 2012; Accepted 24 May 2012
Academic Editor: Jin L. Kuang
Copyright q 2012 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.
λ−1/2
Let wλ x : 1 − x2 and Pλ,n x be the ultraspherical polynomials with respect to
wλ x. Then, we denote the Stieltjes polynomials with respect to wλ x by Eλ,n1 x satisfying
1
1
wλ xPλ,n xEλ,n1 xxm dx 0, 0 ≤ m < n 1, −1 wλ xPλ,n xEλ,n1 xxm dx /
0, m n 1.
−1
In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials
Eλ,n1 x and the product Eλ,n1 xPλ,n x. Especially, we estimate the even-order derivative
values of Eλ,n1 x and Eλ,n1 xPλ,n x at the zeros of Eλ,n1 x and the product Eλ,n1 xPλ,n x,
respectively. Moreover, we estimate asymptotic representations for the odd derivatives values of
Eλ,n1 x and Eλ,n1 xPλ,n x at the zeros of Eλ,n1 x and Eλ,n1 xPλ,n x on a closed subset of
−1, 1, respectively. These estimates will play important roles in investigating convergence and
divergence of the higher-order Hermite-Fejér interpolation polynomials.
1. Introduction
Consider the generalized Stieltjes polynomials Eλ,n1 x defined up to a multiplicative
constant by
1
−1
wλ xPλ,n xEλ,n1 xxk dx 0,
k 0, 1, 2, . . . , n, n 1,
1.1
where wλ x 1 − x2 λ−1/2 , λ > −1/2, and Pλ,n x is the nth ultraspherical polynomial for
the weight function wλ x.
The polynomials Eλ,n1 x, introduced by Stieltjes and studied by Szegö, have been
used in numerical integration, whereas the polynomials Pλ,n xEλ,n1 x have been used in
extended Lagrange interpolation. In this paper, we will prove pointwise and asymptotic
estimates for the higher-order derivatives of Eλ,n1 x and Pλ,n xEλ,n1 x. It is well known
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Journal of Applied Mathematics
that these kind of estimates are useful for studying interpolation processes with multiple
nodes.
In 1934, G. Szegö 1 showed that the zeros of the generalized Stieltjes polynomials
Eλ,n1 x are real and inside −1, 1 and interlace with the zeros of Pλ,n x whenever 0 λ 2. Recently, several authors 2–8 studied further interesting properties for these Stieltjes
polynomials. Ehrich and Mastroianni 3, 4 gave accurate pointwise bounds of Eλ,n1 x 0 λ 1 and the product Fλ,2n1 : Eλ,n1 xPλ,n x 0 λ 1 on −1, 1, and they estimated
x and Fλ,2n1
x at the zeros of Eλ,n1 x and Fλ,2n1 x,
asymptotic representations for Eλ,n1
respectively. In 6, pointwise upper bounds of Eλ,n1
x, Eλ,n1 x, Fλ,2n1
x, and Fλ,2n1 x
are obtained using the asymptotic differential relations of the first and the second order for
the Stieltjes polynomials Eλ,n1 x 0 λ 1 and Fλ,2n1 x 0 λ 1. Also the values of
Eλ,n1 x and Fλ,2n1 x at the zeros of Eλ,n1 x and Fλ,2n1 x are estimated in 6. Moreover,
using the results of 6, the Lebesgue constants of Hermite-Fejér interpolatory process are
estimated in 7.
r
r
In this paper, we find pointwise upper bounds of Eλ,n1 x and Fλ,2n1 x for two cases
of an odd order and of even order. Using these relations, we investigate asymptotic properties
of derivatives of the Stieltjes polynomials Eλ,n1 x and Fλ,2n1 x and we also estimate the
2
2
values of Eλ,n1 x and Fλ,2n1 x at the zeros of Eλ,n1 x and Fλ,2n1 x, respectively. Especi2
r
r
ally, for the value of Fλ,2n1 x at the zeros of Fλ,2n1 x, we will estimate Pλ,n x and Eλ,n1 x
for an odd r at the zeros of Eλ,n1 x and Pλ,n x, respectively. Finally, we investigate asym21
21
ptotic representations for the values of Eλ,n1 and Fλ,2n1 x at the zeros of Eλ,n1 x and
Fλ,2n1 x on a closed subset of −1, 1, respectively. These estimates will play important roles
in investigating convergence and divergence of the higher-order Hermite-Fejér interpolation
polynomials.
This paper is organized as follows. In Section 2, we will introduce the main results. In
Section 3, we will introduce the known results in order to prove the main results. Finally, we
will prove the results in Section 4.
2. Main Results
We first introduce some notations, which we use in the following.
For the ultraspherical
0, we use the normalization Pλ,n 1 n2λ−1
polynomials Pλ,n , λ /
and then we know that
n
λ
Pλ,n 1 ∼ n2λ−1 . We denote the zeros of Pλ,n by xν,n , ν 1, . . . , n, and the zeros of Stieltjes
λ
polynomials Eλ,n1 by ξμ,n1 , μ 1, . . . , n 1. We denote the zeros of Fλ,2n1 : Pλ,n Eλ,n1 by
λ
y
, ν 1, . . . , 2n 1. All nodes are ordered by increasing magnitude. We set ϕx :
√ν,2n1
1 − x2 , and, for any two sequences {bn }n and {cn }n of nonzero real numbers or functions,
we write bn cn , if there exists a constant C > 0, independent of n and x such that bn Ccn
for n large enough and write bn ∼ cn if bn cn and cn bn . We denote by Pn the space of
polynomials of degree at most n.
For the Chebyshev polynomial Tn x, note that for λ 0 and λ 1
2n
Tn1 x − Tn−1 x
π
2
E1,n1 x Tn1 x.
π
E0,n1 x Therefore, we will consider Eλ,n1 x for 0 < λ < 1.
2.1
Journal of Applied Mathematics
3
λ
λ
Theorem 2.1. Let 0 < λ < 1 and r 1 be a positive integer. Then, for all x ∈ ξ1,n1 , ξn1,n1 ,
r
Eλ,n1 x nr1−λ ϕ1−r−λ x.
2.2
r
max Eλ,n1 x ∼ n2r ,
2.3
Moreover, one has
x∈−1,1
λ
λ
and especially one has, for x ∈ −1, ξ1,n1 ∪ ξn1,n1 , 1,
r
Eλ,n1 x ∼ n2r .
2.4
λ
λ
Theorem 2.2. Let 0 < λ < 1 and r 1 be a positive integer. Then, for all x ∈ ξ1,n1 , ξn1,n1 ,
r
Fλ,2n1 x nr ϕ1−2λ−r x.
2.5
r
max Fλ,2n1 x n2λ2r−1 ,
2.6
Moreover, one has
x∈−1,1
λ
λ
and especially, for x ∈ −1, ξ1,n1 ∪ ξn1,n1 , 1,
r
Fλ,2n1 x ∼ n2λ2r−1 .
2
2.7
2
In the following, we also estimate the values of Eλ,n1 x and Fλ,2n1 x, 1 at the
λ
λ
zeros {ξμ,n1 } of Eλ,n1 x and the zeros {yν,2n1 } of Fλ,2n1 x, respectively.
Theorem 2.3. Let 0 < λ < 1 and r 2 be an even integer. For 1 μ n 1, one has
r
λ
λ
Eλ,n1 ξμ,n1 nr ϕ−r ξμ,n1 .
2.8
Theorem 2.4. Let 0 < λ < 1 and r 2 be an even integer. For 1 ν 2n 1, one has
r
λ
λ
Fλ,2n1 yν,2n1 nr−1λ ϕ−r−λ yν,2n1 .
2.9
21
21
Finally, we obtain the asymptotic representations for the values of Eλ,n1 x and
Fλ,2n1 x at the zeros of Eλ,n1 x and Fλ,2n1 x on a closed subset of −1, 1, respectively.
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Journal of Applied Mathematics
λ
Theorem 2.5. Let 0 < λ < 1 and 0 < ε < 1. Suppose |ξμ,n1 | 1 − ε. Then,
a
21
λ
λ
λ
ξμ,n1 O n21 ,
Eλ,n1 ξμ,n1 −1 n 12 ϕ−2 ξμ,n1 Eλ,n1
2.10
b
2
Pλ,n
λ
λ
λ
ξμ,n1 −1 n n 2λ ϕ−2 ξμ,n1 Pλ,n ξμ,n1 O n2λ2−3 .
2.11
In addition,
2
Pλ,n
λ
λ
λ
ξμ,n1 −1 n 12 ϕ−2 ξμ,n1 Pλ,n ξμ,n1 O n2λ−2 .
2.12
λ
Theorem 2.6. Let 0 < λ < 1 and 0 < ε < 1. Suppose |xν,n | 1 − ε. Then,
a
2
λ
λ
λ
Eλ,n1 xν,n −1 n 12 ϕ−2 xν,n Eλ,n1 xν,n O n2 ,
2.13
b
21
Pλ,n
λ
λ
λ
xν,n −1 n n 2λ ϕ−2 xν,n Pλ,n
xν,n O nλ2−2 .
2.14
In addition,
21
Pλ,n
λ
λ
λ
xν,n −1 n 12 ϕ−2 xν,n Pλ,n
xν,n O nλ2−1 .
2.15
λ
Theorem 2.7. Let 0 < λ < 1 and 0 < ε < 1. Suppose |yν,2n1 | 1 − ε. Then, one has, for a positive
integer 1,
21
λ
λ
λ
yν,2n1 O nλ2 ,
Fλ,2n1 yν,2n1 c −1 n 12 ϕ−2 yν,2n1 Fλ,2n1
where c 4 − 2−1 .
3. The Known Results
In this section, we will introduce the known results in 4, 6, 9 to prove main results.
2.16
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5
Proposition 3.1. a Let λ > −1/2. Then, Pλ,n x satisfies the second-order differential equation as follows:
1 − x2 Pλ,n x − 2λ 1xPλ,n
x nn 2λPλ,n x 0.
3.1
b Let λ > −1/2. Then,
Pλ,n
x 2λPλ1,n−1 x.
3.2
c Let λ > −1/2. Then, for 1 μ n 1,
λ
λ
Pλ,n ξμ,n1 n2λ−1 ϕ−2 ξμ,n1 .
3.3
d Let λ > −1/2. Then, for 1 ν n,
λ λ
Pλ,n xν,n ∼ nλ ϕ−λ−1 xν,n .
3.4
e Let λ > −1/2. Then, for x ∈ −1, 1 and r 0,
r Pλ,n x nλr−1 ϕ−λ−r x.
3.5
f Let 0 λ 1. Then, for 1 ν n,
λ λ
Eλ,n1 xν,n nϕ−1 xν,n .
3.6
g Let λ > −1/2 and r 0. Then, Pλ,n x satisfies the higher-order differential equation as follows:
r2
r1
r
1 − x2 Pλ,n x − 2λ 2r 1xPλ,n x n2 2λn − r2λ − r 2 Pλ,n x 0.
3.7
Proof. a It is from 9, 4.2.1. b It is from 9, 4.7.14. c It is from 6, Lemma 3.4. d It
is from 9, 8.9.7. e For r 0, it follows from 9, 7.33.5, and, for r 1, it comes from
b and the case of r 0. f It is from 6, Lemma 3.3 3.23. g Equation 3.7 comes from
a.
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Journal of Applied Mathematics
λ
λ
λ
Proposition 3.2 see 4. Let 0 < λ < 1. Let ξμ,n1 : cos θμ,n1 , μ 1, . . . , n 1 and yν,2n1 :
λ
cos ψν,2n1 , ν 1, . . . , 2n 1. Then, for μ 0, 1, . . . , n 2 and ν 0, 1, . . . , 2n 2,
λ
λ
λ
λ
θμ,n1 − θμ1,n1 ∼ ψν,2n1 − ψν1,2n1 ∼ n−1 ,
λ
λ
λ
3.8
λ
where ψ0,2n1 : θ0,n1 : π and ψ2n2,2n1 : θn2,n1 : 0.
Proposition 3.3 6, Proposition 2.3. Let 0 < λ < 1. Then, for all x ∈ −1, 1,
1 − x2 Eλ,n1 x − xEλ,n1
x n 12 Eλ,n1 x Iλ,n x,
3.9
where
Iλ,n x 8
λ
γn
n1/2
λ
n 1 − νναν,n Tn1−2ν x.
3.10
ν1
Then Iλ,n x is a polynomial of degree n − 1 satisfying
max |Iλ,n x| n2 .
x∈−1,1
3.11
Proposition 3.4 4, Theorem 2.1. Let 0 < λ < 1. Then, for n 0,
|Eλ,n1 x| n1−λ ϕ1−λ x 1
− 1 x 1.
3.12
Furthermore, Eλ,n1 1 1.
Proposition 3.5 6, Theorem 2.5. Let 0 < λ < 1.
λ
λ
a For all x ∈ ξ1,n1 , ξn1,n1 ,
Eλ,n1 x n2−λ ϕ−λ x.
λ
3.13
λ
Moreover, one has, for x ∈ −1, ξ1,n1 ∪ ξn1,n1 , 1,
λ
Eλ,n1 x ∼ n2 .
3.14
Eλ,n1 x n3−λ ϕ−1−λ x.
3.15
λ
b For all x ∈ ξ1,n1 , ξn1,n1 ,
Journal of Applied Mathematics
7
λ
λ
Moreover, one has, for x ∈ −1, ξ1,n1 ∪ ξn1,n1 , 1,
Eλ,n1 x ∼ n4 .
3.16
Proposition 3.6 6, Corollary 2.6. Let 0 < λ < 1. Then, for all x ∈ −1, 1,
1 − x2 Fλ,2n1 x − xFλ,2n1
x 2n2 21 λn 1 Fλ,2n1 x Jλ,n x.
λ
3.17
λ
Here, Jλ,n x is a polynomial of degree of 2n 1 defined in 4.37 such that, for x ∈ ξ1,n1 , ξn1,n1 ,
λ
|Jλ,n x| n2 ϕ1−2λ x
3.18
|Jλ,n x| n12λ .
3.19
λ
and, for x ∈ −1, ξ1,n1 ∪ ξn1,n1 , 1,
Proposition 3.7 6, Corollary 2.7. Let 0 < λ < 1.
λ
λ
a For all x ∈ ξ1,n1 , ξn1,n1 ,
Fλ,2n1 x nϕ−2λ x.
λ
3.20
λ
Moreover, one has, for x ∈ −1, ξ1,n1 ∪ ξn1,n1 , 1,
λ
Fλ,2n1 x ∼ n12λ .
3.21
Fλ,2n1 x n2 ϕ−1−2λ x.
3.22
λ
b For all x ∈ ξ1,n1 , ξn1,n1 ,
λ
λ
Moreover, one has, for x ∈ −1, ξ1,n1 ∪ ξn1,n1 , 1,
Fλ,2n1 x ∼ n32λ .
3.23
Proposition 3.8 4, Lemma 5.5. Let 0 < λ < 1. Then, for μ 1, 2, . . . , n 1,
λ
λ
Eλ,n1 ξμ,n1 ∼ n2−λ ϕ−λ ξμ,n1
3.24
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Journal of Applied Mathematics
and, for ν 1, 2, . . . , 2n 1,
λ
λ
Fλ,2n1 yν,2n1 ∼ nϕ−2λ yν,2n1 .
3.25
We now estimate the second derivatives at the zeros of Eλ,n1 and Fλ,2n1 .
Proposition 3.9 6, Theorem 2.9. Let 0 < λ < 1. Then, for μ 1, 2, . . . , n 1,
λ
λ
Eλ,n1 ξμ,n1 n2 ϕ−2 ξμ,n1
3.26
λ
λ
Fλ,2n1 yν,2n1 n1λ ϕ−2−λ yν,2n1 .
3.27
and, for ν 1, 2, . . . , 2n 1,
4. The Proofs of Main Results
In this section, we let 0 < λ < 1 and m n 1/2. A representation of Stieltjes polynomials
Eλ,n1 x is cf. 1, 10
⎧ λ
λ
⎨αn/2,n cos θ, n even
γn
λ
λ
Eλ,n1 cos θ α0,n cosn 1θ α1,n cosn − 1θ · · · 1 λ
⎩ α
2
,
n odd,
2 n1/2,n
4.1
where
λ
ν
λ
α0,n f0,n 1,
λ
λ
αμ,n fν−μ,n 0,
ν 1, 2, . . . ,
λ
λ
1−
,
1−
ν
nνλ
ν 1, 2, . . . ,
μ0
λ
fν,n :
λ
γn √
π
4.2
√
Γn 2λ
∼ πnλ−1 .
Γn λ 1
In the following, we state the asymptotic differential relation of the higher order of
Eλ,n1 .
Lemma 4.1. Let 0 < λ < 1. Then, for all x ∈ −1, 1 and r 2,
r
r−1
r−2
r−2
1 − x2 Eλ,n1 x 2r − 3xEλ,n1 x r − 22 − n 12 Eλ,n1 x Iλ,n x
λ
4.3
λ
and, for x ∈ ξ1,n1 , ξn1,n1 ,
r−2 Iλ,n x nr ϕ2−r x.
4.4
Journal of Applied Mathematics
9
Here, Iλ,n x is a polynomial of degree n − 1 defined in 3.10;
m
8 λ
n 1 − νναν,n Tn1−2ν x,
λ
Iλ,n x γn
4.5
ν1
such that
r−2 max Iλ,n x n2r−2 .
4.6
x∈−1,1
Proof. For r 2, 4.3 is obtained by r − 2 times differentiation of 3.9. Equation 4.6 follows
by 3.11 and the use of Markov-Bernstein inequality. Now, we prove 4.4. We know that the
Chebyshev polynomial Tn x satisfies the second-order differential equation
1 − x2 Tn x − xTn x n2 Tn x 0,
4.7
so we have, by r − 2 times differentiation of 4.7,
r
r−1
r−2
1 − x2 Tn x − 2r − 3xTn x − r − 22 − n2 Tn x 0.
4.8
Let for a nonnegative integer j 0,
Iλ,n,j x : −
m
8 λ j
1
−
ννα
n
x
.
T
ν,n
n1−2ν
λ
γn
4.9
ν1
Observe that in the view of Szegö’s result cf. 1
λ
λ
λ
α1,n < α2,n < α3,n < · · · < 0,
0
∞
λ
4.10
αν,n < 1.
ν0
j
λ
λ
Then, since |Iλ,n x| Iλ,n,j x note 4.10, we will prove that, for x ∈ ξ1,n1 , ξn1,n1 and
j 0,
Iλ,n,j x nj2 ϕ−j x
4.11
instead of 4.4. Since, from the proof of 6, Proposition 2.3,
0<−
m
8 λ
n 1 − νναν,n n2
λ
γn
4.12
ν1
and, for x cos θ,
sinn 1 − 2νθ −1
n1−2ν x n 1 − 2ν
nϕ x,
sin θ
T
4.13
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Journal of Applied Mathematics
we obtain that Iλ,n,0 x n2 and Iλ,n,1 x n3 ϕ−1 x. Using 4.8, we have, for 2 j n,
Iλ,n,j x −
≤−
m
8 λ
γn
ν1
m
8 λ
γn ν1
λ j
n 1 − νναν,n Tn1−2ν x
λ
n 1 − νναν,n
⎞
j − 2 2 − n 1 − 2ν2 2j − 3 |x| j−1
j−2
×⎝
Tn1−2ν x Tn1−2ν x⎠
1 − x2
1 − x2
⎛
4.14
2j − 3 |x|
n 12
I
Iλ,n,j−2 x.
x
λ,n,j−1
1 − x2
1 − x2
Therefore, 4.11 is proved by the mathematical induction on j. Consequently, we have 4.4.
r
We obtain pointwise upper bounds of Eλ,n1 x for two cases of an odd order and an
even order in the following.
λ
λ
Lemma 4.2. Let 0 < λ < 1 and r 2. Let x ∈ ξ1,n1 , ξn1,n1 . If r is even, then one has
r
x nr ϕ−r x|Eλ,n1 x| nr ϕ−r x,
Eλ,n1 x nr−2 ϕ−r xEλ,n1
4.15
and, if r is odd, then one has
r
x nr−1 ϕ−1−r x|Eλ,n1 x| nr ϕ−r x.
Eλ,n1 x nr−1 ϕ1−r xEλ,n1
λ
4.16
λ
Proof. Let r 2. From 4.3 and 4.4, we have, for x ∈ ξ1,n1 , ξn1,n1 ,
r
r−1
r−2
Eλ,n1 x ϕ−2 xEλ,n1 x n2 ϕ−2 xEλ,n1 x nr ϕ−r x
4.17
x n2 ϕ−2 x|Eλ,n1 x| n2 ϕ−2 x,
Eλ,n1 x ϕ−2 xEλ,n1
4.18
and especially
λ
λ
that is, we have 4.15 for r 2. From Proposition 3.2, we see 1 ξ1,n1 , 1 − ξn1,n1 1/n, so
λ
λ
we have, for x ∈ ξ1,n1 , ξn1,n1 ,
ϕ−1 x n.
4.19
Journal of Applied Mathematics
11
Then, from 4.17 with r 3 and 4.18, we know that
3
x n2 ϕ−4 x|Eλ,n1 x| n3 ϕ−3 x,
Eλ,n1 x n2 ϕ−2 xEλ,n1
4.20
that is, we have 4.16 for r 3. Assume that 4.15 and 4.16 hold for 3, 4, . . . , r − 1 times
differentiation. Let r be an even number. Then, we have, from 4.17, 4.19, and the assumptions for r − 1 and r − 2,
r
x nr ϕ−r x|Eλ,n1 x| nr ϕ−r x,
Eλ,n1 x nr−2 ϕ−r xEλ,n1
4.21
that is, we have 4.15. Similarly, we also have 4.16 for an odd r.
Lemma 4.3. Let −1 < x1 < x2 < · · · < xn < 1 and
P x : x − x1 x − x2 · · · x − xn .
4.22
Then, P x is a polynomial of degree of n − 1 and has distinct real n − 1 zeros in −1, 1. Moreover, if
one lets −1 < y1 < y2 < · · · < yn−1 < 1 be the zeros of P x, then {yi }ni1 is interlaced with the zeros
of P x that is, xi < yi < xi1 , i 1, 2, . . . , n − 1.
Proof. Since the sign of P xi is −1n−i , it is proved.
r
Lemma 4.4. Let r be a nonnegative integer. Then, Eλ,n1 x has distinct n1−r real zeros on −1, 1.
r
be the zeros of the polynomial Eλ,n1 x with
If one lets {xn1 r, i}n1−r
i1
−1 < xn1 r, 1 < xn1 r, 2 < · · · < xn1 r, n 1 − r < 1,
4.23
then one has, for 1 r n and k 1, . . . , n 1 − r,
xn1 0, k < xn1 r, k.
4.24
r
Proof. From Lemma 4.3, we know that Eλ,n1 has distinct real n 1 − r zeros on −1, 1. By the
interlaced zeros property of Lemma 4.3, we see that, for k 1, . . . , n 1 − r,
xn1 r − 1, k < xn1 r, k < xn1 r − 1, k 1.
4.25
Thus, 4.24 is proved.
Proof of Theorem 2.1. Let r 1. Equation 2.2 comes from 4.15, 4.16, 3.12, and 3.13.
From Propositions 3.4, 3.5, and 4.19, we have
r
max Eλ,n1 x ∼ n2r ,
x∈−1,1
r 1, 2.
4.26
12
Journal of Applied Mathematics
Hence, using the Markov-Bernstein inequality, we have 2.3. To prove 2.4, we will use the
mathematical induction. We use 3.14. The formula 2.3 holds for r 1 from 3.14. We
λ
λ
suppose that, for x ∈ −1, ξ1,n1 ∪ ξn1,n1 , 1 and r 2,
r−1
Eλ,n1 x ∼ n2r−1 .
4.27
λ
Then, by Lemma 4.4 and 3.8, we have, for x ∈ ξn1,n1 , 1 and r 2,
r
Eλ,n1 x
r−1
Eλ,n1 x
n2−r
k1
1
1
λ
x − xn1 r − 1, k
ξn1,n1 − xn1 r − 1, n 2 − r
1
λ
ξn1,n1 − xn1 0, n 2 − r
1
λ
ξn1,n1
λ
− ξn,n1
4.28
1
λ
λ
ξn1,n1 − ξn2−r,n1
n2 .
Here, the last inequality is obtained by Proposition 3.2, that is,
λ
λ
λ
λ
4.29
Eλ,n1 x Eλ,n1 xn2 ∼ n2r .
4.30
ξn1,n1 − ξn,n1 cos θn1,n1 − cos θn,n1 ∼ n−2 .
λ
Therefore, we have, for x ∈ ξn1,n1 , 1 and r 2,
r
r−1
λ
Hence, from 2.3, we have 2.4. For x ∈ −1, ξ1,n1 , the proof is similar.
λ
λ
Lemma 4.5. Let be a nonnegative integer and x ∈ ξ1,n1 , ξn1,n1 . Then,
xEλ,n1 xP x
n1 ϕ−−2λ x,
λ,n
Iλ,n xPλ,n x n1λ ϕ−−λ x,
n2 ϕ−2λ−−1 x.
E
λ,n1 xPλ,n x
4.31a
4.31b
4.31c
Journal of Applied Mathematics
13
Proof. a When 0, it is obvious from 3.5 and 3.12. Now, suppose 1. From 3.12,
2.2, and 3.5, we have
xEλ,n1 xP x
λ,n
−1qr
q
r1
Eλ,n1 xPλ,n x
nqr1 ϕ−qr−2λ x n1 ϕ−−2λ x.
4.32
−1qr
b From 4.4 and 3.5, we have
q
r
Iλ,n xPλ,n x
Iλ,n xPλ,n x qr
1λ −−λ
n
ϕ
4.33
x.
c Similarly to the proof of a, we have, from 2.2 and 3.5,
q1
r1
E
xP
x
xP
x
E
λ,n1
λ,n
λ,n
λ,n1
qr
2 −2λ−−1
n
ϕ
4.34
x.
Lemma 4.6. Let 0 < λ < 1. Then, for all x ∈ −1, 1 and r 2,
r
r−1
1 − x2 Fλ,2n1 x 2r − 5xFλ,2n1 x
r−2
r−2
r − 22 − n 12 − nn 2λ Fλ,2n1 x Jλ,n x
λ
4.35
λ
and, for x ∈ ξ1,n1 , ξn1,n1 ,
r−2 Jλ,n x nr ϕ−2λ−r3 x.
4.36
Here, Jλ,n x is a polynomial of degree of 2n 1 defined as follows:
Jλ,n x 2λxEλ,n1 xPλ,n
x
2 1 − x2 Eλ,n1
xPλ,n
x Iλ,n xPλ,n x.
4.37
Furthermore, one has
r−2 Jλ,n 1 n2λ2r−3 .
4.38
14
Journal of Applied Mathematics
Proof. Similarly to the proof of Lemma 4.1, 4.35 is obtained by r − 2 times differentiation of
the second-order differential relation with respect to Fλ,2n1 x, that is, 3.17. So it is sufficient
to prove 4.36 and 4.38. From 4.37, we know that
r−2
r−2
Jλ,n x 2λxEλ,n1 xPλ,n
x
r−2
2 1 − x2 Eλ,n1
Iλ,n xPλ,n xr−2 .
xPλ,n
x
λ
4.39
λ
By Lemma 4.5 a and b, we have, for x ∈ ξ1,n1 , ξn1,n1 ,
r−2 2λxEλ,n1 xP x
nr−1 ϕ−r2−2λ x,
λ,n
Iλ,n xPλ,n xr−2 nr−1λ ϕ−r2−λ x.
λ
4.40
λ
From 4.19 and Lemma 4.5 c, we have, for x ∈ ξ1,n1 , ξn1,n1 and r 4,
r−2 2 1 − x2 E
λ,n1 xPλ,n x
r−2 r−3 2
1−x
Eλ,n1 xPλ,n x
Eλ,n1 xPλ,n x
r−4 Eλ,n1
xPλ,n
x
4.41
nr ϕ−2λ−r3 x nr−1 ϕ−2λ−r2 x nr−2 ϕ−2λ−r3 x
nr ϕ−2λ−r3 x.
When r 2, 3, we can similarly obtain that
r−2 2 1 − x2 E
nr ϕ−2λ−r3 x.
xP
x
λ,n1
λ,n
4.42
Therefore, we have 4.36. On the other hand, from 2.4, 4.6, and 3.2, we know that, for a
nonnegative integer ,
Eλ,n1 1 ∼ n2 ,
Pλ,n 1
Iλ,n 1 n22 ,
∼ Pλ,n− 1 ∼ n
2λ−1
.
4.43
Journal of Applied Mathematics
15
Then, similarly to the proof of Lemma 4.5, we obtain that
xEλ,n1 xP x
λ,n
Iλ,n xPλ,n x n22λ1 ,
x1
E
xP
x
λ,n1
λ,n
n
22λ3
x1
n22λ1 ,
4.44
.
x1
Therefore, we have 4.38.
λ
λ
Lemma 4.7. Let 0 < λ < 1. Then, for r 2, if r is even, one has, for x ∈ ξ1,n1 , ξn1,n1 ,
r
x nr ϕ−r x|Fλ,2n1 x| nr ϕ1−2λ−r x
Fλ,2n1 x nr−2 ϕ−r xFλ,2n1
4.45
and, if r is odd, one has
r
x nr−1 ϕ−1−r x|Fλ,2n1 x| nr ϕ1−2λ−r x.
Fλ,2n1 x nr−1 ϕ1−r xFλ,2n1
4.46
Proof. Using 4.35 and 4.36, we obtain the result similarly to the proof of Lemma 4.2.
Proof of Theorem 2.2. Equation 2.5 comes from Lemma 4.7 and Proposition 3.7. We will show
2.6 and 2.7. From Proposition 3.7, we see
max Fλ,2n1
x ∼ n12λ .
4.47
x∈−1,1
Hence, using Markov-Bernstein inequality for Fλ,2n1
∈ P2n , we have 2.6. Now, we show
2.7. By Proposition 3.7 a, it is true for r 1. We suppose that, for r 2,
r−1
Fλ,2n1 x ∼ n2r2λ−3 ,
λ
λ
x ∈ −1, ξ1,n1 ∪ ξn1,n1 , 1 .
4.48
λ
As the proof of Theorem 2.1, we have, for r 2 and for x ∈ ξn1,n1 , 1,
r
Fλ,2n1 x
r−1
Fλ,2n1 x
4.49
n2 .
Therefore, we see that by induction with Proposition 3.7, 2.7 holds for every r 1, 2, 3, . . ..
λ
λ
Corollary 4.8. Let 0 < λ < 1 and r 2. Then, for x ∈ −1, ξ1,n1 ∪ ξn1,n1 , 1,
r−2 Jλ,n x n2r2λ−3 .
4.50
16
Journal of Applied Mathematics
Proof. Corollary 4.8 comes from 4.35, 2.7, and 3.8.
Proof of Theorem 2.3. Equation 2.8 comes from 4.15 and 3.24.
Lemma 4.9. For 1 μ n 1 and 1 ν 2n 1,
λ
λ
Pλ,n ξμ,n1 ∼ nλ−1 ϕ−λ ξμ,n1 ,
λ λ
Eλ,n1 xν,n ∼ n1−λ ϕ1−λ xν,n .
4.51
4.52
Proof. Since we know from 3.24 and 3.25 that
λ
λ
λ
λ
nϕ−2λ ξμ,n1 ∼ Fλ,2n1
ξμ,n1 Eλ,n1
ξμ,n1 Pλ,n ξμ,n1 λ
λ
∼ n2−λ ϕ−λ ξμ,n1 Pλ,n ξμ,n1 ,
4.53
4.51 is obviously proved. Similarly, since we have, from 3.4 and 3.25,
λ
λ λ
λ nϕ−2λ xν,n ∼ Fλ,2n1
xν,n Eλ,n1 xν,n Pλ,n
xν,n λ −λ−1
∼n ϕ
λ
xν,n
λ Eλ,n1 xν,n ,
4.54
4.52 is obtained.
Lemma 4.10. Let 0 < λ < 1 and r 1. Let r be an odd integer.
a For 1 μ n 1,
r λ λ
Pλ,n ξμ,n1 n2λr−2 ϕ−r−1 ξμ,n1 .
4.55
r
λ λ
Eλ,n1 xν,n nr ϕ−r xν,n .
4.56
λ
λ
Pλ,n ξμ,n1 n2λ−1 ϕ−2 ξμ,n1 .
4.57
b For 1 ν n,
Proof. a We know, from 3.3,
So 4.55 holds for k 1. Assume that, for k 1, 2, . . . , ,
2k−1 λ λ
ξμ,n1 n2λ2k−3 ϕ−2k ξμ,n1 .
Pλ,n
4.58
Journal of Applied Mathematics
17
Then, we have from 3.5, 3.7, and 4.58 that
21 λ ξμ,n1 Pλ,n
2 λ 2−1 λ λ
λ
ϕ−2 ξμ,n1 Pλ,n ξμ,n1 n2 ϕ−2 ξμ,n1 Pλ,n
ξμ,n1 4.59
λ
n2λ2−1 ϕ−2−2 ξμ,n1 .
Therefore, we have the result using the mathematical induction.
b For an odd integer r 1, we have from 3.6, 4.16, and 4.52
r
λ Eλ,n1 xν,n λ λ λ
nr−1 ϕ1−r x Eλ,n1
xν,n Eλ,n1 xν,n nr ϕ−r xν,n
4.60
λ
nr ϕ−r xν,n .
Lemma 4.11. Let r 2. If r is even, then
r λ λ
Pλ,n xν,n nλr−2 ϕ−λ−r−1 xν,n .
4.61
Proof. It is easily proved from 3.4 and 3.7.
Lemma 4.12. Let k be a positive integer. Then, one has, for 1 ν 2n 1,
2k λ
λ
Jλ,n yν,2n1 n2k1λ ϕ−2k−λ yν,2n1 .
4.62
Proof. From 4.37, we know that
Jλ,n x 2λxEλ,n1 xPλ,n
x
2 1 − x2 Eλ,n1
xPλ,n
x Iλ,n xPλ,n x.
λ
λ
4.63
λ
λ
From Lemma 4.5 a and b, we know that for x ∈ y1,2n1 , y2n1,2n1 ξ1,n1 , ξn1,n1 2k n2k1 ϕ−2k−2λ x
xEλ,n1 xP x
λ,n
Iλ,n xPλ,n x2k n2k1λ ϕ−2k−λ x.
4.64
18
Journal of Applied Mathematics
2k
xPλ,n
x
On the other hand, we estimate |1 − x2 Eλ,n1
follows:
| splitting into three terms as
2k 1 − x2 E
λ,n1 xPλ,n x
q1
r1
1 − x2 Eλ,n1 xPλ,n x
4.65
qr2k
2k−1 2k−2 E
.
O1 Eλ,n1
xPλ,n
x
xP
x
λ,n1
λ,n
λ
λ
Here, from Lemma 4.5 c, we have for x ∈ y1,2n1 , y2n1,2n1 2k−1 E
n2k1 ϕ−2λ−2k x,
λ,n1 xPλ,n x
2k−2 n2k ϕ−2λ−2k1 x n2k1 ϕ−2λ−2k x.
E
xP
x
λ,n
λ,n1
4.66
For the first term, we also split into two terms as follows:
q1
r1
1 − x2 Eλ,n1 xPλ,n x
qr2k
q1
r1
1 − x2 Eλ,n1 xPλ,n x
qr2k,q: even,r: even
qr2k,q: odd,r: odd
q1
r1
1 − x2 Eλ,n1 xPλ,n x
4.67
: A1 x A2 x.
From 2.2 and 4.55, we know that for even q and r
q1 λ λ
Eλ,n1 ξμ,n1 nq2−λ ϕ−q−λ ξμ,n1 ,
r1 λ λ
Pλ,n ξμ,n1 n2λr−1 ϕ−r−2 ξμ,n1 .
4.68
Also, we know from 4.56 and 3.5 that for even q and r
q1 λ λ
Eλ,n1 xν,n nq1 ϕ−q−1 xν,n ,
r1 λ λ
Pλ,n xν,n nλr ϕ−λ−r−1 xν,n .
4.69
Journal of Applied Mathematics
19
Then, we have
λ
A1 ξμ,n1 λ
ϕ2 ξμ,n1
q1 λ
r1
λ
Eλ,n1 ξμ,n1 Pλ,n ξμ,n1 qr2k,q: even,r: even
λ
n2k1λ ϕ−2k−λ ξμ,n1 ,
λ λ
A1 xν,n ϕ2 xν,n
4.70
q1 λ r1 λ Eλ,n1 xν,n Pλ,n xν,n qr2k,q: even,r: even
λ
n2k1λ ϕ−λ−2k xν,n .
Similarly, for odd q and r, we have, by 2.8 and 3.5,
q1 λ
r1
λ
λ
Eλ,n1 ξμ,n1 Pλ,n ξμ,n1 n2k1λ ϕ−2k−λ−2 ξμ,n1
4.71
and, by 2.2 and 4.61,
q1 λ r1 λ λ
Eλ,n1 xν,n Pλ,n xν,n n2k1λ ϕ−λ−2k−2 xν,n .
4.72
Thus, we have
λ
A2 ξμ,n1 λ
ϕ2 ξμ,n1
qr2k,q: odd,r: odd
q1 λ
r1
λ
Eλ,n1 ξμ,n1 Pλ,n ξμ,n1 λ
n2k1λ ϕ−2k−λ ξμ,n1 ,
λ λ
A2 xν,n ϕ2 xν,n
4.73
qr2k,q: odd,r: odd
λ
n2k1λ ϕ−λ−2k xν,n .
Therefore, we have the result.
q1 λ r1 λ Eλ,n1 xν,n Pλ,n xν,n 20
Journal of Applied Mathematics
Proof of Theorem 2.4. When r 2, 2.9 holds from 3.27. Let even r > 2, and suppose that
2.9 holds for r − 2. Since we know by 4.35, 2.5, 4.62, and 4.19
r
λ
λ
ϕ2 yν,2n1 Fλ,2n1 yν,2n1 r−1
r−2 λ
r−2
λ
λ
Fλ,2n1 yν,2n1 n2 Fλ,2n1 yν,2n1 Jλ,n yν,2n1 4.74
r−2
λ
λ
n2 Fλ,2n1 yν,2n1 nr−1λ ϕ−r2−λ yν,2n1 ,
we obtain, using mathematical induction,
r
λ
λ
λ
ϕ2 yν,2n1 Fλ,2n1 yν,2n1 nr−1λ ϕ−r2−λ yν,2n1 .
4.75
Therefore, 2.9 is proved.
Proof of Theorem 2.5. a From 4.3, we know that
λ
3
λ
λ
λ
ξμ,n1 O Eλ,n1
ξμ,n1
ϕ2 ξμ,n1 Eλ,n1 ξμ,n1 −n 12 Eλ,n1
λ
λ
ξμ,n1 .
O Eλ,n1 ξμ,n1 O Iλ,n
4.76
Therefore, we have, by 2.2, 3.24, and 3.26,
3
λ
λ
λ
Eλ,n1 ξμ,n1 −n 12 ϕ−2 ξμ,n1 Eλ,n1
ξμ,n1 O n3 .
4.77
Suppose that, for an integer 2,
2−1
λ
λ
λ
Eλ,n1 ξμ,n1 −1−1 n 12−1 ϕ−2−1 ξμ,n1 Eλ,n1
ξμ,n1 O n2−1 .
4.78
Then, from 4.3, we obtain
λ
21
λ
2−1
λ
2
λ
ϕ2 ξμ,n1 Eλ,n1 ξμ,n1 − n 12 Eλ,n1 ξμ,n1 O Eλ,n1 ξμ,n1
2−1
λ
2−1
λ
ξμ,n1 .
O Eλ,n1 ξμ,n1 O Iλ,n
4.79
Therefore, by 2.2, 2.8, and 4.4, we have
21
λ
λ
2−1
λ
Eλ,n1 ξμ,n1 −n 12 ϕ−2 ξμ,n1 Eλ,n1 ξμ,n1 O n21
−1 n 12 ϕ−2
λ
ξμ,n1
λ
Eλ,n1
ξμ,n1 O n21 .
4.80
Journal of Applied Mathematics
21
b Similarly to the proof of a, by 4.51, 3.3, and 3.7, we can obtain
2
Pλ,n
λ
λ
λ
ξμ,n1 −1 n n 2λ ϕ−2 ξμ,n1 Pλ,n ξμ,n1 O n2λ2−3 .
4.81
In addition, we see that, from 4.51,
2
Pλ,n
λ
λ
λ
ξμ,n1 −1 n 12 ϕ−2 ξμ,n1 Pλ,n ξμ,n1
λ
O n2−1 Pλ,n ξμ,n1 O n2λ2−3
4.82
λ
λ
−1 n 12 ϕ−2 ξμ,n1 Pλ,n ξμ,n1 O n2λ−2 .
Proof of Theorem 2.6. a From 3.9, we know that
λ
λ
λ
λ
λ
xν,n O Iλ,n xν,n .
ϕ2 xν,n Eλ,n1 xν,n −n 12 Eλ,n1 xν,n O Eλ,n1
4.83
Therefore, by 4.4 and 4.56, we have
λ
λ
λ
Eλ,n1 xν,n −n 12 ϕ−2 xν,n Eλ,n1 xν,n O n2 .
4.84
Then, we obtain from 4.3, 2.2, and 4.56 that
λ
2
λ
2−2
λ
ϕ2 xν,n Eλ,n1 xν,n −n 12 Eλ,n1 xν,n O n2 .
4.85
Therefore, we have the result inductively.
b From 3.7, we know that
λ
3
λ
λ
λ
λ
ϕ2 xν,n Pλ,n xν,n −nn 2λPλ,n
xν,n O Pλ,n
xν,n O Pλ,n
xν,n .
4.86
Therefore, by 3.4 and 4.61, we have
3
λ
λ
λ
Pλ,n xν,n −nn 2λϕ−2 xν,n Pλ,n
xν,n O nλ .
4.87
Suppose that, for an integer 2,
2−1
Pλ,n
λ
λ
λ
xν,n −1−1 n−1 n 2λ−1 ϕ−2−1 xν,n Pλ,n
xν,n O nλ2−4 .
4.88
Then from 3.5, 3.7, and 4.61
21
Pλ,n
λ
λ
λ
xν,n −1 n n 2λ ϕ−2 xν,n Pλ,n
xν,n O nλ2−2 .
4.89
22
Journal of Applied Mathematics
Here, we see that for 2
{nn 2λ} n 12 O n2−1 .
4.90
Hence, we obtain, from 3.4,
21
Pλ,n
λ
λ
λ
xν,n −1 n 12 ϕ−2 xν,n Pλ,n
xν,n
λ
xν,n O nλ2−2
O n2−1 Pλ,n
4.91
λ
λ
xν,n O nλ2−1 .
−1 n 12 ϕ−2 xν,n Pλ,n
λ
Lemma 4.13. Let 0 < ε < 1 and |yν,2n1 | 1 − ε. Then, for a nonnegative integer 0,
21
Jλ,n
λ
λ
λ
yν,2n1 221 −11 n 122 ϕ−2 yν,2n1 F yν,2n1 O nλ22 .
4.92
λ
Proof. Let |yν,2n1 | 1 − ε. From 4.37 and Lemma 4.5, we see that
21
Jλ,n
λ
yν,2n1
21
2
2 1 − x2 Eλ,n1
− 4x2 1 Eλ,n1
xPλ,n
x
xPλ,n
x
2−1
− 22 1 Eλ,n1
xPλ,n
x
2λ xEλ,n1 xPλ,n
x
21
Iλ,n xPλ,n x21 21 2 1 − x2 Eλ,n1
xPλ,n
x
λ
4.93
λ
xyν,2n1
O n22λ .
xyν,2n1
Here, we let 22 1Eλ,n1
xPλ,n
x2−1 0 when 0. To estimate the first term, we
split it into two terms as follows:
21
Eλ,n1
xPλ,n
x
0i21, i: even
0i21, i: odd
2 1 i1
22−i
Eλ,n1 xPλ,n
x.
i
4.94
Journal of Applied Mathematics
23
λ
Then, using |xν,n | 1 − ε, from 2.13 and 2.15, we obtain
0i21,i: odd
2 1 i1 λ 22−i λ xν,n
Eλ,n1 xν,n Pλ,n
i
λ
λ
xν,n
−11 n 122 ϕ−2−2 xν,n Fλ,2n1
0i21,i: odd
2 1
O nλ22 4.95
i
λ
λ
xν,n O nλ22
22 −11 n 122 ϕ−2−2 xν,n Fλ,2n1
and, from 4.56 and from 4.61,
0i21,
2 1 i1 λ 22−i λ xν,n O nλ21 .
Eλ,n1 xν,n Pλ,n
i
i: even
4.96
λ
Thus, we have, for |xν,n | 1 − ε,
21 Eλ,n1
xPλ,n
x
2 −1
2
1
λ
xxν,n
n 1
22 −2−2
ϕ
λ
xν,n
Fλ,2n1
λ
xν,n
O n
λ22
4.97
.
λ
Similarly, noting |ξμ,n | 1 − ε, from 2.10 and 2.12
0i21,i: even
2 1 i1 λ 22−i λ ξμ,n1
Eλ,n1 ξμ,n1 Pλ,n
i
λ
λ
ξμ,n1 O nλ22
22 −11 n 122 ϕ−2−2 ξμ,n1 Fλ,2n1
4.98
and, from 2.8 and 4.55,
0i21, i: odd
2 1 i1 λ 22−i λ ξμ,n1 O n2λ21 .
Eλ,n1 ξμ,n1 Pλ,n
i
4.99
λ
Then, we have, for |ξμ,n1 | 1 − ε,
Eλ,n1
xPλ,n
x
2 −1
2
21 1
λ
xξμ,n1
n 1
22 −2−2
ϕ
λ
ξμ,n1
Fλ,2n1
λ
ξμ,n1
O n
λ22
4.100
.
24
Journal of Applied Mathematics
λ
Therefore, we have, for |yν,2n1 | 1 − ε,
21 Eλ,n1
xPλ,n
x
2 −1
2
1
λ
xyν,2n1
n 1
22 −2−2
ϕ
λ
yν,2n1
Fλ,2n1
λ
yν,2n1
O n
λ22
4.101
.
λ
Thus, we have, for |yν,2n1 | 1 − ε,
21
Jλ,n
λ
λ
λ
yν,2n1 221 −11 n 122 ϕ−2 yν,2n1 Fλ,2n1
yν,2n1 O nλ22 .
4.102
Proof of Theorem 2.7. From 4.35, 3.25, 3.27, and Lemma 4.13, we have
λ
3
λ
ϕ2 yν,2n1 Fλ,2n1 yν,2n1
λ
λ
yν,2n1 Jλ,n
yν,2n1
−2n 12 Fλ,2n1
λ
λ
yν,2n1 O Fλ,2n1
yν,2n1
O nFλ,2n1
4.103
λ
yν,2n1 O nλ2 .
−3n 12 Fλ,2n1
Suppose that
2−1
λ
λ
λ
yν,2n1 O nλ2−2 .
Fλ,2n1 yν,2n1 c−1 −1−1 n 12−2 ϕ−22 yν,2n1 Fλ,2n1
4.104
Then, we obtain from 4.35
λ
21
λ
ϕ2 yν,2n1 Fλ,2n1 yν,2n1
λ
λ
yν,2n1 O nλ2
2c−1 22−1 −1 n 12 ϕ−22 yν,2n1 Fλ,2n1
λ
λ
yν,2n1 O nλ2 .
c −1 n 12 ϕ−22 yν,2n1 Fλ,2n1
Therefore, 2.16 is proved.
Acknowledgment
The authors thank the referees for many kind suggestions and comments.
4.105
Journal of Applied Mathematics
25
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2 S. Ehrich, “Asymptotic properties of Stieltjes polynomials and Gauss-Kronrod quadrature formulae,”
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