Convergence and Divergence of Higher-Order Hermite or ...downloads.hindawi.com/journals/isrn/2012/904169.pdf · with respect to the higher-order Hermite and Hermite-Fejer interpolation

International Scholarly Research NetworkISRN Mathematical AnalysisVolume 2012, Article ID 904169, 31 pagesdoi:10.5402/2012/904169

Research ArticleConvergence and Divergence of Higher-OrderHermite or Hermite-Fejér InterpolationPolynomials with Exponential-Type Weights

Hee Sun Jung,1 Gou Nakamura,2Ryozi Sakai,3 and Noriaki Suzuki3

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

2 Science Division, Center for General Education, Aichi Institute of Technology,Yakusa-cho, Toyota 470-0392, Japan

3 Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Correspondence should be addressed to Hee Sun Jung, [email protected]

Received 30 December 2011; Accepted 26 February 2012

Academic Editors: B. Ricceri and W. Yu

Copyright q 2012 Hee Sun Jung et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Let R � �−∞,∞�, and let wρ�x� � |x|ρe−Q�x�, where ρ > −1/2 and Q ∈ C1�R� : R → R� � �0,∞�is an even function. Then we can construct the orthonormal polynomials pn�w2ρ;x� of degreen for w2ρ�x�. In this paper for an even integer ν ≥ 2 we investigate the convergence theoremswith respect to the higher-order Hermite and Hermite-Fejér interpolation polynomials and relatedapproximation process based at the zeros {xk,n,ρ}nk�1 of pn�w2ρ;x�. Moreover, for an odd integerν ≥ 1, we give a certain divergence theorem with respect to the higher-order Hermite-Fejérinterpolation polynomials based at the zeros {xk,n,ρ}nk�1 of pn�w2ρ;x�.

1. Introduction

Let R � �−∞,∞�, and let Q ∈ C1�R� : R → R� � �0,∞� be an even function. Consider theweight w�x� � exp�−Q�x��, and define, for ρ > −1/2,

wρ�x� :� |x|ρw�x�, x ∈ R. �1.1�

Suppose that∫∞0 x

nwρ�x�dx < ∞, for all n � 0, 1, 2, . . .. Then we can construct the orthonormal

2 ISRN Mathematical Analysis

polynomials pn,ρ�x� � pn�w2ρ;x� of degree n for w2ρ�x�; that is,

∫∞

−∞pn,ρ�x�pm,ρ�x�w2ρ�x�dx � δm,n �Kronecker delta�. �1.2�

We write pn,ρ�x� by

pn,ρ�x� � γnxn � · · ·, γn � γn,ρ > 0, �1.3�

and denote the zeros of pn,ρ�x� by

−∞ < xn,n,ρ < xn−1,n,ρ < · · · < x2,n,ρ < x1,n,ρ < ∞. �1.4�

Let Pn denote the class of polynomials with degree at most n. For f ∈ C�R� we definethe higher-order Hermite-Fejér interpolation polynomial Ln�ν, f ;x� ∈ Pνn−1 based at the zeros{xk,n,ρ}nk�1 as follows:

L�i�n

(ν, f ;xk,n,ρ

)� δ0,if

(xk,n,ρ

)for k � 1, 2, . . . , n, i � 0, 1, . . . , ν − 1. �1.5�

We note that Ln�1, f ;x� is the Lagrange interpolation polynomial, Ln�2, f ;x� is theordinary Hermite-Fejér interpolation polynomial, and Ln�4, f ;x� is the Krylov-Stayermannpolynomial. For the general cases Kanjin and Sakai �1, 2� started to investigate the so-calledFreud-type weights. The fundamental polynomials hk,n,ρ�ν;x� ∈ Pνn−1 for the higher-orderHermite-Fejér interpolation polynomial Ln�ν, f ;x� are defined as follows:

hk,n,ρ�ν;x� � lνk,n,ρ�x�ν−1∑

i�0

ei�ν, k, n�(x − xk,n,ρ

)i,

lk,n,ρ�x� �pn(w2ρ;x

)

(x − xk,n,ρ

)p′n(w2ρ;xk,n,ρ

) ,

hk,n,ρ(ν;xp,n,ρ

)� δk,p, h

�i�k,n,ρ

(ν;xp,n,ρ

)� 0,

k, p � 1, 2, . . . , n, i � 1, 2, . . . , ν − 1.

�1.6�

Using them, we can write

Ln(ν, f ;x

)�

n∑

k�1

f(xk,n,ρ

)hk,n,ρ�ν;x�. �1.7�

ISRN Mathematical Analysis 3

Furthermore, we extend the operator Ln�ν, f ;x�. Let l be a nonnegative integer, and letν − 1 � l. For f ∈ Cl�R� we define the �l, ν�-order Hermite-Fejér interpolation polynomialsLn�l, ν, f ;x� ∈ Pνn−1 as follows: for each k � 1, 2, . . . , n,

Ln(l, ν, f ;xk,n,ρ

)� f

(xk,n,ρ

),

L�j�n

(l, ν, f ;xk,n,ρ

)� f �j�

(xk,n,ρ

), j � 1, 2, . . . , l,

L�j�n

(l, ν, f ;xk,n,ρ

)� 0, j � l � 1, l � 2, . . . , ν − 1.

�1.8�

Especially, Ln�0, ν, f ;x� is equal to Ln�ν, f ;x�, and for each P ∈ Pνn−1 we see Ln�ν−1, ν, P ;x� �P�x�; that is, for f ∈ C�ν−1��R�, Ln�ν − 1, ν, f ;x� is the Hermite interpolation polynomial. Thefundamental polynomials hs,k,n,ρ�l, ν;x� ∈ Pνn−1, k � 1, 2, . . . , n, of Ln�l, ν, f ;x� are defined by

hs,k,n,ρ�l, ν;x� � lνk,n,ρ�x�ν−1∑

i�s

es,i�ν, k, n�(x − xk,n,ρ

)i,

h�j�s,k,n,ρ

(l, ν;xp,n,ρ

)� δs,jδk,p, j, s � 0, 1, . . . , ν − 1, p � 1, 2, . . . , n.

�1.9�

Then we have

Ln(l, ν, f ;x

)�

n∑

k�1

l∑

s�0

f �s�(xk,n,ρ

)hs,k,n,ρ�l, ν;x�. �1.10�

For the ordinary Hermite and Hermite-Fejér interpolation polynomial Ln�1, 2, f ;x�,Ln�2, f ;x� and the related approximation process, Lubinsky �3� gave some interestingconvergence theorems.

Our purpose in this paper is to study Ln�ν, f ;x� and Ln�l, ν, f ;x� as certain analogiesof the Lubinsky theorems in �3� and the related approximation process for the exponential-type weights. Kasuga and Sakai �4–8� investigated the convergence and divergence theoremsfor the Freud-type weights. Then for an even integer ν ≥ 2 we give the convergence theoremsfor them; moreover, for an odd integer ν � 1, we obtain a certain divergence theoremwith respect to Ln�ν, f ;x�. In Section 1, we give the preliminaries for these studies, and inSection 2 we write some preliminary description. In Section 3 we report our theorems withsome lemmas, and in Section 4 we prove the theorems. Finally, in Section 5, for an odd integerν � 1 we obtain a certain divergence theorem with respect to Ln�ν, f ;x�.

In what follows we abbreviate several notations as xk,n :� xk,n,ρ, hkn�x� :� hk,n,ρ�ν, x�,lkn�x� :� lk,n,ρ�x�, hskn�x� :� hs,k,n,ρ�ν, x� and pn�x� :� pn,ρ�x� if there is no confusion. Forarbitrary nonzero real-valued functions f�x� and g�x�, we write f�x� ∼ g�x� if there existconstants C1, C2 > 0 independent of x such that C1g�x� � f�x� � C2g�x� for all x. Forarbitrary positive sequences {cn}∞n�1 and {dn}∞�1, we define cn ∼ dn similarly.

Throughout this paper C,C1, C2, . . . denote positive constants independent of n, x, t, orpolynomials Pn�x�, and the same symbol does not necessarily denote the same constant indifferent occurrences.


2. Preliminaries

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.

Definition 2.1 �see �9��. Let Q : R → R� be a continuous even function satisfying thefollowing properties.

�a� Q′�x� is continuous in R and Q�0� � 0.

�b� Q′′�x� exists and is positive in R \ {0}.�c� limx→∞Q�x� � ∞.�d� The function

T�x� :�xQ′�x�Q�x�

, x /� 0, �2.1�

is quasi-increasing in �0,∞�, with T�x� � Λ > 1 for x ∈ R� \ {0}.�e� There exists C1 > 0 such that

Q′′�x�|Q′�x�| � C1

|Q′�x�|Q�x�

, a.e. x ∈ R \ {0}. �2.2�

Then we say thatw � exp�−Q� is in the classF�C2�. Besides, if there exist a compactsubinterval J�� 0� of R and C2 > 0 such that

Q′′�x�|Q′�x�| � C2

|Q′�x�|Q�x�

, a.e. x ∈ R \ J, �2.3�

then we say that w � exp�−Q� is in the class F�C2��.

Example 2.2. We present some typical examples of Q�x� satisfying w � exp�−Q� ∈ F�C2��.

�1� If a continuous exponential Q�x� satisfies

1 < Λ1 ��xQ′�x��′

Q′�x�� Λ2 �2.4�

with some constants Λi, i � 1, 2, then w�x� � exp�−Q�x�� is called a Freud weight.The class F�C2�� contains the Freud weights.

�2� �See �9�� For α > 1 and a nonnegative integer r, we put

Q�x� � Qr,α�x� :� expr(|x|α) − expr�0�, �2.5�


where, for r � 1,

expr�x� :� exp(exp

(exp

(· · · exp x) · · · )) �r-times� �2.6�

and exp0�x� :� x.

�3� �See �10�� For m � 0, α � 0 with α �m > 1, we put

Q�x� � Qr,α,m�x� :� |x|m{expr

(|x|α) − α∗expr�0�}, �2.7�

where for r > 0 we suppose α∗ � 0 if α � 0; α∗ � 1 otherwise. For r � 0 we supposem > 1 and α � 0.

�4� �See �10�� For α > 1, we put

Q�x� � Qα�x� :� �1 � |x|�|x|α − 1. �2.8�

For x > 0, we define the Mhaskar-Rakhmanov-Saff number ax by the equation

x �2π

∫1

0

axuQ′�axu�

�1 − u2�1/2du. �2.9�

We have the following estimates for the coefficients es,i�ν, k, n��ei�ν, k, n� :� e0,i�ν, k, n�� in�1.6� or �1.9�.

Lemma 2.3 �see �11, Theorem 2.6��. Let w�x� � exp�−Q�x�� ∈ F�C2��. For each k � 1, 2, . . . , nand s � 0, 1, . . . , ν − 1, we have e0,0�ν, k, n� � e0�ν, k, n� � 1,

|es,i�ν, k, n�| � C

⎛

⎜⎝

n√a22n − x2k,n

⎞

⎟⎠

i−s

, s � 1, 2, . . . , ν − 1, i � s, s � 1, . . . , ν − 1. �2.10�

If we consider the higher-order Hermite-Fejér interpolation polynomial Ln�ν, f ;x� ona certain finite interval, then we can see a remarkable difference between the parity of ν, forexample, the Lagrange interpolation polynomial Ln�1, f ;x� and the ordinary Hermite-Fejérinterpolation polynomial Ln�2, f ;x� ��12–17��. Also, we can see a similar phenomenon in thecase of the infinite intervals ��1, 2, 4–8��. To describe these aspects, however, we need a furtherstrengthened definition for ν � 2 than Definition 2.1.

Definition 2.4. Let w�x� � exp�−Q�x�� ∈ F�C2��, and let ν � 1 be an integer. Assume thatQ�x� is a ν-times continuously differentiable function on R and satisfies the following.

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


�b� There exist constants Ci > 0 such that

∣∣∣Q�i�1��x�

∣∣∣ � Ci

∣∣∣Q�i��x�

∣∣∣|Q′�x�|Q�x�

, x ∈ R \ {0}, i � 1, 2, . . . , ν. �2.11�

�c� There exist 0 � δ < 1 and c1 > 0 such that

Q�ν�1��x� � C(1x

)δ, x ∈ �0, c1�. �2.12�

Then we say that w�x� � exp�−Q�x�� is in the class Fν�C2��.�d� Suppose one of the following.

�d-1� Q′�x�/Q�x� is quasi-increasing on a certain positive interval �c2,∞�.�d-2� Q�ν�1��x� is nondecreasing on a certain positive interval �c2,∞�.�d-3� There exist constants C > 0 and 0 � δ < 1 such that Q�ν�1��x� � C�1/x�δ on

�0,∞�.Then one says that w�x� � exp�−Q�x�� is in the class F̃ν�C2��.

Example 2.5 �cf. �10, Theorem 3.1��. Let ν be a positive integer, and let Qr,α,m be defined in�2.7�.

�1� Let m and α be nonnegative even integers with m � α > 1. Then w�x� �exp�−Qr,α,m� ∈ Fν�C2��.�a� If r > 0, then we see that Q′r,α,m�x�/Qr,α,m�x� is quasi-increasing on a certain

positive interval �c1,∞� and Qr,0,m�x� is nondecreasing on �0,∞�.�b� If r � 0, then we see that Q0,0,m�x�, m � 2, is nondecreasing on �0,∞�.

Hence, w�x� � exp�−Qr,α,m� ∈ F̃ν�C2��.�2� Let m � α − ν > 0. Then w�x� � exp�−Qr,α,m� ∈ Fν�C2��, and one has the following.

�c� If r � 2 and α > 0, then there exists a constant c1 > 0 such thatQ′r,α,m�x�/Qr,α,m�x� is quasi-increasing on �c1,∞�.

�d� Let r � 1. If α � 1, then there exists a constant c2 > 0 such thatQ′1,α,m�x�/Q1,α,m�x� is quasi-increasing on �c2,∞�, and, if 0 < α < 1, thenQ′1,α,m�x�/Q1,α,m�x� is quasi-decreasing on �c2,∞�.

�e� Let r � 1 and 0 < α < 1, then Q�ν�1�1,α,m�x� is nondecreasing on a certain positiveinterval on �c2,∞�.Hence, w�x� � exp�−Qr,α,m� ∈ F̃ν�C2��.

Definition 2.6. One uses the following notation:

ϕu�x� �

⎧⎪⎨

⎪⎩

auu

1 − |x|/a2u√1 − |x|/au � δu

, |x| � au;

ϕu�au�, au < |x|,δu � �uT�au��−2/3, u > 0. �2.13�


Lemma 2.7 �see �18, Corollary 4.5��. Let wρ�x� � |x|ρexp�−Q�x��, exp�−Q� ∈ F̃ν�C2��. Ifxk,n /� 0 and |xk,n| � an�1 � δn�, then e0�ν, k, n� � 1 and, for i � 1, 2, . . . , ν − 1,

|ei�ν, k, n�| � C{T�an�an

�∣∣Q′�xk,n�

∣∣ �

1|xk,n|

}〈i〉( na2n − |xk,n| �

T�an�an

)i−〈i〉, �2.14�

where

〈i〉 �{1, i: odd,0, i: even.

�2.15�

For xk,n � 0 one has

e0�ν, k, n� � 1, |ei�ν, k, n�| � C(

n

an

)i, i � 1, 2, . . . , ν − 1. �2.16�

Remark 2.8. In �19, Theorem 2.2� we see that x1,n < an if n is large enough. ThereforeLemma 2.7 holds for all xk,n, k � 1, 2, . . . , n.

Levin and Lubinsky �see �9, Lemma 3.7�� showed that there exists C > 0 such that forsome ε > 0 and for large enough t,

T�at� � Ct2−ε. �2.17�

In �20� we have the following estimations.

Lemma 2.9 �see �20, Theorem 1.6��. Let w � exp�−Q� ∈ F�C2��.�1� Let T�x� be unbounded. Then, for any η > 0, there exists C�η� > 0 such that, for t � 1,

at � C(η)tη. �2.18�

�2� Let λ :� C1 be the constant in Definition 2.1(e), that is,

Q′′�x�Q′�x�

� λQ′�x�

Q�x�, a.e. x ∈ R \ {0}. �2.19�

If λ > 1, then there exists C�λ, η� such that

T�at� � C(λ, η

)t2�η�λ−1�/�λ�1�, t � 1, �2.20�

and, if 0 < λ � 1, then for any η > 0 there exists C�λ, η� such that

T�at� � C(λ, η

)tη, t � 1. �2.21�


Remark 2.10. �1� If T�x� is bounded, then w is called a Freud-type weight, and, if T�x� isunbounded, then w is called an Erdős-type weight.

�2� In �2.20� and �2.21�, we set 0 < η < 2 and

ε �

⎧⎪⎨

⎪⎩

2 − η, 0 < λ � 1,2(2 − η)

�λ � 1�, λ > 1.

�2.22�

Then �2.17� holds.�3� If

lim supx→∞

Q�x�Q′′�x�

�Q′�x��2� 1, �2.23�

then we have �2.21�. Note that all the examples in Example 2.5 satisfy this inequality.�4� For the Freud-type exponent Q�x� � |x|m,m > 1, we have

T�at� � m, at ∼ t1/m. �2.24�

�5� The inequality �2.18� implies

0 < C � nan

{T�an�an

}ν−1. �2.25�

3. Theorems

In the rest of this paper we assume the following for the weight w.

Assumption 3.1. Consider the weightwρ�x� � |x|ρ exp�−Q�x��, exp�−Q�x�� ∈ F̃ν�C2��, ρ � 0.�a� �cf. �20, Theorem 1.4�� If T�x� is bounded, then we suppose, for δ in �2.12�,

an � Cn1/�1�ν−δ�. �3.1�

�b� There exist 0 � γ < 1 and C�γ� > 0 such that

T�an� � C(γ)nγ ; �3.2�

here, if T�x� is bounded, that is, a Freud-type weight, then we set γ � 0, and if T�x�is unbounded, that is, an Erdős-type weight, then we set 0 < γ < 1. Define

εn :�

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

ann,

T�an�an

< 1,

1n1−γ

,T�an�an

� 1.�3.3�


Remark 3.2. �1� If T�x� is unbounded, then �3.1� holds because of Lemma 2.9 �2.21�.�2� �3.2� holds for

γ �2(η � λ − 1)

λ � 1, 0 < λ < 3. �3.4�

�3� In �3.3�we note that εn log n → 0 as n → ∞.

We shall state our theorems. Put

Xn(ν, f ;x

):�

n∑

j�1

f(xj,n,ρ

)lνj,n,ρ�x�

ν−2∑

i�0

ei(ν, j, n

)(x − xj,n,ρ

)i,

Yn(ν, f ;x

):�

n∑

j�1

f(xj,n,ρ

)lνj,n,ρ�x�eν−1

(ν, j, n

)(x − xj,n,ρ

)ν−1,

Zn(l, ν, f ;x

):�

n∑

j�1

l∑

s�1

f �s�(xj,n,ρ

)lνj,n,ρ�x�

ν−1∑

i�s

esi(ν, j, n

)(x − xj,n,ρ

)i.

�3.5�

Furthermore, we consider the class G � {gs ∈ C�R�, s � l � 1, l � 2, . . . , ν − 1} and construct thefollowing interpolation polynomial:

Wn�l, ν, G;x� :�n∑

j�1

ν−1∑

s�l�1

gs(xj,n,ρ

)lνj,n,ρ�x�

ν−1∑

i�s

esi(ν, j, n

)(x − xj,n,ρ

)i. �3.6�

Then we have

Ln(ν, f ;x

)� Xn

(ν, f ;x

)� Yn

(ν, f ;x

),

Ln(l, ν, f ;x

)� Ln

(ν, f ;x

)� Zn

(l, ν, f ;x

),

L̃n(l, ν, f ⊕G;x) :� Ln

(ν, f ;x

)� Zn

(l, ν, f ;x

)�Wn�l, ν, G;x�.

�3.7�

Define

Φ�x� :�1

�1 �Q�x��2/3T�x�. �3.8�

Here we note that, for some d > 0,

Φ�x� ∼ Q�x�1/3

xQ′�x�, |x| � d > 0. �3.9�

Moreover, we define

Φn�x� :� max{δn, 1 − |x|

an

}, n � 1, 2, 3, . . . . �3.10�


Lemma 3.3. Let w � exp�−Q� ∈ F�C2��. Let L > 0 be fixed. Then one has the following:

�a� (See [9, Lemma 3.5(a)]) Uniformly for t > 0,

aLt ∼ at. �3.11�

�b� (See [9, Lemma 3.5(b)]) Uniformly for t > 0,

Q�j��aLt� ∼ Q�j��at�, j � 0, 1. �3.12�

Moreover,

T�aLt� ∼ T�at�. �3.13�

�c� (See [9, Lemma 3.11 (3.52)]) Uniformly for t > 0,

∣∣∣∣1 −aLtat

∣∣∣∣ ∼1

T�at�. �3.14�

�d� (See [9, Lemma 3.4 (3.17), (3.18)]) Uniformly for t > 0, one has

Q�at� ∼ t√T�at�

, Q′�at� ∼t√T�at�at

. �3.15�

Lemma 3.4. For x ∈ R, we have

Φ�x� � CΦn�x�, n � 1. �3.16�

Proof. Let x � au, u � 1. By Lemma 3.3�d�we have

u ∼ Q�au�√T�au�. �3.17�

So, we have

δ−1u ∼ Q�au�2/3T�au� �auQ

′�au�

Q�au�1/3�

xQ′�x�Q1/3�x�

∼ Φ−1�au�. �3.18�


Now, if u � n/2, then

1 − auan

� 1 − an/2an

∼ 1T�an�

(by Lemma 3.2�c�

)

� 1�nT�an��2/3

� δn(by �2.17�

).

�3.19�

So, we have

Φn�x� � 1 − auan

� 1 − aua2u

∼ 1T�au�

(by Lemma 3.2�c�

)

� 1�uT�au��2/3

� δu ∼ Φ�x�(by �2.17� and �3.18�

).

�3.20�

Let n/2 < u < n. Then we have

Φn�x� � δn ∼ δu ∼ Φ�x�(by Lemmas 3.2�a�, �2.17�, and �3.18�

). �3.21�

Let C�f� denote a positive constant depending only on f .

Assumption 3.5. Letwρ�x� � |x|ρw�x� andw�x� � exp�−Q�x�� ∈ Fν�C2��, ρ � 0. Suppose thefollowing.

�A-1� Let f ∈ C�R� satisfy that, for a given 0 < δ < 1,

∣∣f�x�∣∣wν−δ�x�

{∣∣Q′�x�∣∣ �

1|x|

}� C

(f), x ∈ R, �3.22�

where we suppose lim sup|x|→ 0|f�x�/x| � C�f�.

�A-2� Let f ∈ Cl�R� for a certain 0 < l � ν − 1. Then we suppose that f satisfies

∣∣∣f �s��x�∣∣∣{Φ−3/4�x�wρ�x�

}ν� C

(f), x ∈ R, s � 1, 2, . . . , l. �3.23�

�A-3� Let G � {gs ∈ C�R�, s � l � 1, l � 2, . . . , ν − 1}. Then we suppose that there exists aconstant C > 0 such that

∣∣gs�x�∣∣{Φ−3/4�x�wρ�x�

}ν� C�G�, x ∈ R, s � l � 1, l � 2, . . . , ν − 1. �3.24�


Remark 3.6. In �3.22�, we have the following.

�1� f�0� � 0 and

∣∣f�x�

∣∣{Φ−3/4�x�wρ�x�

}ν(∣∣Q′�x�

∣∣ �

1|x|

)� C

(f), x ∈ R \ {0}. �3.25�

�2� For some positive constant C, we have |Q′�x�| � 1/|x| � C. Hence, from �3.22�, itfollows that

∣∣f�x�

∣∣{Φ−3/4�x�wρ�x�

}ν� C

(f), x ∈ R. �3.26�

We have a chain of results under Assumption 3.1.

Proposition 3.7. Let ν � 1, 2, 3, . . .. For f�x� satisfying �3.26�, one has

∥∥∥∥

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}νXn

(ν, f ;x

)∥∥∥∥L∞�R�

� C(f). �3.27�


∥∥∥∥

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}νYn(ν, f ;x

)∥∥∥∥L∞�R�

� C(f)εn log n, �3.28�

where εn is defined by �3.3�.


∥∥∥∥

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}νZn(l, ν, f ;x

)∥∥∥∥L∞�R�

� C(f)an logn

n, �3.29�

and for f�x� satisfying �3.24� one has

∥∥∥∥

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}νWn�l, ν, G;x�

∥∥∥∥L∞�R�

� C�G�an logn

n. �3.30�

Proposition 3.10. Let ν � 1, 2, 3, . . .. Let P ∈ Pm be fixed. Then one has∥∥∥∥

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}ν�Ln�ν, P, x� − P�x��

∥∥∥∥L∞�R�

−→ 0 as n −→ ∞. �3.31�

Proposition 3.11. Let ν � 2, 4, 6, . . .. Let P ∈ Pm with P�0� � 0 be fixed. Then one has∥∥∥∥

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}ν�Xn�ν, P ;x� − P�x��

∥∥∥∥L∞�R�

−→ 0 as n −→ ∞. �3.32�


Theorem 3.12. Let ν � 2, 4, 6, . . .. For f�x� satisfying �3.22�, one has

∥∥∥∥

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}ν(Ln(ν, f ;x

) − f�x�)∥∥∥∥L∞�R�

−→ 0 as n −→ ∞. �3.33�

Theorem 3.13. Let ν � 2, 4, 6, . . .. For f�x� satisfying �3.22� and �3.23�, one has

∥∥∥∥

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}ν(Ln(l, ν, f ;x

) − f�x�)∥∥∥∥L∞�R�

−→ 0 as n −→ ∞. �3.34�

Corollary 3.14. Let ν � 2, 4, 6, . . .. For f�x� satisfying �3.22�, one has

∥∥∥∥

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}ν(Xn

(ν, f ;x

) − f�x�)∥∥∥∥L∞�R�

−→ 0 as n −→ ∞. �3.35�

Theorem 3.15. Let ν � 2, 4, 6, . . .. For f�x� satisfying �3.22�, �3.23� and �3.24�, one has

∥∥∥∥

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}ν(L̃n(l, ν, f ⊕G;x) − f�x�

)∥∥∥∥L∞�R�

−→ 0 as n −→ ∞. �3.36�

Define

In[k,Hn

(f)]

:�∫∞

−∞k�x�Hn

(f ;x

)dx, �3.37�

where Hn�f� equals to one of the following:

Ln(ν, f

), Ln

(l, ν, f

), L̃n

(l, ν, f ⊕G), Xn

(ν, f

). �3.38�

One also defines

I[k, f

]:�∫∞

−∞k�x�f�x�dx. �3.39�

Theorem 3.16. Let ν � 2, 4, 6, . . .. Let f�x� satisfy �3.22�, �3.23� and �3.24�. If

∫∞

−∞k�x�

{Φ3/4�x�w�x�

(|x| � an

n

)ρ}−νdx < ∞, n � 1, 2, 3, . . . , �3.40�

then we have

limn→∞

In[k,Hn

(f)]

� I[k, f

]. �3.41�


For example, we can take k�x� in the following way. If T�x� is unbounded, we have,for Δ ∈ R,

∫∞

1�1 � |x|�−ΔΦ3ν/4�x�dx �

∫∞

1�1 � |x|�−Δ

(1

T�x�Q�x�2/3

)3ν/4dx < ∞ �3.42�

�see �10, Lemma 2.1�b��. Then we set

k�x� :� �1 � |x|�ρν−Δ{Φ3/2�x�w�x�

}ν. �3.43�

If T�x� is bounded andw�x� � exp�−|x|β�, β > 1, then we takeΔ as νβ/2�Δ > 1 so that �3.43�can hold. Then k�x� defined by �3.43� satisfies �3.40�.

4. Proof of Theorems

For constants C,C1 > 0, the same symbol does not necessarily denote the same constant indifferent occurrences.

Lemma 4.1. One has the following. �1� (See [19, Theorem 2.3]) Let w�x� � exp�−Q�x�� ∈ F�C2�and ρ > −1/2. Then, uniformly for n � 1 one has

supx∈R

∣∣pn,ρ�x�w�x�∣∣(|x| � an

n

)ρ∣∣∣x2 − a2n∣∣∣1/4 ∼ 1, �4.1�

and for w�x� � exp�−Q�x�� ∈ F�C2��,

supx∈R

∣∣pn,ρ�x�w�x�∣∣(|x| � an

n

)ρ∼ a−1/2n �nT�an��1/6. �4.2�

�2� (See [19, Theorem 2.5(d)]) Let w ∈ F�C2�� and ρ > −1/2. Let 1 � j � n − 1 andx ∈ �xj�1,n, xj,n�. Then

∣∣pn�x�w�x�∣∣(|x| � an

n

)ρ∼ min{∣∣x − xj,n

∣∣,∣∣x − xj�1,n

∣∣}ϕn(xj,n

)−1∣∣∣a2n − x2j,n∣∣∣−1/4

. �4.3�

�3� (See [19, Theorem 2.5(c)]) Let w ∈ F�C2�� and ρ > −1/2. Then one has

maxx∈R

∣∣∣∣∣pn�x�w�x��|x| � an/n�ρ

(x − xj,n

)p′n(xj,n

)w(xj,n

)(∣∣xj,n∣∣ � an/n

)ρ

∣∣∣∣∣

� maxx∈R

∣∣∣∣ljn�x�w�x�(|x| � an

n

)ρ∣∣∣∣w−1(xj,n

)(∣∣xj,n

∣∣ �ann

)ρ∼ 1.

�4.4�


�4� (See [19, Theorem 2.5(a)]) Let w ∈ F�C2�� and ρ > −1/2. For 1 � j � n we have

∣∣p′nw∣∣(xj,n

)(∣∣xj,n

∣∣ �ann

)ρ∼ ϕ−1n

(xj,n

)∣∣∣a2n − x2j,n

∣∣∣−1/4

. �4.5�

Lemma 4.2 �see �19, Theorem 2.2��. Let w�x� � exp�−Q�x�� ∈ F�C2�� and ρ > −1/2. For thezeros xj,n � xj,n,ρ, one has the following:

�1� For n � 1 and 1 � j � n − 1,

xj,n − xj�1,n ∼ ϕn(xj,n

),

ϕn(xj,n

) ∼ ϕn(xj�1,n

)�see ��19, Lemma A.1�A.3��.

�4.6�

�2� For the minimum positive zero x�n/2�,n (�n/2� is the largest integer � n/2), one has

x�n/2�,n ∼ ann−1, �4.7�

and for large enough n,

1 − x1,nan

∼ δn. �4.8�

�3� (See [19, Lemma 4.7]) bn � γn−1/γn ∼ an ∼ x1,n, where γn is defined by �1.3�.

Lemma 4.3. Let w ∈ F�C2��. Then there exist C1, C2 > 0 such that

supx∈R

∣∣pn,ρ�x�w�x�∣∣(|x| � an

n

)ρΦ1/4�x� � C1 sup

x∈R

∣∣pn,ρ�x�w�x�∣∣(|x| � an

n

)ρΦ1/4n �x�

� C2a−1/2n .�4.9�

Proof. The first inequality follows from Lemma 3.4. We show the second inequality. Noting�4.8�, from Lemma 4.1 �4.1�we have

C � sup|x|�x1,n

∣∣pn,ρ�x�w�x�∣∣(|x| � an

n

)ρ∣∣∣x2 − a2n∣∣∣1/4

∼ sup|x|�x1,n

∣∣pn,ρ�x�w�x�∣∣(|x| � an

n

)ρa1/2n Φ

1/4n �x�.

�4.10�


From �4.2�we see that

a−1/2n � C sup|x|>x1,n

∣∣pn,ρ�x�w�x�

∣∣(|x| � an

n

)ρδ1/4n

∼ C sup|x|>x1,n

∣∣pn,ρ�x�w�x�

∣∣(|x| � an

n

)ρΦ1/4n �x�.

�4.11�

Therefore we have the result.

Proof of Proposition 3.7. We recall the definition of Xn�ν, f ;x�:

Xn(ν, f ;x

)�

n∑

j�1

f(xj,n

)lνjn�x�

ν−2∑

i�0

ei(ν, j, n

)(x − xj,n

)i

�ν−2∑

i�0

n∑

j�1

f(xj,n

)lνjn�x�

(x − xj,n

)iei(ν, j, n

).

�4.12�

Using Lemma 2.3, we may estimate, for i � 0, 1, . . . , ν − 2 and 1 � j � n,

Ai,j�x� :�(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν∣∣f(xj,n

)∣∣∣∣ljn�x�∣∣ν∣∣x − xj,n

∣∣i

⎛

⎜⎝

n√a22n − x2j,n

⎞

⎟⎠

i

. �4.13�

Let �xm�1,n � xm,n�/2 < x � xm,n or xm,n � x < �xm−1,n � xm,n�/2. For simplicity, we assumex > 0, and let x0,n :� x1,n � δϕn�x1,n� for a fixed δ > 0 small enough. Then we can assume thatthere exists a constant c > 0 such that

x0,n :� x1,n � δϕn�x1,n� < an − cδn. �4.14�

Assume that x < x0,n. Then we first estimate Ai,m�x�. By Lemma 4.1�3� and the definition ofϕn�x�, we have

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν|lmn�x�|ν � CΦ3ν/4�x�

(w�xm,n�

(|xm,n| � an

n

)ρ)ν, �4.15�

|x − xm,n|i⎛

⎜⎝

n√a22n − x2m,n

⎞

⎟⎠

i

� C

⎛

⎜⎝

nϕn�xm,n�√a22n − x2m,n

⎞

⎟⎠

i

� C(√

1 − |xm,n|/a2n√1 − |xm,n|/an

)i, �4.16�

and by Remark 3.6

∣∣f�xm,n�∣∣ � C

(f)Φ3ν/4�xm,n�

(w�xm,n �

(|xm,n| � an

n

)ρ)−ν. �4.17�


Therefore, we have

Ai,m�x� � C(f)Φ3ν/4�x�Φ3ν/4�xm,n�

(√1 − |xm,n|/a2n

√1 − |xm,n|/an

)i

� C(f)Φ3ν/4�x�

(max

{1 − |xm,n|

an, δn

})3ν/4(√1 − |xm,n|/a2n√1 − |xm,n|/an

)i

� C(f)Φ3ν/4�x�

(1 − |xm,n|

an

)3ν/4−i/2⎛

⎝

√

1 − |xm,n|a2n

⎞

⎠

i

� C(f).

�4.18�

Next, we estimate∑

1�j�n,j /�m Ai,j�x�. For 1 � j � n, j /�m, we have, by Lemma 4.3 andLemma 4.1�4�,

(Φ3/4�x�w�x��|x| � an/n�ρ

)ν∣∣ljn�x�∣∣ν∣∣x − xj,n

∣∣i

� Φν/2�x�

∣∣∣∣∣pn�x�Φ1/4�x�w�x��|x| � an/n�ρp′n(xj,n

)w(xj,n

)(∣∣xj,n∣∣ � an/n

)ρ

∣∣∣∣∣

ν (w(xj,n

)(∣∣xj,n∣∣ � an/n

)ρ)ν

∣∣x − xj,n∣∣ν−i

� Φν/2�x�a−ν/2n ϕνn

(xj,n

)∣∣∣a2n − x2j,n∣∣∣ν/4

(w(xj,n

)(∣∣xj,n∣∣ � an/n

)ρ)ν

∣∣x − xj,n∣∣ν−i

.

�4.19�

Then, since we know from Remark 3.6 that

∣∣f(xj,n

)∣∣ � C(f)Φ3ν/4

(xj,n

)(w�xj,n�

(∣∣xj,n∣∣ �

ann

)ρ)−ν, �4.20�

we have, by Lemma 3.4,

∑

j /�m

Ai,j�x� � C(f)∑

j /�m

Φν/2n �x�Φ3ν/4n

(xj,n

)ϕνn(xj,n

)

×(

1 −∣∣xj,n

∣∣

an

)ν/4(1

∣∣x − xj,n∣∣

)ν−i⎛

⎜⎝

n√a22n − x2j,n

⎞

⎟⎠

i

.

�4.21�

By the definition of ϕn�x� and by Lemma 3.3�a� a2n ∼ an, we have

ϕνn(xj,n

)

⎛

⎜⎝

n√a22n − x2j,n

⎞

⎟⎠

i

� C(ann

)ν−i(

1 −∣∣xj,n

∣∣

a2n

)ν−i/2(

1 −∣∣xj,n

∣∣

an

)−ν/2. �4.22�


By Lemma 4.2�1�, we have

1∣∣x − xj,n

∣∣ � C

1∑

j�s�m�1 or m−1�s�j ϕn�xs,n�

� C n/an∑j�s�m�1 or m−1�s�j�1 − |xs,n|/a2n�/�1 − |xs,n|/an � δn�1/2

� C n/an∑j�s�m�1 or m−1�s�j�1 − |xs,n|/an�/�1 − |xs,n|/an � δn�1/2

� C n/an∣∣m − j∣∣(1 −max{|x|, ∣∣xj,n

∣∣}/an

)1/2 .

�4.23�

Here, we note from �4.14� and �4.8� that

Φn�x� ∼(1 − |x|

an

), Φn

(xj,n

) ∼(

1 −∣∣xj,n

∣∣

an

)

. �4.24�

Thus, we have, for j /�m,

Φν/2n �x�Φ3ν/4n

(xj,n

)ϕνn(xj,n

)(

1 −∣∣xj,n

∣∣

an

)ν/4(1

|x − xj,n|

)ν−i⎛

⎜⎝

n√a22n − x2j,n

⎞

⎟⎠

i

� C(1 − |x|

an

)ν/2(

1 − |xj,n|an

)ν/2(

1 − |xj,n|a2n

)ν−i/2

×⎛

⎝ 1

|m − j|(1 −max{|x|, ∣∣xj,n∣∣}/an

)1/2

⎞

⎠

ν−i

� C(

1|m − j|

)ν−i.

�4.25�

Therefore, we have

∑

1�j�n,j /�m

Ai,j�x� � C(f)∑

j /�m

(1

∣∣m − j∣∣

)ν−i� C

(f), �4.26�

because of ν − i � 2.

Remark 4.4. If we consider the estimate of∑n

i�1 Ai,ν−1�x� with �4.13�, then we obtain

n∑

i�1

Ai,ν−1�x� � C(f)log n. �4.27�


We continue the proof Proposition 3.7. We need to estimate∑n

j�1 Ai,j�x� for |x| > x0,n.Now, suppose x > x0,n. Then similarly to �4.23�, we have

1∣∣x0,n − xj,n

∣∣ � C

1∑

1�s�j ϕn�xs,n�� C n/an

j�1 − x0,n/an�1/2� C 1

jδ1/2n

n

an. �4.28�

Similarly to �4.21�, we have using �4.22�

∑Ai,j�x� � C

(f)∑

j

Φν/2�x�Φ3ν/4n(xj,n

)ϕνn(xj,n

)

×(

1 −∣∣xj,n

∣∣

an

)ν/4(1

|x0,n − xj,n|

)ν−i⎛

⎜⎝

n√a22n − x2j,n

⎞

⎟⎠

i

� C(f)∑

j

Φν/2n �x�

(

1 −∣∣xj,n

∣∣

an

)ν/2(

1 − |xj,n|a2n

)ν−i/2(1

jδ1/2n

)ν−i

� C(f)∑

j

1jν−i

C(f),

�4.29�

because of ν − i � 2 and Φν/2�x�δ−�ν−i�/2n � δν/2n δ−�ν−i�/2n � C. Consequently we achieve theresult.

Remark 4.5. The above proof implies the following: there exists a constant C > 0 such that

n∑

j�1

∣∣hj,n,ρ�ν;x�∣∣ �

n∑

j�1

∣∣ljn�x�∣∣ν

ν−2∑

i�0|ei�ν, k, n�|

∣∣x − xj,n∣∣i � C, x ∈ R. �4.30�

Proof of Proposition 3.8. We use the same method to prove of Proposition 3.7. So, we let�xm�1,n � xm,n�/2 < x � xm,n or xm,n � x < �xm−1,n � xm,n�/2. For simplicity, we assumex > 0, and let x0,n :� x1,n � δϕn�x1,n� for a fixed δ > 0 small enough and there exists a constantδ1 > 0 such that x0,n :� x1,n � δϕn�x1,n� < an − δ1δn. Assume that x < x0,n. Since f�0� � 0,we may leave out the term with xj,n � 0. Hence we consider only the term of xj,n /� 0. NotingAssumption 3.1, �3.2�, and Lemma 2.7, we estimate

∑nj�1 Bν−1,j�x�, where

Bν−1,j�x� :�(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν

× ∣∣f(xj,n)∣∣∣∣ljn�x�

∣∣ν∣∣x − xj,n∣∣ν−1

{T�an�an

�∣∣Q′

(xj,n

)∣∣ �1

∣∣xj,n∣∣

}(n

a2n −∣∣xj,n

∣∣

)ν−2.

�4.31�


First we estimate Bν−1,m�x�. By Lemma 4.2, �4.16�, and the definition of ϕn�x�, we have

|x − xm,n|ν−1⎛

⎜⎝

n√a22n − x2m,n

⎞

⎟⎠

ν−2

� Cϕn�xm,n�(√

1 − |xm,n|/a2n√1 − |xm,n|/an

)ν−2

� Cann

(1 − |xm,n|

a2n

)1/2(√1 − |xm,n|/a2n√1 − |xm,n|/an

)ν−1.

�4.32�

Since here |Q′�xm,n�| � 1/|xm,n| � C for some positive constant C, we know that(T�an�an

�∣∣Q′�xm,n�

∣∣ �

1|xm,n|

)� Cmax

{1,

T�an�an

}(∣∣Q′�xm,n�

∣∣ �

1|xm,n|

), �4.33�

and by �3.25�

∣∣f�xm,n�∣∣ � C

(f)Φ3ν/4�xm,n�

(w�xm,n�

(|xm,n| � an

n

)ρ)−ν(∣∣Q′�xm,n�∣∣ �

1|xm,n|

)−1. �4.34�

Therefore, using �3.3� and �4.15�, we have

Bν−1,m�x� � C(f)max

{1,

T�an�an

}Φ3ν/4n �x�Φ

3ν/4n �xm,n�

ann

(1 − |xm,n|

a2n

)1/2

×(√

1 − |xm,n|/a2n√1 − |xm,n|/an

)ν−1

� C(f)εnΦ3ν/4n �x�Φ

3ν/4n �xm,n�

(1 − |xm,n|

a2n

)1/2(√1 − |xm,n|/a2n√1 − |xm,n|/an

)ν−1.

�4.35�

Noting �4.24�, we have

Bν−1,m�x� � C(f)εnΦ3ν/4n �x�

(1 − |xm,n|

an

)�ν�2�/4(1 − |xm,n|

a2n

)ν/2� C

(f)εn. �4.36�

Next, we estimate∑

j /�m Bν−1,j�x�. Noting �4.19� and �4.22�, we have

(Φ3/4n �x�w�x�

(|x| � an

n

)ρ)ν∣∣ljn�x�∣∣ν∣∣x − xj,n

∣∣ν−1

� Φν/2n �x�a−ν/2n ϕ

νn

(xj,n

)∣∣∣a2n − x2j,n∣∣∣ν/4

(w(xj,n

)(∣∣xj,n∣∣ � an/n

)ρ)ν∣∣x − xj,n

∣∣ ,

ϕνn(xj,n

)

⎛

⎜⎝

n√a22n − x2j,n

⎞

⎟⎠

ν−2

� C(ann

)2(

1 − |xj,n|an

)−ν/2(

1 − |xj,n|a2n

)ν/2�1.

�4.37�


Then by �4.33� and �3.25� �noting �4.34��, using the notation �3.3� and �4.24�, we have

Bν−1,j�x� � C(f)max

{1,

T�an�an

}(ann

)2Φν/2n �x�Φ

3ν/4n

(xj,n

)a−ν/2n

∣∣∣a2n − x2j,n

∣∣∣ν/4

× 1∣∣x − xj,n∣∣

(

1 −∣∣xj,n

∣∣

an

)−ν/2(

1 −∣∣xj,n

∣∣

a2n

)ν/2�1

� C(f)εn

annΦν/2n �x�Φ

3ν/4n

(xj,n

)(

1 −∣∣xj,n

∣∣

an

)−ν/4(

1 −∣∣xj,n

∣∣

a2n

)ν/2�11

∣∣x − xj,n

∣∣

� C(f)εn

ann

(1 − |x|

an

)ν/2(

1 −∣∣xj,n

∣∣

an

)ν/2(

1 − |xj,n|a2n

)ν/2�11

∣∣x − xj,n

∣∣ .

�4.38�

Therefore, since we know from �4.23� that

1∣∣x − xj,n

∣∣ � Cn/an

∣∣m − j∣∣(1 −max{|x|, ∣∣xj,n∣∣}/an

)1/2 , �4.39�

noting �4.24�, we have

∑

j /�m

Bν−1,j�x� � C(f)εn

(1 − |x|

an

)ν/2∑

j /�m

(1 − ∣∣xj,n

∣∣/an)ν/2(1 − ∣∣xj,n

∣∣/a2n)ν/2�1

∣∣m − j∣∣(1 −max{|x|, ∣∣xj,n∣∣}/an

)1/2

� C(f)εn

(1 − |x|

an

)�ν−1�/2∑

j /�m

(1 − |xj,n|/an

)�ν−1�/2(1 − ∣∣xj,n∣∣/a2n

)ν/2�1∣∣m − j∣∣

� C(f)εn∑

j /�m

1∣∣m − j∣∣ � C

(f)εn log n.

�4.40�

Finally, we estimate∑n

j�1 Bν−1,j�x� for x > x0,n. Suppose x > x0,n. Then similar to the abovecomputations, we have


annΦν/2�x�Φ3ν/4n

(xj,n

)(

1 −∣∣xj,n

∣∣

an

)−ν/4(

1 −∣∣xj,n

∣∣

a2n

)ν/2�11

∣∣x0,n − xj,n∣∣

� C(f)εn

annΦν/2�x�

(

1 −∣∣xj,n

∣∣

an

)ν/2(

1 −∣∣xj,n

∣∣

a2n

)ν/2�11

∣∣x0,n − xj,n∣∣ .

�4.41�

Then, since we know by �4.28� that

1∣∣x0,n − xj,n

∣∣ � C1

jδ1/2n

n

an, �4.42�


we have

Bν−1,j�x� � C(f)εnΦν/2�x�

(

1 −∣∣xj,n

∣∣

an

)ν/2(

1 −∣∣xj,n

∣∣

a2n

)ν/2�11

jδ1/2n. �4.43�

Then, since Φν/2�x�δ−1/2n < C for x > x0,n, we have

n∑

j�1


n∑

j�1

1j

� C(f)εn log n. �4.44�

Therefore, the result is proved.

Proof of Proposition 3.9. By Lemma 2.3 we see that, for a constant C1 > 0,

∣∣Zn(l, ν, f ;x

)∣∣ � C1n∑

j�1

l∑

s�1

∣∣∣f �s�(xj,n

)∣∣∣∣∣ljn�x�

∣∣νν−1∑

i�s

⎛

⎜⎝

n√a22n − x2j,n

⎞

⎟⎠

i−s∣∣x − xj,n

∣∣i

� C1ann

n∑

j�1

l∑

s�1

∣∣∣f �s�(xj,n

)∣∣∣∣∣ljn�x�

∣∣νν−1∑

i�s

⎛

⎜⎝

n√a22n − x2j,n

⎞

⎟⎠

i

∣∣x − xj,n∣∣i

� C1ann

l∑

s�1

ν−1∑

i�0

n∑

j�1

∣∣∣f �s�(xj,n

)∣∣∣∣∣ljn�x�

∣∣ν∣∣x − xj,n∣∣i

⎛

⎜⎝

n√a22n − x2j,n

⎞

⎟⎠

i

�:ann

l∑

s�1

ν−1∑

i�0

Ci,s�x�,

�4.45�

where,

Ci,s�x� :�n∑

j�1

∣∣∣f �s�(xj,n

)∣∣∣∣∣ljn�x�

∣∣ν∣∣x − xj,n∣∣i

⎛

⎜⎝

n√a22n − x2j,n

⎞

⎟⎠

i

. �∗�

We set

Xi,s,n(ν, f ;x

):�

n∑

j�1

f �s�(xj,n

)lνjn�x�ei

(ν, j, n

)(x − xj,n

)i, i � 0, 1, . . . , ν − 1, �4.46�

and hereafter we wrote �∗� as |X|i,s,n�ν, f ;x�. Now, we repeat the proof of Proposition 3.7 byexchanging f�x� with f �s��x�, s � 1, 2, . . . , ν − 1, and we note �3.23� �and �3.26��. Then, for0 � i � ν − 2 we obtain

{Φ3/4n �x�w�x�

(|x| � an

n

)}ν|X|i,s,n

(ν, f ;x

)� C1C

(f). �4.47�


For i � ν − 1, we use the estimate for lmn�x� in the proof of Proposition 3.7; furthermore weuse Remark 4.4. Then we have

{Φ3/4n �x�w�x�

(|x| � an

n

)}ν|X|ν−1,s,n

(ν, f ;x

)� C1C

(f)log n. �4.48�

Consequently, we have

{Φn�x�3/4w�x�

(|x| � an

n

)}ν∣∣Zn

(l, ν, f ;x

)∣∣ � C1C(f)an log n

n. �4.49�

Similarly, we have

{Φ�x�3/4w�x�

(|x| � an

n

)}ν|Wn�l, ν, G;x�| � C1C�G�

an log nn

. �4.50�

Proof of Proposition 3.10. Let P ∈ Pm be fixed. From Proposition 3.9 we see

∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν{Ln�ν, P� − P}

∥∥∥∥L∞�R�

�∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν{Ln�ν, P� − Ln�ν − 1, ν, P�}

∥∥∥∥L∞�R�

�∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)νZn�ν − 1, ν, P�

∥∥∥∥L∞�R�

−→ 0 as n −→ ∞.

�4.51�

Proof of Proposition 3.11. Let P ∈ Pm with P�0� � 0. Then P satisfies the condition �A-1�. ByProposition 3.8, we see

∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)νYn�ν, P�

∥∥∥∥L∞�R�

−→ 0 as n −→ ∞. �4.52�

So, from Propositions 3.10 and 3.8 �noting �3.3��we have

∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν�Xn�ν, P� − P�

∥∥∥∥L∞�R�

� C1∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν�Ln�ν, P� − P�

∥∥∥∥L∞�R�

�∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)νYn�ν, P�

∥∥∥∥L∞�R�

−→ 0 as n −→ ∞.

�4.53�


Proof of Theorem 3.12. Since f satisfies �3.22�, we see

lim|x|→∞

∣∣f�x�

∣∣w�x�ν−δ � 0. �4.54�

For a given ε > 0, there exists a polynomial P ∈ Pm with P�0� � 0 such that

supx∈R

∣∣f�x� − P�x�∣∣

{Φ−3/4�x�w�x�

(|x| � an

n

)ρ}ν< ε. �4.55�

In fact, by �21, Theorem 1.4�, there exists a polynomial R ∈ Pm such that

supx∈R

∣∣f�x� − R�x�∣∣wν−δ�x� < ε2. �4.56�

Let P�x� :� R�x� − R�0�. Noting that

{Φ−3/4�x�w�x�

(|x| � an

n

)ρ}ν� Cwν−δ�x�, x ∈ R for some C > 0 �4.57�

and f�0� � 0, we have

supx∈R

∣∣f�x� − P�x�∣∣{Φ−3/4�x�w�x�

(|x| � an

n

)ρ}ν

� supx∈R

∣∣f�x� − P�x�∣∣wν−δ�x�

� supx∈R

∣∣f�x� − R�x�∣∣wν−δ�x� � ∣∣f�0� − R�0�∣∣wν−δ�0� < ε;

�4.58�

that is, we have �4.55�. Here, we know that

Ln(ν, f

)�x� − f�x� � Xn

(ν, f − P) � �Xn�ν, P� − P� �

(P − f) � Yn

(ν, f

). �4.59�


Therefore, by Propositions 3.7, 3.11, and 3.8, we have for n large enough

∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν(Ln(ν, f

) − f)∥∥∥∥L∞�R�

� C∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)νXn

(ν, f − P)

∥∥∥∥L∞�R�

�∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν�Xn�ν, P� − P�

∥∥∥∥L∞�R�

�∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν(P − f)

∥∥∥∥L∞�R�

�∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)νYn(ν, f

)∥∥∥∥L∞�R�

�: In � IIn � IIIn � IVn.

�4.60�

Here we see that, by Proposition 3.7 with ε :� C�f − P� �constant depending only on f − P�and �4.55�,

In � Cε, IIIn � ε, �4.61�

and for n � n0 large enough, we have

IIn � Cε,(by Proposition 3.9

),

IVn � Cε,(by Proposition 3.6

).

�4.62�

Consequently, noting �3.3�, we have

limn→∞

{In � IIn � IIIn � IVn} � 0. �4.63�

Then we have Theorem 3.12.

Proof of Theorem 3.13. From Theorem 3.12 and Proposition 3.9, we have

∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν(Ln(l, ν, f

) − f)∥∥∥∥L∞�R�

� C1∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν(Ln(ν, f

) − f)∥∥∥∥L∞�R�

�∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)νZn(l, ν, f

)∥∥∥∥L∞�R�

−→ 0 as n −→ ∞.

�4.64�


Proof of Corollary 3.14. From Theorem 3.12 and Proposition 3.8, we have

∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν(Xn

(ν, f

) − f)∥∥∥∥L∞�R�

� C1∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν�Ln�ν, f� − f�

∥∥∥∥L∞�R�

�∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)νYn�ν, f�

∥∥∥∥L∞�R�

−→ 0 as n −→ ∞.

�4.65�

Proof of Theorem 3.15. From Theorem 3.13 and Proposition 3.10 we have

∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν(L̃n(l, ν, f ⊕G) − f

)∥∥∥∥L∞�R�

� C∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν(Ln(l, ν, f

) − f)∥∥∥∥L∞�R�

�∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)νWn�l, ν, G�

∥∥∥∥L∞�R�

−→ 0 as n −→ ∞.

�4.66�

Proof of Theorem 3.16. We use Theorems 3.12, 3.13, and Corollary 3.14, and Theorem 3.15.Then we have

∣∣In[k,Hn

(f); f] − I[k, f]∣∣

�∫∞

−∞k�x�

{Hn

(f)�x� − f�x�}dx

� C∥∥∥∥

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)ν{Hn

(f)�x� − f�x�}

∥∥∥∥L∞�R�

×∫∞

−∞

(Φ3/4�x�w�x�

(|x| � an

n

)ρ)νk�x�dx −→ 0 as n −→ ∞.

�4.67�

5. Divergence Theorem

If ν is a positive odd integer, then we obtain the unboundedness of Ln�ν, f ;x�. We define

Λn�ν,R� � maxx∈R

n∑

k�1

|hkn�x�|. �5.1�

Theorem 5.1 �cf. �1, Theorem 2��. Let ν > 0 be an odd integer. Then there exists a constant C > 0and n0 > 0 such that for n � n0

Λn�ν,R� � C log n. �5.2�


Let −∞ � a < b � ∞, and let us define

Λn�ν, �a, b�� maxa�x�b

n∑

k�1

|hkn�x�|. �5.3�

Then we will show that for n � n0

Λn�ν, �a, b�� C log n. �5.4�

Remark 5.2. For the interpolation polynomial Ln�ν, f ;x�, we can see a remarkable differencebetween the cases of an odd number ν and of an even number ν. Let us consider anycontinuous function f ∈ C��a, b��, 0 < a < b. Then we can extend f to a continuous functionf∗ ∈ C�R� which satisfies �3.22� and f�x� � f∗�x�, x ∈ �a, b�. Then from Theorem 3.12 for aneven positive integer ν, we see

∥∥Ln(ν, f∗;x

) − f�x�∥∥L∞��a,b�� −→ 0 as n −→ ∞. �5.5�

On the other hand, the standard argument �cf, �22, Theorem 4.3�� leads us to the following.Theorem 5.1 means that there exists a certain function f∗ ∈ C�R� such that

∥∥Ln(ν, f∗;x

) − f∗�x�∥∥L∞��a,b�� 0 as n −→ ∞; �5.6�

that is, for f�x� :� f∗�x�, a � x � b we see that the interpolation polynomials Ln�ν, f∗;x� donot converge to f . We also remark that the polynomial Ln�ν, f∗;x� interpolates f ∈ C�a, b� atonly {xj,n ∈ �a, b�, 1 � j � n}.

To prove the theorem we use the following lemma.

Lemma 5.3 �see �18, Theorem 11��. For j � 0, 1, 2, . . ., there is a polynomial Ψj�x� of degree j suchthat �−1�jΨj�−ν� > 0 for ν � 1, 3, 5, . . ., and the following relation holds. Let 0 < ε < 1. Then one hasan expression for �1/ε��an/n� � |xk,n| � εan and s � 0, 1, . . . , �ν − 1�/2:

e2s�ν, k, n� � �−1�s{

1�2s�!

}Ψs�−ν�βs�n, k�

(n

an

)2s{1 � ηkn�ν, s�

}, �5.7�

where for βs�n, k�, n � 1, 2, 3, . . ., k � 1, 2, . . . , n, s � 0, 1, . . . , �ν − 1�/2, there exist the constantsC1, C2 such that

0 < C1 � βs�n, k� � C2, �5.8�


and ηkn�ν, s� satisfies

∣∣ηkn�ν, s�

∣∣ � ε∗n, ε∗n � ε∗n�ε� −→ 0 as ε −→ 0, �5.9�

for k with �1/ε��an/n� � |xk,n| � εan, and for s � 0, 1, 2, . . ..

Proof of Theorem 5.1. To get a lower bound of Λn�ν,R�, it suffices to consider a lower bound�5.4� ofΛn�ν, �a, b��with 0 < a < b < ∞. Let c :� a/2, d :� 2b−a and we consider the intervalsI :� �a, b�, J � �c, d�. If we consider n large enough, then we have �1/ε��an/n� � c < 2b − a �εan. See the expression

|hkn�x�| �∣∣∣lνkn�x�eν−1�ν, k, n��x − xk,n�ν−1

∣∣∣ −

∣∣∣∣∣lνkn�x�

ν−2∑

i�0

ei�ν, k, n��x − xk,n�i∣∣∣∣∣. �5.10�

Then we have

Λn�ν, �a, b�� maxx∈I

∑

xkn∈J

∣∣∣lνkn�x�eν−1�ν, k, n��x − xk,n�ν−1∣∣∣

−maxx∈I

∑

xkn∈J

∣∣∣∣∣lνkn�x�

ν−2∑

i�0

ei�ν, k, n��x − xk,n�i∣∣∣∣∣

�: maxx∈I

Fn�x� −maxx∈I

Gn�x�.

�5.11�

It follows from Remark 4.5 that maxa�x�b Gn�x� � C. Hence, it is enough to show thatmaxx∈IFn�x� � C logn. Let xk�1,n, xk,n ∈ J ; then by Lemma 4.2 and the definition of ϕn�xk,n�there exists 0 < α � β < ∞ such that

αann

� |xk,n − xk�1,n| � βann. �5.12�

For x ∈ I we consider only xk,n ∈ J such that, for some positive integer jk,

(jk − 1

)αann

� |xk,n − x| � jkβann. �5.13�

Then for each x ∈ I we define

Γ�x� :�{jk;xk,n ∈ J,

(jk − 1

)αann

� |xk,n − x| � jkβann

}. �5.14�


Here, we will see that

{j; 1 � j � d − c

2βn

an− 1

}⊂ Γ�x�. �5.15�

Let

xm�1,n < x � xm,n,xk�c��1,n < c � xk�c�,n, xk�d�,n � d < xk�d�−1,n.

�5.16�

Then we have

x − xm�1,n � xm,n − xm�1,n � βann,

x − xm�2,n � xm,n − xm�2,n � 2βann,

...

x − xk�c�,n � xm,n − xk�c�,n � j�c�βann,

xm,n − x � xm,n − xm�1,n � βann,

xm−1,n − x � xm−1,n − xm�1,n � 2βann,

...

xk�d�,n − x � xk�d�,n − xm�1,n � j�d�βann,

�5.17�

where j�c� and j�d� are integers. On the other hand,

(xk�d�,n − xk�c�,n

)

(βan/n

) �(d − c − 2βan/n

)

(βan/n

) �d − cβ

n

an− 2. �5.18�

Therefore, we have

max(j�c�, j�d�

)� 1

2

(d − cβ

n

an− 2

)�

d − c2β

n

an− 1. �5.19�

Now, we take a positive integer N�n� such that

N�n� � d − c2β

n

an− 1 < N�n� � 1. �5.20�


Consequently, we have the following. By Lemmas 4.2�1�, Lemma 4.1�1�, �2�, �4�, andLemma 2.3, we have

Fn�x� �∑

xk,n∈J|lkn�x�|ν|eν−1�ν, k, n�||xk,n − x|ν−1

�∑

xk,n∈J

( ∣∣pn�x�

∣∣wρ�x�

|x − xk,n|∣∣p′n�xk,n�

∣∣wρ�xk,n�

wρ�xk,n�wρ�x�

)ν|eν−1�ν, k, n�||x − xk,n|ν−1

�∑

xk,n∈J

( ∣∣pn�x�

∣∣wρ�x�

∣∣p′n�xk,n�

∣∣wρ�xk,n�

wρ�xk,n�wρ�x�

)ν|eν−1�ν, k, n�| 1|x − xk,n|

� C∑

jk∈Γ�x�

(a−1/2n

�n/an�a−1/2n

wρ�d�wρ�a�

)ν|eν−1�ν, k, n�| n

jkβan

� C∑

jk∈Γ�x�

1jk

(ann

)ν−1|eν−1�ν, k, n�|.

�5.21�

Here, using Lemma 5.3 and �5.15�, we have

Fn�x� � C∑

jk∈Γ�x�

1jk

(ann

)ν−1( nan

)ν−1� C

∑

1�j�N�n�

1j. �5.22�

Here, there exists 0 < η < 1 such that nη � N�n�. Therefore we see

Fn�x� � C∑

1�j�nη

1j

� C log n. �5.23�

So we have �5.4�, and consequently the proof of Theorem 5.1 is completed.

Acknowledgment

The authors thank the referees for many kind suggestions and comments.

References

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