# Interpolation Polynomials of Entire Functions for Erdös-Type Weights

Authors: Gou Nakamura; Ryozi Sakai; Noriaki Suzuki
Journal: Journal of Mathematics
 (2013)

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## Abstract

Letℝ=(−∞,∞), and let Q∈C1(ℝ):ℝ→[0,∞) be an even function. In this paper, we consider some Lagrange interpolation polynomials and the Gauss-Jacobi quadrature formula of entire functions associated with Erdös-type weights w(x)=e−Q(x), x∈ℝ, and we will estimate the error terms.


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## Body

## 1. Introduction

For a weightw, we suppose that ∫0∞‍xnw2(x)dx<∞ for all n=0,1,2,…. Then we define a unique sequence of orthonormal polynomials {pn(w;x)} as follows:
(1)∫-∞∞‍pn(w;x)pm(w;x)w(x)dx=δn,m(theKroneckersymbol),
where pn(w;x):=γnxn+⋯,γn>0 is a polynomial of degree n. Furthermore, we denote the zeros of pn(w;x) by xk,n  (k=1,2,…,n),  x1,n>x2,n>⋯>xn,n. Hence,
(2)pn(w;x)=γn(x-xn,n)⋯(x-x2,n)(x-x1,n).
For a given function f, the Lagrange interpolation polynomial Ln(w;f) with the weight w is defined to be a unique polynomial of degree at most n-1 which coincides with f at xk,n,  k=1,2,…,n. Then we have the representation
(3)Ln(w;f;ξ)=∑k=1n‍f(xk,n)lk,n(ξ),
where
(4)lk,n(ξ)=pn(w;ξ)pn′(w;xk,n)(ξ-xk,n),k=1,2,…,n.
The Gauss-Jacobi quadrature formula to the function f is defined by
(5)Qn(w;f):=∑k=1n‍λn(w;xk,n)f(xk,n),
where the coefficients λn(w;xk,n) are the Christoffel numbers, which are given by
(6)λn(w;x):=(∑k=1n-1‍pk(w;x)2)-1.We denote byW a class of all weight functions of the form w=exp(-2Q) such that Q is an even differentiable function on ℝ, except possibly at x=0, increasing for x>0, and there exists ρ<1 such that xρQ′(x) is increasing. In addition, the unique positive sequence {qn} determined by qnQ′(qn)=n satisfies the condition q2n/qn≥C>1,n=1,2,…, for some constant C.For an entire functionf and R>0, we set
(7)Mf(R):=max|z|=R|f(z)|,z∈ℂ.
In [1, 2], Al-Jarrah showed the following results (see also [3–5]).Theorem 1.
Letw∈W and f be an entire function.(1)
There exists a constantA∈(0,1), depending on Q only, such that if f satisfies
(8)limsupR→∞logMf(R)2Q(R)≤A,
then, uniformly on compact subsets of ℂ,
(9)limsupn→∞|f(ξ)-Ln(w;f;ξ)|1/n<1,limsupn→∞|∫ℝ‍f(x)w(x)dx-Qn(w;f)|1/n<1.(2)
If the entire functionf satisfies
(10)limR→∞logMf(R)2Q(R)=0,
then, uniformly on compact subsets of ℂ,
(11)limn→∞|f(ξ)-Ln(w;f;ξ)|1/n=0,limn→∞|∫ℝ‍f(x)w(x)dx-Qn(w;f)|1/n=0.In this paper, we extend the above results to Erdös-type weights. Note that every weight inW is not Erdös-type (see Remark 4). In Section 2, we recall the definition of ℱ(lip(1/2)+), which is the class of weights we discuss here. Our theorems are stated in Section 3.


## 2. Preliminaries

For any nonzero real valued functionsf(x) and g(x), we write f(x)~g(x) if there exists a constant C>1 such that C-1g(x)≤f(x)≤Cg(x) for all x. Throughout this paper, C denotes a positive constant independent of n,x,t. The same symbol does not necessarily denote the same constant in different occurrences. We say that f:ℝ+:=[0,∞)→ℝ+ is quasi-increasing if there exists C>0 such that f(x)≤Cf(y),0<x<y.First, we recall a class of weights.Definition 2 (see cf. [6, Definition 1.2]).
LetQ:ℝ→[0,∞) be a continuous even function. We write w=exp(-Q)∈ℱ(lip(1/2)+) if Q satisfies the following properties.(a)
Q′(x) is continuous in ℝ, with Q(0)=0.(b)
Q′(x) is nondecreasing in ℝ.(c)
limx→∞Q(x)=∞.(d)
The functionT(x):=xQ′(x)/Q(x), x≠0, is quasi-increasing in (0,∞) and T(x)⩾Λ>1, x∈ℝ∖{0}.(e)
There existsε0∈(0,1) such that for x∈ℝ∖{0}, T(x)~T(x(1-(ε0/T(x)))).(f)
For anyε>0, there exists δ>0 such that for any x∈ℝ∖{0},
(12)∫x-δ|x|/T(x)x+δ|x|/T(x)‍Q′(s)-Q′(x)|s-x|3/2ds≤ε|Q′(x)|T(x)|x|.Example 3.
Letw=exp(-Q)∈ℱ(lip(1/2)+).(1)
IfT(x) is bounded, then the weight w(x) is called a Freud-type weight (see [5]). We see that w(x)=exp(-|x|α), α>1, is a Freud-type weight.(2)
IfT(x) is unbounded, then w(x) is called an Erdös-type weight. The following Qs give Erdös-type weights w=exp(-Q) (see [7, Example 1.2], [8, Theorem 3.1]). Let expl(x):=exp(exp(exp⋯expx)⋯)(ltimes) for l=1,2,…. Then for α+β>1, α>0, and β≥0, we set
(13)Q(x)=Ql,α,β(x):=|x|β(expl(|x|α)-expl(0)).Also, for α>1, put Q(x)=Qα(x):=(1+|x|)|x|α-1.Remark 4.
Every weightw in W is Freud type. In fact, by definition, q2k≥Ckq1. On the other hand, if we assume that w is Erdös type, then for any given M>0, there exists a constant C>0 such that Q′(x)/Q(x)=T(x)/x≥(M+1)/x for x>C. By integration, we see
(14)xM+1≤CQ(x)
for x>C. Since qnQ(qn)=n, qnM+1≤CQ(qn)=Cn/qn; that is, qn≤Cn1/(M+2). If we take M>0 sufficiently large, this contradicts the fact that q2k≥Ckq1 (cf. [7, Lemma 3.2]).Foru>0, we define the Mhaskar-Rakhmanov-Saff number au by
(15)u=2π∫01‍autQ′(aut)(1-t2)1/2dt.Lemma 5.
Letw=exp(-Q)∈ℱ(lip(1/2)+).(1)
([6, Lemma  3.4 (3.18)]) For t>0, Q(at)~t/T(at).(2)
([6, Theorem 1.19 (f)]) There exists n0>0 such that for n⩾n0, 1-x1,n/an~1/(nT(an))2/3. In particular x1,n≤an holds.(3)
([6, Theorem 15.2 (15.8)]) There exists C>0 such that γn-1/γn≤Can for all n=1,2,….For an entire functionf and n=1,2,…, we set
(16)En(ξ):=f(ξ)-Ln(w;f;ξ).
Then by the residue theorem, for ξ∈Dn, we have
(17)En(ξ)=pn(w;ξ)2πi∫Cn‍f(z)dzpn(w;z)(z-ξ),
where Dn is a simply connected domain containing all zeros of pn(w) and Cn is its boundary. Note that for any ξ∈ℂ, there exists n such that ξ∈Dn. For each m=1,2,… we also set
(18)Ln,m(w;f;ξ)≔∑j=1n‍∑s=0m-1‍1(m-1-s)!(f(z)γnm∏k≠j(z-xk,n)m)(m-1-s)|z=xj,n×pn(w;ξ)m(ξ-xj,n)s+1,En,m(ξ):=f(ξ)-Ln,m(w;f;ξ).
Then Ln,m(w;f;ξ) is a polynomial of degree at most nm-1 such that
(19)Ln,m(j)(w;f;xk,n)=f(j)(xk,n),k=1,2,…,n,j=1,2,…,m-1
holds. By the residue theorem again, we see
(20)En,m(ξ)=pn(w;ξ)m2πi∫Cn‍f(z)dzpn(w;z)m(z-ξ)
for ξ∈Dn.For the Gauss-Jacobi quadrature, we set(21)En,*:=∫ℝ‍f(x)w(x)dx-Qn(w;f).
Then, we also see
(22)En,*=∑k=n∞‍γk+1γk12πi∫Ck‍f(z)dzpk(w;z)pk+1(w;z),
(cf. [1–5]).


## 3. Theorems and Proofs

We state our theorems. Throughout this section, we assumew=exp(-Q)∈ℱ(lip(1/2)+), and let
(23)η:=limsupt→∞Q(at)T(at)t.
Note that η is determined finitely by Lemma 5(1).Theorem 6.
(1) If an entire functionf satisfies
(24)limsupR→∞logMf(5R)T(R)Q(R)≤1η,
then
(25)limsupn→∞|f(ξ)-Ln(w;f;ξ)|1/n<1
uniformly on compact subsets; that is, for any compact subset K in ℂ, there exists a constant 0<c<1 such that
(26)limsupn→∞|f(ξ)-Ln(w;f;ξ)|1/n≤c
for ξ∈K. Moreover, for any m=1,2,…,
(27)limsupn→∞|f(ξ)-Ln,m(w;f;ξ)|1/n<1
holds also uniformly on compact subsets.
(2) If an entire functionf satisfies
(28)limsupR→∞logM(10R)T(R)Q(R)≤1η,
then for any j=1,2,3,…,
(29)limsupn→∞|f(j)(ξ)-Ln(w;f(j);ξ)|1/n<1
uniformly on compact subsets. Moreover, for any m=1,2,…,
(30)limsupn→∞|f(j)(ξ)-Ln,m(w;f(j);ξ)|1/n<1
holds also uniformly on compact subsets.As for the Gauss-Jacobi quadrature, we have the following theorem.Theorem 7.
Letf be an entire function. If
(31)limsupR→∞logMf(λ0R)T(R)Q(R)≤1η
holds for some λ0>1 large enough, then
(32)limsupn→∞|∫ℝ‍f(x)w(x)dx-Qn(w;f)|1/n≤c<1,
for a constant c>0. Further, if
(33)limsupR→∞logMf(2λ0R)T(R)Q(R)≤1η
holds, then for any j=1,2,3,…,
(34)limsupn→∞|∫ℝ‍f(j)(x)w(x)dx-Qn(w;f(j))|1/n≤c<1.Theorem 8.
Letf be an entire function. Suppose that
(35)limR→∞logMf(λR)T(R)Q(R)=0,
for every λ>0 large enough. Then
(36)limn→∞|f(ξ)-Ln(w;f;ξ)|1/n=0
holds uniformly on compact subsets of ℂ and for any j=1,2,3,…,
(37)limn→∞|f(j)(ξ)-Ln,m(w;f(j);ξ)|1/n=0
also holds uniformly on compact subsets of ℂ. Moreover for any j=1,2,3,…,
(38)limn→∞|∫ℝ‍f(j)(x)w(x)dx-Qn(w;f(j))|1/n=0
holds true.Remark 9.
The order of an entire functionf is given by
(39)μ(f):=limsupR→∞loglogMf(R)logR.
Also the type of f with order μ<∞ is given by
(40)τ(f):=τμ(f)=limsupR→∞logMf(R)Rμ.
If w=exp(-Q) is an Erdös-type weight and μ(f),τ(f)<∞, then (35) holds for every λ>0. In fact, by (14),
(41)limR→∞logMf(λR)T(R)Q(R)=limR→∞logMf(λR)(λR)μ(λR)μT(R)Q(R)=0.
The following lemma was proved in [2, Lemmas 3.3 and 3.4]. Its proof is available for our case.Lemma 10.
Letw=exp(-Q)∈ℱ(lip(1/2)+), and let pn(w;z) be the nth orthonormal polynomial generated by w. Then
(42)|pn(w;z)|≤γn2n/2x1,nn,
for all z such that |z|≤x1,n, and
(43)|pn(w;z)|-1≤1γn|z|nexp(nx1,n22(|z|2-x1,n2))
for all z such that |z|>x1,n. Also we have
(44)∑j=1[n/2]‍xj,n2=∑j=1n-1‍(γj-1γj)2.Proof of Theorem6.
(1) We estimate the error formEn(ξ) given by the formula (17). For a given compact subset K, choose a number n such that |ξ|≤x1,n for ξ∈K. We may take the path of integration Cn to be the circle |z|=5an because x1,n≤an. By (42), we see |pn(w;ξ)|≤γn2n/2ann. From the definition of η and (24), for any ε>0, there exists Nε>0 such that for n≥Nε,
(45)Q(an)T(an)≤(η+ε)n,logMf(5an)≤(1η+ε)T(an)Q(an).
Hence by (43), for n large enough,
(46)|En(ξ)|≤Cγn2n/2ann1γn|5an|n×exp(nx1,n22((5an)2-x1,n2))Mf(5an)2π∫Cn‍|dz|4an≤C2n/215nexp{n48}exp((1η+ε)Q(an)T(an))≤C2n/215nexp{n48}exp((1η+ε)(η+ε)n).
Here we select ε>0 such as
(47)148+(1η+ε)(η+ε)≤2524.
Then we have for n≥Nε and any ξ∈K,
(48)|En(ξ)|≤C(25)nexp(25n24)=C((25)exp(2524))n.
Since (2/5)exp(25/24)=:c<1, we conclude (26). In the above proof, we only exchange (48) for
(49)|En,m(ξ)|≤C(25)nmexp(49n+nm48)≤C((25)exp(2524))nm.
Then (27) follows.
(2) Since(50)f(j)(z)=j!2πi∫C2R‍f(ζ)(ζ-z)j+1dζ,
where C2R={ζ;|ζ|=2R}, we see Mf(j)(R)≤2(j!)Mf(2R)/Rj. Hence, (28) gives us
(51)limsupR→∞logMf(j)(5R)T(R)Q(R)≤limsupR→∞log(2(j!)/Rj)+log(Mf(10R))T(R)Q(R)≤limsupR→∞logMf(10R)T(R)Q(R)≤1η.
Thus, (29) and (30) follow from (26) and (27), respectively.Proof of Theorem7.
First, we show the casej=0. From Lemma 5(3) and (44), we have
(52)∑k=1[n/2]‍xk,n2≤C∑k=1n-1‍ak2≤Can-12(n-1),
where C>0 is an absolutely constant. Also by Lemma 5(3), we see
(53)1γn≤Canγn-1≤⋯≤Cn(anan-1⋯a1)≤(Can)n.
Now let |z|>x1,n. Since
(54)|pn(w;z)|-1≤1γn|z|nexp(1|z|2-x1,n2∑j=1[n/2]‍xj,n2),
(see [3, page 131]), (52) gives us
(55)|pn(w:z)|-1≤1γn|z|nexp(Can-12(n-1)|z|2-x1,n2).
Therefore, for |z|>x1,n+1, we conclude that
(56)|pn(w;z)pn+1(w;z)|-1≤1γnγn+1|z|2n+1exp(2Can2n|z|2-x1,n+12).
We consider the circle Cn={z;|z|=λ0an}, where we will choose λ0>0 large enough later. Now we estimate
(57)In:=γn+1γn12πi∫Cn‍f(z)dzpn(w;z)pn+1(w;z).
From Lemma 5(1), (53), (56), and our assumption (31), we have for δ>0 small enough,
(58)|In|≤1γn21|λ0an|2n+1Mf(λ0an)exp(2Can2n|λ0an|2-x1,n2)λ0an≤(Can)2nexp{(1/η+δ)(η+δ)n}(λ0an)2nexp(2Cnλ02-1)≤((Cλ0)2exp((1η+δ)(η+δ))exp(Cλ02-1))n.
Here we select λ0 as
(59)0<c:=(Cλ0)2exp((1η+δ)(η+δ)+Cλ02-1)<1.
Then we have
(60)|En,*|≤∑j=n∞‍|Ij|≤cn1-c,
which shows that
(61)limsupn→∞|En,*|1/n≤climsupn→∞(11-c)1/n=c<1.
For the case of j⩾1, as in the proof of (2) of Theorem 6, (33) gives us
(62)limsupR→∞logMf(j)(λ0R)T(R)Q(R)≤limsupR→∞Mf(2λ0R)T(R)Q(R)≤1η.
Hence, (34) follows from (32).Proof of Theorem8.
In (17), we take the path of integration Cn to be the circle |z|=λan  (λ>0). Then we have
(63)|En(ξ)|≤C((2λ)exp(λ2λ2-1))n.
In this right-hand side, we can take λ as large as we like because of our assumption (35). Hence, we have (36). Similarly as above, (37) follows from (36). Also Theorem 7 shows (38).



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