# Convergence and Divergence of  Higher-Order  Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Authors: Hee Sun Jung; Gou Nakamura; Ryozi Sakai; Noriaki Suzuki
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

Letℝ=(-∞,∞), 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(wρ2;x) of degree n for wρ2(x). In this paper for an even integer ν≥2
we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejér interpolation polynomials and related approximation process based at the zeros {xk,n,ρ}k=1n of pn(wρ2;x). Moreover, for an odd integer  ν≥1, we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros {xk,n,ρ}k=1n of pn(wρ2;x).


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## Body

## 1. Introduction

Letℝ=(-∞,∞), and let Q∈C1(ℝ):ℝ→ℝ+=[0,∞) be an even function.  Consider the weight w(x)=exp(-Q(x)), and define, for ρ>-1/2,
(1.1)wρ(x):=|x|ρw(x),x∈R.
Suppose that ∫0∞xnwρ(x)dx<∞, for all n=0,1,2,…. Then we can construct the orthonormal polynomials pn,ρ(x)=pn(wρ2;x) of degree n for wρ2(x); that is,
(1.2)∫-∞∞pn,ρ(x)pm,ρ(x)wρ2(x)dx=δm,n(Kroneckerdelta).
We write pn,ρ(x) by
(1.3)pn,ρ(x)=γnxn+⋯,γn=γn,ρ>0,
and denote the zeros of pn,ρ(x) by
(1.4)-∞<xn,n,ρ<xn-1,n,ρ<⋯<x2,n,ρ<x1,n,ρ<∞.Let𝒫n denote the class of polynomials with degree at most n. For f∈C(ℝ) we define the higher-order Hermite-Fejér interpolation polynomial Ln(ν,f;x)∈𝒫νn-1 based at the zeros {xk,n,ρ}k=1n as follows:
(1.5)Ln(i)(ν,f;xk,n,ρ)=δ0,if(xk,n,ρ)fork=1,2,…,n,i=0,1,…,ν-1.
We note that Ln(1,f;x) is the Lagrange interpolation polynomial, Ln(2,f;x) is the ordinary Hermite-Fejér interpolation polynomial, and Ln(4,f;x) is the   Krylov-Stayermann  polynomial.  For the general cases Kanjin and Sakai [1, 2] started to investigate the so-called Freud-type weights. The fundamental polynomials hk,n,ρ(ν;x)∈𝒫νn-1 for the higher-order Hermite-Fejér interpolation polynomial Ln(ν,f;x) are defined as follows:
(1.6)hk,n,ρ(ν;x)=lk,n,ρν(x)∑i=0ν-1ei(ν,k,n)(x-xk,n,ρ)i,lk,n,ρ(x)=pn(wρ2;x)(x-xk,n,ρ)pn′(wρ2;xk,n,ρ),hk,n,ρ(ν;xp,n,ρ)=δk,p,hk,n,ρ(i)(ν;xp,n,ρ)=0,k,p=1,2,…,n,i=1,2,…,ν-1.
Using them, we can write
(1.7)Ln(ν,f;x)=∑k=1nf(xk,n,ρ)hk,n,ρ(ν;x).
Furthermore, we extend the operator Ln(ν,f;x). Let l be a nonnegative integer, and let ν-1⩾l. For f∈Cl(ℝ) we define the (l,ν)-order Hermite-Fejér interpolation polynomials Ln(l,ν,f;x)∈𝒫νn-1 as follows: for each k=1,2,…,n,
(1.8)Ln(l,ν,f;xk,n,ρ)=f(xk,n,ρ),Ln(j)(l,ν,f;xk,n,ρ)=f(j)(xk,n,ρ),j=1,2,…,l,Ln(j)(l,ν,f;xk,n,ρ)=0,j=l+1,l+2,…,ν-1.
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. The fundamental polynomials hs,k,n,ρ(l,ν;x)∈𝒫νn-1, k=1,2,…,n, of Ln(l,ν,f;x) are defined by
(1.9)hs,k,n,ρ(l,ν;x)=lk,n,ρν(x)∑i=sν-1es,i(ν,k,n)(x-xk,n,ρ)i,hs,k,n,ρ(j)(l,ν;xp,n,ρ)=δs,jδk,p,j,s=0,1,…,ν-1,p=1,2,…,n.
Then we have
(1.10)Ln(l,ν,f;x)=∑k=1n∑s=0lf(s)(xk,n,ρ)hs,k,n,ρ(l,ν;x).
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 interesting convergence theorems.Our purpose in this paper is to studyLn(ν,f;x) and Ln(l,ν,f;x) as certain analogies of 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 theorems for the Freud-type weights. Then for an even integer ν≥2 we give the convergence theorems for them; moreover, for an odd integer ν⩾1, we obtain a certain divergence theorem with respect to Ln(ν,f;x). In Section 1, we give the preliminaries for these studies, and in Section 2 we write some preliminary description. In Section 3 we report our theorems with some lemmas, and in Section 4 we prove the theorems. Finally, in Section 5, for an odd integer ν⩾1 we obtain a certain divergence theorem with respect to Ln(ν,f;x).In what follows we abbreviate several notations asxk,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. For arbitrary nonzero real-valued functions f(x) and g(x), we write f(x)~g(x) if there exist constants C1,C2>0 independent of x such that C1g(x)⩽f(x)⩽C2g(x) for all x. For arbitrary positive sequences {cn}n=1∞ and {dn}=1∞, we define cn~dn similarly.Throughout this paperC,C1,C2,… denote positive constants independent of n,x,t, or polynomials Pn(x), and the same symbol does not necessarily denote the same constant in different occurrences.


## 2. Preliminaries

A functionf: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]).
LetQ:ℝ→ℝ+ be a continuous even function satisfying the following 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(2.1)T(x):=xQ′(x)Q(x),x≠0,
is quasi-increasing in (0,∞), with T(x)⩾Λ>1 for x∈ℝ+∖{0}.(e)
There existsC1>0 such that
(2.2)Q′′(x)|Q′(x)|⩽C1|Q′(x)|Q(x),a.e.x∈R∖{0}.
Then we say that w=exp(-Q) is in the class ℱ(C2). Besides, if there exist a compact subinterval J(∋0) of ℝ and C2>0 such that
(2.3)Q′′(x)|Q′(x)|⩾C2|Q′(x)|Q(x),a.e.x∈R∖J,
then we say that w=exp(-Q) is in the class ℱ(C2+).Example 2.2.
We present some typical examples ofQ(x) satisfying w=exp(-Q)∈ℱ(C2+). (1)
If a continuous exponentialQ(x) satisfies
(2.4)1<Λ1⩽(xQ′(x))′Q′(x)⩽Λ2
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
(2.5)Q(x)=Qr,α(x):=exp⁡r(|x|α)-exp⁡r(0),
where, for r⩾1,
(2.6)exp⁡r(x):=exp⁡(exp⁡(exp⁡(⋯exp⁡x)⋯))(r-times)
and exp0(x):=x.(3)
(See [10]) For m⩾0,α⩾0 with α+m>1, we put
(2.7)Q(x)=Qr,α,m(x):=|x|m{exp⁡r(|x|α)-α*exp⁡r(0)},
where for r>0 we suppose α*=0 if α=0; α*=1 otherwise. For r=0 we suppose m>1 and α=0.(4)
(See [10]) For α>1, we put
(2.8)Q(x)=Qα(x):=(1+|x|)|x|α-1.
For x>0, we define the Mhaskar-Rakhmanov-Saff number ax by the equation(2.9)x=2π∫01axuQ′(axu)(1-u2)1/2du.
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]).
Letw(x)=exp(-Q(x))∈ℱ(C2+). For each  k=1,2,…,n and  s=0,1,…,ν-1, we have  e0,0(ν,k,n)=e0(ν,k,n)=1,
(2.10)|es,i(ν,k,n)|⩽C(na2n2-xk,n2)i-s,s=1,2,…,ν-1,i=s,s+1,…,ν-1.If we consider the higher-order Hermite-Fejér interpolation polynomialLn(ν,f;x) on a certain finite interval, then we can see a remarkable difference between the parity of ν, for example, the Lagrange interpolation polynomial Ln(1,f;x) and the ordinary Hermite-Fejér interpolation polynomial Ln(2,f;x) ([12–17]). Also, we can see a similar phenomenon in the case of the infinite intervals ([1, 2, 4–8]). To describe these aspects, however, we need a further strengthened definition for ν⩾2 than Definition 2.1.Definition 2.4.
Letw(x)=exp(-Q(x))∈ℱ(C2+),and let ν⩾1 be an integer. Assume that Q(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 constantsCi>0 such that(2.11)|Q(i+1)(x)|⩽Ci|Q(i)(x)||Q′(x)|Q(x),x∈R∖{0},i=1,2,…,ν.(c)
There exist0⩽δ<1 and c1>0 such that
(2.12)Q(ν+1)(x)⩽C(1x)δ,x∈(0,c1].
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 constantsC>0 and 0⩽δ<1 such that Q(ν+1)(x)⩽C(1/x)δ on (0,∞).Then one says thatw(x)=exp(-Q(x)) is in the class ℱ̃ν(C2+).Example 2.5 (cf. [10, Theorem 3.1]).
Letν be a positive integer, and let Qr,α,m be defined in (2.7).(1)
Letm and α be nonnegative even integers with m+α>1.Then w(x)=exp(-Qr,α,m)∈ℱν(C2+).
(a)
Ifr>0,then we see that Qr,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)
Ifr=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)
Letm+α-ν>0. Then w(x)=exp(-Qr,α,m)∈ℱν(C2+), and one has the following.
(c)
Ifr⩾2 and α>0, then there exists a constant c1>0 such that Qr,α,m′(x)/Qr,α,m(x) is quasi-increasing on (c1,∞).(d)
Letr=1. If α⩾1, then there exists a constant c2>0 such that Q1,α,m′(x)/Q1,α,m(x) is quasi-increasing on (c2,∞), and, if 0<α<1, then Q1,α,m′(x)/Q1,α,m(x) is quasi-decreasing on (c2,∞).(e)
Letr=1 and 0<α<1, then Q1,α,m(ν+1)(x) is nondecreasing on a certain positive interval on (c2,∞).Hence,w(x)=exp(-Qr,α,m)∈ℱ̃ν(C2+).Definition 2.6.
One uses the following notation:(2.13)φu(x)={auu1-|x|/a2u1-|x|/au+δu,|x|⩽au;φu(au),au<|x|,δu=(uT(au))-2/3,u>0.Lemma 2.7 (see [18, Corollary 4.5]).
Letwρ(x)=|x|ρexp(-Q(x)),  exp(-Q)∈ℱ̃ν(C2+). If  xk,n≠0 and  |xk,n|⩽an(1+δn), then  e0(ν,k,n)=1 and, for  i=1,2,…,ν-1,
(2.14)|ei(ν,k,n)|⩽C{T(an)an+|Q′(xk,n)|+1|xk,n|}〈i〉(na2n-|xk,n|+T(an)an)i-〈i〉,
where
(2.15)〈i〉={1,i: odd,0,i: even.
For xk,n=0 one has
(2.16)e0(ν,k,n)=1,|ei(ν,k,n)|⩽C(nan)i,i=1,2,…,ν-1.Remark 2.8.
In [19, Theorem 2.2] we see that x1,n<an if n is large enough. Therefore Lemma 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 for some ε>0 and for large enough t,
(2.17)T(at)⩽Ct2-ε.In [20] we have the following estimations.Lemma 2.9 (see [20, Theorem 1.6]).
Letw=exp(-Q)∈ℱ(C2+).  (1)
LetT(x) be unbounded. Then, for any  η>0, there exists  C(η)>0 such that, for  t⩾1,(2.18)at⩽C(η)tη.(2)
Letλ:=C1 be the constant in Definition 2.1(e), that is,
(2.19)Q′′(x)Q′(x)⩽λQ′(x)Q(x),a.e.x∈R∖{0}.
If λ>1, then there exists C(λ,η) such that
(2.20)T(at)⩽C(λ,η)t2(η+λ-1)/(λ+1),t⩾1,
and, if 0<λ⩽1, then for any η>0 there exists C(λ,η) such that
(2.21)T(at)⩽C(λ,η)tη,t⩾1.Remark 2.10.
(1) If T(x) is bounded, then w is called a Freud-type weight, and, if T(x) is unbounded, then w is called an Erdős-type weight.
(2) In (2.20) and (2.21), we set 0<η<2 and
(2.22)ε={2-η,0<λ⩽1,2(2-η)(λ+1),λ>1.
Then (2.17) holds.
(3) If
(2.23)limsup⁡x→∞Q(x)Q′′(x)(Q′(x))2⩽1,
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
(2.24)T(at)=m,at~t1/m.
(5) The inequality (2.18) implies
(2.25)0<C⩽nan{T(an)an}ν-1.


## 3. Theorems

In the rest of this paper we assume the following for the weightw.Assumption 3.1.
Consider the weightwρ(x)=|x|ρexp(-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),
(3.1)an⩽Cn1/(1+ν-δ).(b)
There exist0⩽γ<1 and C(γ)>0 such that
(3.2)T(an)⩽C(γ)nγ;here, ifT(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
(3.3)εn:={ann,T(an)an<1,1n1-γ,T(an)an⩾1.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
(3.4)γ=2(η+λ-1)λ+1,0<λ<3.
(3) In (3.3) we note that εnlogn→0 as n→∞.We shall state our theorems. Put(3.5)Xn(ν,f;x):=∑j=1nf(xj,n,ρ)lj,n,ρν(x)∑i=0ν-2ei(ν,j,n)(x-xj,n,ρ)i,Yn(ν,f;x):=∑j=1nf(xj,n,ρ)lj,n,ρν(x)eν-1(ν,j,n)(x-xj,n,ρ)ν-1,Zn(l,ν,f;x):=∑j=1n∑s=1lf(s)(xj,n,ρ)lj,n,ρν(x)∑i=sν-1esi(ν,j,n)(x-xj,n,ρ)i.
Furthermore, we consider the class G={gs∈C(ℝ), s=l+1,l+2,…,ν-1} and construct the following interpolation polynomial:(3.6)Wn(l,ν,G;x):=∑j=1n∑s=l+1ν-1gs(xj,n,ρ)lj,n,ρν(x)∑i=sν-1esi(ν,j,n)(x-xj,n,ρ)i.Then we have(3.7)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).
Define
(3.8)Φ(x):=1(1+Q(x))2/3T(x).
Here we note that, for some d>0,(3.9)Φ(x)~Q(x)1/3xQ′(x),|x|⩾d>0.Moreover, we define(3.10)Φn(x):=max⁡{δn,1-|x|an},n=1,2,3,….Lemma 3.3.
Letw=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,(3.11)aLt~at.(b)
(See [9, Lemma 3.5(b)]) Uniformly for t>0,(3.12)Q(j)(aLt)~Q(j)(at),j=0,1.Moreover,(3.13)T(aLt)~T(at).(c)
(See [9, Lemma 3.11 (3.52)]) Uniformly for t>0,
(3.14)|1-aLtat|~1T(at).(d)
(See [9, Lemma 3.4 (3.17), (3.18)]) Uniformly for t>0, one has(3.15)Q(at)~tT(at),Q′(at)~tT(at)at.Lemma 3.4.
Forx∈ℝ, we have
(3.16)Φ(x)⩽CΦn(x),n⩾1.Proof.
Letx=au,u⩾1. By Lemma 3.3(d) we have
(3.17)u~Q(au)T(au).
So, we have
(3.18)δu-1~Q(au)2/3T(au)=auQ′(au)Q(au)1/3=xQ′(x)Q1/3(x)~Φ-1(au).
Now, if u⩽n/2, then
(3.19)1-auan⩾1-an/2an~1T(an)(byLemma3.2(c))⩾1(nT(an))2/3=δn(by(2.17)).
So, we have
(3.20)Φn(x)=1-auan⩾1-aua2u~1T(au)(byLemma3.2(c))⩾1(uT(au))2/3=δu~Φ(x)(by(2.17)and(3.18)).
Let n/2<u<n. Then we have
(3.21)Φn(x)⩾δn~δu~Φ(x)(byLemmas3.2(a),(2.17),and(3.18)).LetC(f) denote a positive constant depending only on f.Assumption 3.5.
Letwρ(x)=|x|ρw(x) and w(x)=exp(-Q(x))∈ℱν(C2+), ρ⩾0. Suppose the following.(A-1)
Letf∈C(ℝ) satisfy that, for a given 0<δ<1,(3.22)|f(x)|wν-δ(x){|Q′(x)|+1|x|}⩽C(f),x∈R,where we supposelimsup|x|→0|f(x)/x|⩽C(f).(A-2)
Letf∈Cl(ℝ) for a certain 0<l⩽ν-1. Then we suppose that f satisfies
(3.23)|f(s)(x)|{Φ-3/4(x)wρ(x)}ν⩽C(f),x∈R,s=1,2,…,l.(A-3)
LetG={gs∈C(ℝ),s=l+1,l+2,…,ν-1}. Then we suppose that there exists a constant C>0 such that(3.24)|gs(x)|{Φ-3/4(x)wρ(x)}ν⩽C(G),x∈R,s=l+1,l+2,…,ν-1.Remark 3.6.
In (3.22), we have the following.(1)
f(0)=0 and
(3.25)|f(x)|{Φ-3/4(x)wρ(x)}ν(|Q′(x)|+1|x|)⩽C(f),x∈R∖{0}.(2)
For some positive constantC, we have |Q′(x)|+1/|x|⩾C. Hence, from (3.22), it follows that
(3.26)|f(x)|{Φ-3/4(x)wρ(x)}ν⩽C(f),x∈R.
We have a chain of results under Assumption 3.1.Proposition 3.7.
Letν=1,2,3,…. For  f(x) satisfying (3.26), one has
(3.27)‖{Φ3/4(x)w(x)(|x|+ann)ρ}νXn(ν,f;x)‖L∞(R)⩽C(f).Proposition 3.8.
Letν=2,4,6,…. For  f(x) satisfying (3.25), one has
(3.28)‖{Φ3/4(x)w(x)(|x|+ann)ρ}νYn(ν,f;x)‖L∞(R)⩽C(f)εnlog⁡n,
where εn is defined by (3.3).Proposition 3.9.
Letν=1,2,3,…. For  f(x) satisfying (3.23), one has
(3.29)‖{Φ3/4(x)w(x)(|x|+ann)ρ}νZn(l,ν,f;x)‖L∞(R)⩽C(f)anlog⁡nn,
and for f(x) satisfying (3.24) one has
(3.30)‖{Φ3/4(x)w(x)(|x|+ann)ρ}νWn(l,ν,G;x)‖L∞(R)⩽C(G)anlog⁡nn.Proposition 3.10.
Letν=1,2,3,…. Let  P∈𝒫m be fixed. Then one has
(3.31)‖{Φ3/4(x)w(x)(|x|+ann)ρ}ν(Ln(ν,P,x)-P(x))‖L∞(R)⟶0asn⟶∞.Proposition 3.11.
Letν=2,4,6,…. Let  P∈𝒫m with  P(0)=0 be fixed. Then one has
(3.32)‖{Φ3/4(x)w(x)(|x|+ann)ρ}ν(Xn(ν,P;x)-P(x))‖L∞(R)⟶0asn⟶∞.Theorem 3.12.
Letν=2,4,6,…. For  f(x) satisfying (3.22), one has
(3.33)‖{Φ3/4(x)w(x)(|x|+ann)ρ}ν(Ln(ν,f;x)-f(x))‖L∞(R)⟶0asn⟶∞.Theorem 3.13.
Letν=2,4,6,…. For  f(x) satisfying (3.22) and (3.23), one has
(3.34)‖{Φ3/4(x)w(x)(|x|+ann)ρ}ν(Ln(l,ν,f;x)-f(x))‖L∞(R)⟶0asn⟶∞.Corollary 3.14.
Letν=2,4,6,…. For  f(x) satisfying (3.22), one has
(3.35)‖{Φ3/4(x)w(x)(|x|+ann)ρ}ν(Xn(ν,f;x)-f(x))‖L∞(R)⟶0asn⟶∞.Theorem 3.15.
Letν=2,4,6,…. For  f(x) satisfying (3.22), (3.23) and (3.24), one has
(3.36)‖{Φ3/4(x)w(x)(|x|+ann)ρ}ν(L̃n(l,ν,f⊕G;x)-f(x))‖L∞(R)⟶0asn⟶∞.Define(3.37)In[k,Hn(f)]:=∫-∞∞k(x)Hn(f;x)dx,
where Hn(f) equals to one of the following:
(3.38)Ln(ν,f),Ln(l,ν,f),L̃n(l,ν,f⊕G),Xn(ν,f).
One also defines
(3.39)I[k,f]:=∫-∞∞k(x)f(x)dx.Theorem 3.16.
Letν=2,4,6,…. Let  f(x) satisfy (3.22), (3.23) and (3.24). If
(3.40)∫-∞∞k(x){Φ3/4(x)w(x)(|x|+ann)ρ}-νdx<∞,n=1,2,3,…,
then we have
(3.41)lim⁡n→∞In[k,Hn(f)]=I[k,f].For example, we can takek(x) in the following way. If T(x) is unbounded, we have, for Δ∈ℝ,(3.42)∫1∞(1+|x|)-ΔΦ3ν/4(x)dx⩽∫1∞(1+|x|)-Δ(1T(x)Q(x)2/3)3ν/4dx<∞
(see [10, Lemma 2.1(b)]). Then we set
(3.43)k(x):=(1+|x|)ρν-Δ{Φ3/2(x)w(x)}ν.
If T(x) is bounded and w(x)=exp(-|x|β), β>1, then we take Δ 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 constantsC,C1>0, the same symbol does not necessarily denote the same constant in different 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
(4.1)sup⁡x∈R|pn,ρ(x)w(x)|(|x|+ann)ρ|x2-an2|1/4~1,
and for w(x)=exp(-Q(x))∈ℱ(C2+),
(4.2)sup⁡x∈R|pn,ρ(x)w(x)|(|x|+ann)ρ~an-1/2(nT(an))1/6.
(2) (See [19, Theorem 2.5(d)]) Let w∈ℱ(C2+) and ρ>-1/2. Let 1⩽j⩽n-1 and x∈[xj+1,n,xj,n].  Then
(4.3)|pn(x)w(x)|(|x|+ann)ρ~min⁡{|x-xj,n|,|x-xj+1,n|}φn(xj,n)-1|an2-xj,n2|-1/4.
(3) (See [19, Theorem 2.5(c)]) Let w∈ℱ(C2+) and ρ>-1/2. Then one has
(4.4)max⁡x∈R|pn(x)w(x)(|x|+an/n)ρ(x-xj,n)pn′(xj,n)w(xj,n)(|xj,n|+an/n)ρ|=max⁡x∈R|ljn(x)w(x)(|x|+ann)ρ|w-1(xj,n)(|xj,n|+ann)ρ~1.
(4) (See [19, Theorem 2.5(a)]) Let w∈ℱ(C2+) and ρ>-1/2. For 1⩽j⩽n we have
(4.5)|pn′w|(xj,n)(|xj,n|+ann)ρ~φn-1(xj,n)|an2-xj,n2|-1/4.Lemma 4.2 (see [19, Theorem 2.2]).
Letw(x)=exp(-Q(x))∈ℱ(C2+) and  ρ>-1/2. For the zeros  xj,n=xj,n,ρ, one has the following:  (1)
Forn⩾1 and  1⩽j⩽n-1,(4.6)xj,n-xj+1,n~φn(xj,n),φn(xj,n)~φn(xj+1,n)(see([19,LemmaA.1(A.3)])).(2)
For the minimum positive zerox[n/2],n ([n/2] is the largest integer ⩽n/2), one has(4.7)x[n/2],n~ann-1,and for large enoughn,(4.8)1-x1,nan~δn.(3)
(See [19, Lemma 4.7]) bn=γn-1/γn~an~x1,n, where γn is defined by (1.3).Lemma 4.3.
Letw∈ℱ(C2+). Then there exist  C1,C2>0 such that
(4.9)sup⁡x∈R|pn,ρ(x)w(x)|(|x|+ann)ρΦ1/4(x)⩽C1sup⁡x∈R|pn,ρ(x)w(x)|(|x|+ann)ρΦn1/4(x)⩽C2an-1/2.Proof.
The first inequality follows from Lemma3.4. We show the second inequality. Noting (4.8), from Lemma 4.1 (4.1) we have
(4.10)C⩾sup⁡|x|⩽x1,n|pn,ρ(x)w(x)|(|x|+ann)ρ|x2-an2|1/4~sup⁡|x|⩽x1,n|pn,ρ(x)w(x)|(|x|+ann)ρan1/2Φn1/4(x).
From (4.2) we see that
(4.11)an-1/2⩾Csup⁡|x|>x1,n|pn,ρ(x)w(x)|(|x|+ann)ρδn1/4~Csup⁡|x|>x1,n|pn,ρ(x)w(x)|(|x|+ann)ρΦn1/4(x).
Therefore we have the result.Proof of Proposition3.7.
We recall the definition ofXn(ν,f;x):
(4.12)Xn(ν,f;x)=∑j=1nf(xj,n)ljnν(x)∑i=0ν-2ei(ν,j,n)(x-xj,n)i=∑i=0ν-2∑j=1nf(xj,n)ljnν(x)(x-xj,n)iei(ν,j,n).
Using Lemma 2.3, we may estimate, for i=0,1,…,ν-2 and 1⩽j⩽n,
(4.13)Ai,j(x):=(Φ3/4(x)w(x)(|x|+ann)ρ)ν|f(xj,n)||ljn(x)|ν|x-xj,n|i(na2n2-xj,n2)i.
Let (xm+1,n+xm,n)/2<x⩽xm,n or xm,n⩽x<(xm-1,n+xm,n)/2. For simplicity, we assume x>0, and let x0,n:=x1,n+δφn(x1,n) for a fixed δ>0 small enough. Then we can assume that there exists a constant c>0 such that
(4.14)x0,n:=x1,n+δφn(x1,n)<an-cδn.
Assume that x<x0,n. Then we first estimate Ai,m(x). By Lemma 4.1(3) and the definition of φn(x), we have
(4.15)(Φ3/4(x)w(x)(|x|+ann)ρ)ν|lmn(x)|ν⩽CΦ3ν/4(x)(w(xm,n)(|xm,n|+ann)ρ)ν,(4.16)|x-xm,n|i(na2n2-xm,n2)i⩽C(nφn(xm,n)a2n2-xm,n2)i⩽C(1-|xm,n|/a2n1-|xm,n|/an)i,
and by Remark 3.6(4.17)|f(xm,n)|⩽C(f)Φ3ν/4(xm,n)(w(xm,n)(|xm,n|+ann)ρ)-ν.
Therefore, we have
(4.18)Ai,m(x)⩽C(f)Φ3ν/4(x)Φ3ν/4(xm,n)(1-|xm,n|/a2n1-|xm,n|/an)i=C(f)Φ3ν/4(x)(max⁡{1-|xm,n|an,δn})3ν/4(1-|xm,n|/a2n1-|xm,n|/an)i=C(f)Φ3ν/4(x)(1-|xm,n|an)3ν/4-i/2(1-|xm,n|a2n)i⩽C(f).
Next, we estimate ∑1⩽j⩽n,j≠mAi,j(x). For 1⩽j⩽n, j≠m, we have, by Lemma 4.3 and Lemma 4.1(4),
(4.19)(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)ρpn′(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)an-ν/2φnν(xj,n)|an2-xj,n2|ν/4(w(xj,n)(|xj,n|+an/n)ρ)ν|x-xj,n|ν-i.
Then, since we know from Remark 3.6  that
(4.20)|f(xj,n)|⩽C(f)Φ3ν/4(xj,n)(w(xj,n)(|xj,n|+ann)ρ)-ν,
we have, by Lemma 3.4,
(4.21)∑j≠mAi,j(x)⩽C(f)∑j≠mΦnν/2(x)Φn3ν/4(xj,n)φnν(xj,n)×(1-|xj,n|an)ν/4(1|x-xj,n|)ν-i(na2n2-xj,n2)i.
By the definition of φn(x) and by Lemma 3.3(a) a2n~an, we have
(4.22)φnν(xj,n)(na2n2-xj,n2)i⩽C(ann)ν-i(1-|xj,n|a2n)ν-i/2(1-|xj,n|an)-ν/2.
By Lemma 4.2(1), we have
(4.23)1|x-xj,n|⩽C1∑j⩽s⩽m+1orm-1⩽s⩽jφn(xs,n)⩽Cn/an∑j⩽s⩽m+1orm-1⩽s⩽j(1-|xs,n|/a2n)/(1-|xs,n|/an+δn)1/2⩽Cn/an∑j⩽s⩽m+1orm-1⩽s⩽j(1-|xs,n|/an)/(1-|xs,n|/an+δn)1/2⩽Cn/an|m-j|(1-max⁡{|x|,|xj,n|}/an)1/2.
Here, we note from (4.14) and (4.8)  that
(4.24)Φn(x)~(1-|x|an),Φn(xj,n)~(1-|xj,n|an).
Thus, we have, for j≠m,
(4.25)Φnν/2(x)Φn3ν/4(xj,n)φnν(xj,n)(1-|xj,n|an)ν/4(1|x-xj,n|)ν-i(na2n2-xj,n2)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.
Therefore, we have
(4.26)∑1⩽j⩽n,j≠mAi,j(x)⩽C(f)∑j≠m(1|m-j|)ν-i⩽C(f),
because of ν-i⩾2.Remark 4.4.
If we consider the estimate of∑i=1nAi,ν-1(x) with (4.13), then we obtain
(4.27)∑i=1nAi,ν-1(x)⩽C(f)log⁡n.
We continue the proof Proposition3.7. We need to estimate ∑j=1nAi,j(x) for |x|>x0,n. Now, suppose x>x0,n. Then similarly to (4.23), we have(4.28)1|x0,n-xj,n|⩽C1∑1⩽s⩽jφn(xs,n)⩽Cn/anj(1-x0,n/an)1/2⩽C1jδn1/2nan.
Similarly to (4.21), we have using (4.22)
(4.29)∑Ai,j(x)⩽C(f)∑jΦν/2(x)Φn3ν/4(xj,n)φnν(xj,n)×(1-|xj,n|an)ν/4(1|x0,n-xj,n|)ν-i(na2n2-xj,n2)i⩽C(f)∑jΦnν/2(x)(1-|xj,n|an)ν/2(1-|xj,n|a2n)ν-i/2(1jδn1/2)ν-i⩽C(f)∑j1jν-iC(f),
because of ν-i⩾2 and Φν/2(x)δn-(ν-i)/2⩽δnν/2δn-(ν-i)/2⩽C. Consequently we achieve the result.Remark 4.5.
The above proof implies the following: there exists a constantC>0 such that
(4.30)∑j=1n|hj,n,ρ(ν;x)|⩽∑j=1n|ljn(x)|ν∑i=0ν-2|ei(ν,k,n)||x-xj,n|i⩽C,x∈R.Proof of Proposition3.8.
We use the same method to prove of Proposition3.7. 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 assume x>0, and let x0,n:=x1,n+δφn(x1,n) for a fixed δ>0 small enough and there exists a constant δ1>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. Noting Assumption 3.1, (3.2), and Lemma 2.7, we estimate ∑j=1nBν-1,j(x), where
(4.31)Bν-1,j(x):=(Φ3/4(x)w(x)(|x|+ann)ρ)ν×|f(xj,n)||ljn(x)|ν|x-xj,n|ν-1{T(an)an+|Q′(xj,n)|+1|xj,n|}(na2n-|xj,n|)ν-2.
First we estimate Bν-1,m(x). By Lemma 4.2, (4.16), and the definition of φn(x), we have
(4.32)|x-xm,n|ν-1(na2n2-xm,n2)ν-2⩽Cφn(xm,n)(1-|xm,n|/a2n1-|xm,n|/an)ν-2⩽Cann(1-|xm,n|a2n)1/2(1-|xm,n|/a2n1-|xm,n|/an)ν-1.
Since here |Q′(xm,n)|+1/|xm,n|⩾C for some positive constant C, we know  that
(4.33)(T(an)an+|Q′(xm,n)|+1|xm,n|)⩽Cmax⁡{1,T(an)an}(|Q′(xm,n)|+1|xm,n|),
and by (3.25)
(4.34)|f(xm,n)|⩽C(f)Φ3ν/4(xm,n)(w(xm,n)(|xm,n|+ann)ρ)-ν(|Q′(xm,n)|+1|xm,n|)-1.
Therefore, using (3.3) and (4.15), we have
(4.35)Bν-1,m(x)⩽C(f)max⁡{1,T(an)an}Φn3ν/4(x)Φn3ν/4(xm,n)ann(1-|xm,n|a2n)1/2×(1-|xm,n|/a2n1-|xm,n|/an)ν-1⩽C(f)εnΦn3ν/4(x)Φn3ν/4(xm,n)(1-|xm,n|a2n)1/2(1-|xm,n|/a2n1-|xm,n|/an)ν-1.
Noting (4.24), we have
(4.36)Bν-1,m(x)⩽C(f)εnΦn3ν/4(x)(1-|xm,n|an)(ν+2)/4(1-|xm,n|a2n)ν/2⩽C(f)εn.
Next, we estimate ∑j≠mBν-1,j(x). Noting (4.19) and (4.22), we have
(4.37)(Φn3/4(x)w(x)(|x|+ann)ρ)ν|ljn(x)|ν|x-xj,n|ν-1=Φnν/2(x)an-ν/2φnν(xj,n)|an2-xj,n2|ν/4(w(xj,n)(|xj,n|+an/n)ρ)ν|x-xj,n|,φnν(xj,n)(na2n2-xj,n2)ν-2⩽C(ann)2(1-|xj,n|an)-ν/2(1-|xj,n|a2n)ν/2+1.
Then by (4.33) and (3.25) (noting (4.34)), using the notation (3.3) and (4.24), we have
(4.38)Bν-1,j(x)⩽C(f)max⁡{1,T(an)an}(ann)2Φnν/2(x)Φn3ν/4(xj,n)an-ν/2|an2-xj,n2|ν/4×1|x-xj,n|(1-|xj,n|an)-ν/2(1-|xj,n|a2n)ν/2+1⩽C(f)εnannΦnν/2(x)Φn3ν/4(xj,n)(1-|xj,n|an)-ν/4(1-|xj,n|a2n)ν/2+11|x-xj,n|⩽C(f)εnann(1-|x|an)ν/2(1-|xj,n|an)ν/2(1-|xj,n|a2n)ν/2+11|x-xj,n|.
Therefore, since we know from (4.23)  that
(4.39)1|x-xj,n|⩽Cn/an|m-j|(1-max⁡{|x|,|xj,n|}/an)1/2,
noting (4.24), we have
(4.40)∑j≠mBν-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≠m1|m-j|⩽C(f)εnlog⁡n.
Finally, we estimate ∑j=1nBν-1,j(x) for x>x0,n. Suppose x>x0,n. Then similar to the above computations, we have
(4.41)Bν-1,j(x)⩽C(f)εnannΦν/2(x)Φn3ν/4(xj,n)(1-|xj,n|an)-ν/4(1-|xj,n|a2n)ν/2+11|x0,n-xj,n|⩽C(f)εnannΦν/2(x)(1-|xj,n|an)ν/2(1-|xj,n|a2n)ν/2+11|x0,n-xj,n|.
Then, since we know by (4.28)  that
(4.42)1|x0,n-xj,n|⩽C1jδn1/2nan,
we have
(4.43)Bν-1,j(x)⩽C(f)εnΦν/2(x)(1-|xj,n|an)ν/2(1-|xj,n|a2n)ν/2+11jδn1/2.
Then, since Φν/2(x)δn-1/2<C for x>x0,n, we have
(4.44)∑j=1nBν-1,j(x)⩽C(f)εn∑j=1n1j⩽C(f)εnlog⁡n.
Therefore, the result is proved.Proof of Proposition3.9.
By Lemma2.3 we see that, for a constant C1>0,
(4.45)|Zn(l,ν,f;x)|⩽C1∑j=1n∑s=1l|f(s)(xj,n)||ljn(x)|ν∑i=sν-1(na2n2-xj,n2)i-s|x-xj,n|i⩽C1ann∑j=1n∑s=1l|f(s)(xj,n)||ljn(x)|ν∑i=sν-1(na2n2-xj,n2)i|x-xj,n|i⩽C1ann∑s=1l∑i=0ν-1∑j=1n|f(s)(xj,n)||ljn(x)|ν|x-xj,n|i(na2n2-xj,n2)i=:ann∑s=1l∑i=0ν-1Ci,s(x),
where,
(*)Ci,s(x):=∑j=1n|f(s)(xj,n)||ljn(x)|ν|x-xj,n|i(na2n2-xj,n2)i.
We set
(4.46)Xi,s,n(ν,f;x):=∑j=1nf(s)(xj,n)ljnν(x)ei(ν,j,n)(x-xj,n)i,i=0,1,…,ν-1,
and hereafter we wrote (*) as |X|i,s,n(ν,f;x). Now, we repeat the proof of Proposition 3.7 by exchanging f(x) with f(s)(x), s=1,2,…,ν-1, and we note (3.23) (and (3.26)). Then, for 0⩽i⩽ν-2 we obtain
(4.47){Φn3/4(x)w(x)(|x|+ann)}ν|X|i,s,n(ν,f;x)⩽C1C(f).
For i=ν-1, we use the estimate for lmn(x) in the proof of Proposition 3.7; furthermore we use Remark 4.4. Then we have
(4.48){Φn3/4(x)w(x)(|x|+ann)}ν|X|ν-1,s,n(ν,f;x)⩽C1C(f)log⁡n.
Consequently, we have
(4.49){Φn(x)3/4w(x)(|x|+ann)}ν|Zn(l,ν,f;x)|⩽C1C(f)anlog⁡nn.
Similarly, we have
(4.50){Φ(x)3/4w(x)(|x|+ann)}ν|Wn(l,ν,G;x)|⩽C1C(G)anlog⁡nn.Proof of Proposition3.10.
LetP∈𝒫m be fixed. From Proposition 3.9 we see
(4.51)‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν{Ln(ν,P)-P}‖L∞(R)=‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν{Ln(ν,P)-Ln(ν-1,ν,P)}‖L∞(R)=‖(Φ3/4(x)w(x)(|x|+ann)ρ)νZn(ν-1,ν,P)‖L∞(R)⟶0asn⟶∞.Proof of Proposition3.11.
LetP∈𝒫m with P(0)=0. Then P satisfies the condition (A-1). By Proposition 3.8, we see
(4.52)‖(Φ3/4(x)w(x)(|x|+ann)ρ)νYn(ν,P)‖L∞(R)⟶0asn⟶∞.
So, from Propositions 3.10 and 3.8 (noting (3.3)) we have
(4.53)‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(Xn(ν,P)-P)‖L∞(R)⩽C1‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(Ln(ν,P)-P)‖L∞(R)+‖(Φ3/4(x)w(x)(|x|+ann)ρ)νYn(ν,P)‖L∞(R)⟶0asn⟶∞.Proof of Theorem3.12.
Sincef satisfies (3.22), we see
(4.54)lim⁡|x|→∞|f(x)|w(x)ν-δ=0.
For a given ε>0, there exists a polynomial P∈𝒫m with P(0)=0 such that
(4.55)sup⁡x∈R|f(x)-P(x)|{Φ-3/4(x)w(x)(|x|+ann)ρ}ν<ε.
In fact, by [21, Theorem 1.4], there exists a polynomial R∈𝒫m such that
(4.56)sup⁡x∈R|f(x)-R(x)|wν-δ(x)<ε2.
Let P(x):=R(x)-R(0). Noting  that
(4.57){Φ-3/4(x)w(x)(|x|+ann)ρ}ν⩽Cwν-δ(x),x∈RforsomeC>0
and f(0)=0, we have
(4.58)sup⁡x∈R|f(x)-P(x)|{Φ-3/4(x)w(x)(|x|+ann)ρ}ν⩽sup⁡x∈R|f(x)-P(x)|wν-δ(x)⩽sup⁡x∈R|f(x)-R(x)|wν-δ(x)+|f(0)-R(0)|wν-δ(0)<ε;
that is, we have (4.55). Here, we know that
(4.59)Ln(ν,f)(x)-f(x)=Xn(ν,f-P)+(Xn(ν,P)-P)+(P-f)+Yn(ν,f).
Therefore, by Propositions 3.7, 3.11, and 3.8, we have for n large enough
(4.60)‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(Ln(ν,f)-f)‖L∞(R)⩽C‖(Φ3/4(x)w(x)(|x|+ann)ρ)νXn(ν,f-P)‖L∞(R)+‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(Xn(ν,P)-P)‖L∞(R)+‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(P-f)‖L∞(R)+‖(Φ3/4(x)w(x)(|x|+ann)ρ)νYn(ν,f)‖L∞(R)=:In+IIn+IIIn+IVn.
Here we see that, by Proposition 3.7 with ε:=C(f-P) (constant depending only on f-P) and (4.55),
(4.61)In⩽Cε,IIIn⩽ε,
and for n⩾n0 large enough, we have
(4.62)IIn⩽Cε,(byProposition3.9),IVn⩽Cε,(byProposition3.6).
Consequently, noting (3.3), we have
(4.63)lim⁡n→∞{In+IIn+IIIn+IVn}=0.
Then we have Theorem 3.12.Proof of Theorem3.13.
From Theorem3.12 and Proposition 3.9, we have
(4.64)‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(Ln(l,ν,f)-f)‖L∞(R)⩽C1‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(Ln(ν,f)-f)‖L∞(R)+‖(Φ3/4(x)w(x)(|x|+ann)ρ)νZn(l,ν,f)‖L∞(R)⟶0asn⟶∞.Proof of Corollary3.14.
From Theorem3.12 and Proposition 3.8, we have
(4.65)‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(Xn(ν,f)-f)‖L∞(R)⩽C1‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(Ln(ν,f)-f)‖L∞(R)+‖(Φ3/4(x)w(x)(|x|+ann)ρ)νYn(ν,f)‖L∞(R)⟶0asn⟶∞.Proof of Theorem3.15.
From Theorem3.13 and Proposition  3.10 we have
(4.66)‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(L̃n(l,ν,f⊕G)-f)‖L∞(R)⩽C‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν(Ln(l,ν,f)-f)‖L∞(R)+‖(Φ3/4(x)w(x)(|x|+ann)ρ)νWn(l,ν,G)‖L∞(R)⟶0asn⟶∞.Proof of Theorem3.16.
We use Theorems3.12, 3.13, and Corollary 3.14, and Theorem 3.15. Then we have(4.67)|In[k,Hn(f);f]-I[k,f]|=∫-∞∞k(x){Hn(f)(x)-f(x)}dx⩽C‖(Φ3/4(x)w(x)(|x|+ann)ρ)ν{Hn(f)(x)-f(x)}‖L∞(R)×∫-∞∞(Φ3/4(x)w(x)(|x|+ann)ρ)νk(x)dx⟶0asn⟶∞.


## 5. Divergence Theorem

Ifν is a positive odd integer,  then we obtain the unboundedness of Ln(ν,f;x).  We define
(5.1)Λn(ν,R)=max⁡x∈R∑k=1n|hkn(x)|.Theorem 5.1 (cf. [1, Theorem 2]).
Letν>0 be an odd integer. Then there exists a constant  C>0 and  n0>0 such that for  n⩾n0(5.2)Λn(ν,R)⩾Clog⁡n.Let-∞⩽a<b⩽∞, and let us define
(5.3)Λn(ν,(a,b))=max⁡a⩽x⩽b∑k=1n|hkn(x)|.
Then we will show that for n⩾n0(5.4)Λn(ν,(a,b))⩾Clog⁡n.Remark 5.2.
For the interpolation polynomialLn(ν,f;x), we can see a remarkable difference between the cases of an odd number ν and of an even number ν. Let us consider any continuous function f∈C([a,b]),0<a<b. Then we can extend f to a continuous function f*∈C(ℝ) which satisfies (3.22) and f(x)=f*(x), x∈[a,b]. Then from Theorem 3.12 for an even positive integer ν, we see
(5.5)‖Ln(ν,f*;x)-f(x)‖L∞([a,b])⟶0asn⟶∞.
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
(5.6)‖Ln(ν,f*;x)-f*(x)‖L∞([a,b])↛0asn⟶∞;
that is, for f(x):=f*(x), a⩽x⩽b we see that the interpolation polynomials Ln(ν,f*;x) do not converge to f. We also remark that the polynomial Ln(ν,f*;x) interpolates f∈C[a,b] at only {xj,n∈[a,b],1⩽j⩽n}.
To prove the theorem we use the following lemma.Lemma 5.3 (see [18, Theorem 11]).
Forj=0,1,2,…, there is a polynomial  Ψj(x) of degree  j such that  (-1)jΨj(-ν)>0 for  ν=1,3,5,…, and the following relation holds. Let  0<ε<1. Then one has an expression for  (1/ε)(an/n)⩽|xk,n|⩽εan and  s=0,1,…,(ν-1)/2:
(5.7)e2s(ν,k,n)=(-1)s{1(2s)!}Ψs(-ν)βs(n,k)(nan)2s{1+ηkn(ν,s)},
where for βs(n,k), n=1,2,3,…, k=1,2,…,n, s=0,1,…,(ν-1)/2, there exist the constants C1,C2 such that
(5.8)0<C1⩽βs(n,k)⩽C2,
and ηkn(ν,s) satisfies
(5.9)|ηkn(ν,s)|⩽εn*,εn*=εn*(ε)⟶0asε⟶0,
for k with (1/ε)(an/n)⩽|xk,n|⩽εan, and for s=0,1,2,….Proof of Theorem5.1.
To get a lower bound ofΛn(ν,ℝ), it suffices to consider a lower bound (5.4) of Λn(ν,[a,b]) with 0<a<b<∞. Let c:=a/2,d:=2b-a and we consider the intervals I:=[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
(5.10)|hkn(x)|⩾|lknν(x)eν-1(ν,k,n)(x-xk,n)ν-1|-|lknν(x)∑i=0ν-2ei(ν,k,n)(x-xk,n)i|.
Then we have
(5.11)Λn(ν,[a,b])⩾max⁡x∈I∑xkn∈J|lknν(x)eν-1(ν,k,n)(x-xk,n)ν-1|-max⁡x∈I∑xkn∈J|lknν(x)∑i=0ν-2ei(ν,k,n)(x-xk,n)i|=:max⁡x∈IFn(x)-max⁡x∈IGn(x).
It follows from Remark 4.5 that maxa⩽x⩽bGn(x)⩽C. Hence, it is enough to show that maxx∈IFn(x)⩾Clogn. 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
(5.12)αann⩽|xk,n-xk+1,n|⩽βann.
For x∈I we consider only xk,n∈J such that, for some positive integer jk,
(5.13)(jk-1)αann⩽|xk,n-x|⩽jkβann.
Then for each x∈I we define
(5.14)Γ(x):={jk;xk,n∈J,(jk-1)αann⩽|xk,n-x|⩽jkβann}.
Here, we will see that
(5.15){j;1⩽j⩽d-c2βnan-1}⊂Γ(x).
Let
(5.16)xm+1,n<x⩽xm,n,xk(c)+1,n<c⩽xk(c),n,xk(d),n⩽d<xk(d)-1,n.
Then we have
(5.17)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,
where j(c) and j(d) are integers. On the other hand,
(5.18)(xk(d),n-xk(c),n)(βan/n)⩾(d-c-2βan/n)(βan/n)=d-cβnan-2.
Therefore, we have
(5.19)max⁡(j(c),j(d))⩾12(d-cβnan-2)=d-c2βnan-1.
Now, we take a positive integer N(n) such that
(5.20)N(n)⩽d-c2βnan-1<N(n)+1.
Consequently, we have the following. By Lemmas 4.2(1), Lemma 4.1(1),  (2), (4), and Lemma 2.3, we have
(5.21)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||pn′(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)|pn′(xk,n)|wρ(xk,n)wρ(xk,n)wρ(x))ν|eν-1(ν,k,n)|1|x-xk,n|⩾C∑jk∈Γ(x)(an-1/2(n/an)an-1/2wρ(d)wρ(a))ν|eν-1(ν,k,n)|njkβan⩾C∑jk∈Γ(x)1jk(ann)ν-1|eν-1(ν,k,n)|.
Here, using Lemma 5.3 and (5.15), we have
(5.22)Fn(x)⩾C∑jk∈Γ(x)1jk(ann)ν-1(nan)ν-1⩾C∑1⩽j⩽N(n)1j.
Here, there exists 0<η<1 such that nη⩽N(n). Therefore we see
(5.23)Fn(x)⩾C∑1⩽j⩽nη1j⩾Clog⁡n.
So we have (5.4), and consequently the proof of Theorem 5.1 is completed.



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*Source: 904169-2012-05-16.xml*