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

Abstract

Let ℝ=(−∞,∞), and let 𝑤_𝜌(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 𝑝_𝑛(𝑤^2𝜌;𝑥) of degree 𝑛 for 𝑤^2𝜌(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 {𝑥_{𝑘,𝑛,𝜌}}^𝑛𝑘=1 of 𝑝_𝑛(𝑤^2𝜌;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 {𝑥_{𝑘,𝑛,𝜌}}^𝑛𝑘=1 of 𝑝_𝑛(𝑤^2𝜌;𝑥).

1. Introduction

Let ℝ=(−∞,∞), 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, 𝑤_𝜌(𝑥)∶=|𝑥|^𝜌𝑤(𝑥),𝑥∈ℝ.(1.1) Suppose that ∫^∞0𝑥^𝑛𝑤_𝜌(𝑥)𝑑𝑥<∞, for all 𝑛=0,1,2,…. Then we can construct the orthonormal polynomials 𝑝_𝑛,𝜌(𝑥)=𝑝_𝑛(𝑤^2𝜌;𝑥) of degree 𝑛 for 𝑤^2𝜌(𝑥); that is, ^∞−∞𝑝_𝑛,𝜌(𝑥)𝑝_𝑚,𝜌(𝑥)𝑤^2𝜌(𝑥)𝑑𝑥=𝛿_𝑚,𝑛(Kroneckerdelta).(1.2) We write 𝑝_𝑛,𝜌(𝑥) by 𝑝_𝑛,𝜌(𝑥)=𝛾_𝑛𝑥^𝑛+⋯,𝛾_𝑛=𝛾_𝑛,𝜌>0,(1.3) and denote the zeros of 𝑝_𝑛,𝜌(𝑥) by −∞<𝑥_{𝑛,𝑛,𝜌}<𝑥_{𝑛−1,𝑛,𝜌}<⋯<𝑥_2,𝑛,𝜌<𝑥_1,𝑛,𝜌<∞.(1.4)

Let 𝒫_𝑛 denote the class of polynomials with degree at most 𝑛. For 𝑓∈𝐶(ℝ) we define the higher-order Hermite-Fejér interpolation polynomial 𝐿_𝑛(𝜈,𝑓;𝑥)∈𝒫_𝜈𝑛−1 based at the zeros {𝑥_{𝑘,𝑛,𝜌}}^𝑛𝑘=1 as follows: 𝐿^𝑛(𝑖)𝜈,𝑓;𝑥_{𝑘,𝑛,𝜌}=𝛿_0,𝑖𝑓𝑥_{𝑘,𝑛,𝜌}for𝑘=1,2,…,𝑛,𝑖=0,1,…,𝜈−1.(1.5) We note that 𝐿_𝑛(1,𝑓;𝑥) is the Lagrange interpolation polynomial, 𝐿_𝑛(2,𝑓;𝑥) is the ordinary Hermite-Fejér interpolation polynomial, and 𝐿_𝑛(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 ℎ_{𝑘,𝑛,𝜌}(𝜈;𝑥)∈𝒫_𝜈𝑛−1 for the higher-order Hermite-Fejér interpolation polynomial 𝐿_𝑛(𝜈,𝑓;𝑥) are defined as follows: ℎ_{𝑘,𝑛,𝜌}(𝜈;𝑥)=𝑙^{𝜈𝑘,𝑛,𝜌}(𝑥)^𝜈−1_𝑖=0𝑙_{𝑘,𝑛,𝜌}𝑒_𝑖(𝜈,𝑘,𝑛)𝑥−𝑥_{𝑘,𝑛,𝜌}^𝑖,(𝑥)=𝑝^𝑛𝑤^2𝜌;𝑥ℎ_{𝑘,𝑛,𝜌}𝑥−𝑥_{𝑘,𝑛,𝜌}𝑝^𝑛𝑤^2𝜌;𝑥_{𝑘,𝑛,𝜌},𝑘𝜈,;𝑝𝑥=1_{𝑝,𝑛,𝜌},2=,…𝛿,_𝑘,𝑝𝑛,,𝑖ℎ=1,^{(𝑖)𝑘,𝑛,𝜌}2,𝜈…;𝑥,𝜈−_{𝑝,𝑛,𝜌}1=.0,(1.6) Using them, we can write 𝐿_𝑛(𝜈,𝑓;𝑥)=^𝑛_𝑘=1𝑓𝑥_{𝑘,𝑛,𝜌}ℎ_{𝑘,𝑛,𝜌}(𝜈;𝑥).(1.7) Furthermore, we extend the operator 𝐿_𝑛(𝜈,𝑓;𝑥). Let 𝑙 be a nonnegative integer, and let 𝜈−1⩾𝑙. For 𝑓∈𝐶^𝑙(ℝ) we define the (𝑙,𝜈)-order Hermite-Fejér interpolation polynomials 𝐿_𝑛(𝑙,𝜈,𝑓;𝑥)∈𝒫_𝜈𝑛−1 as follows: for each 𝑘=1,2,…,𝑛, 𝐿_𝑛(𝑗)𝐿_𝑛𝑙,𝜈,𝑓;𝑥_{𝑘,𝑛,𝜌}=𝑓𝑥_{𝑘,𝑛,𝜌},𝐿𝑙,𝜈,𝑓;𝑥_{𝑘,𝑛,𝜌}=𝑓^(𝑗)𝑥_{𝑘,𝑛,𝜌},𝑗=1,2,…,𝑙,_𝑛(𝑗)𝑙,𝜈,𝑓;𝑥_{𝑘,𝑛,𝜌}=0,𝑗=𝑙+1,𝑙+2,…,𝜈−1.(1.8) Especially, 𝐿_𝑛(0,𝜈,𝑓;𝑥) is equal to 𝐿_𝑛(𝜈,𝑓;𝑥), and for each 𝑃∈𝒫_𝜈𝑛−1 we see 𝐿_𝑛(𝜈−1,𝜈,𝑃;𝑥)=𝑃(𝑥); that is, for 𝑓∈𝐶^(𝜈−1)(ℝ), 𝐿_𝑛(𝜈−1,𝜈,𝑓;𝑥) is the Hermite interpolation polynomial. The fundamental polynomials ℎ_{𝑠,𝑘,𝑛,𝜌}(𝑙,𝜈;𝑥)∈𝒫_𝜈𝑛−1, 𝑘=1,2,…,𝑛, of 𝐿_𝑛(𝑙,𝜈,𝑓;𝑥) are defined by ℎ_{𝑠,𝑘,𝑛,𝜌}(𝑙,𝜈;𝑥)=𝑙^{𝜈𝑘,𝑛,𝜌}(𝑥)^𝜈−1_𝑖=𝑠ℎ_{(𝑗)𝑠,𝑘,𝑛,𝜌}𝑒_𝑠,𝑖(𝜈,𝑘,𝑛)𝑥−𝑥_{𝑘,𝑛,𝜌}^𝑖,𝑙,𝜈;𝑥_{𝑝,𝑛,𝜌}=𝛿_𝑠,𝑗𝛿_𝑘,𝑝,𝑗,𝑠=0,1,…,𝜈−1,𝑝=1,2,…,𝑛.(1.9) Then we have 𝐿_𝑛(𝑙,𝜈,𝑓;𝑥)=^𝑛_𝑙𝑘=1_𝑠=0𝑓^(𝑠)𝑥_{𝑘,𝑛,𝜌}ℎ_{𝑠,𝑘,𝑛,𝜌}(𝑙,𝜈;𝑥).(1.10) For the ordinary Hermite and Hermite-Fejér interpolation polynomial 𝐿_𝑛(1,2,𝑓;𝑥), 𝐿_𝑛(2,𝑓;𝑥) and the related approximation process, Lubinsky [3] gave some interesting convergence theorems.

Our purpose in this paper is to study 𝐿_𝑛(𝜈,𝑓;𝑥) and 𝐿_𝑛(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 𝐿_𝑛(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 𝐿_𝑛(𝜈,𝑓;𝑥).

In what follows we abbreviate several notations as 𝑥_𝑘,𝑛∶=𝑥_{𝑘,𝑛,𝜌}, ℎ_𝑘𝑛(𝑥)∶=ℎ_{𝑘,𝑛,𝜌}(𝜈,𝑥), 𝑙_𝑘𝑛(𝑥)∶=𝑙_{𝑘,𝑛,𝜌}(𝑥), ℎ_𝑠𝑘𝑛(𝑥)∶=ℎ_{𝑠,𝑘,𝑛,𝜌}(𝜈,𝑥) and 𝑝_𝑛(𝑥)∶=𝑝_𝑛,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 {𝑐_𝑛}^∞𝑛=1 and {𝑑_𝑛}^∞=1, we define 𝑐_𝑛∼𝑑_𝑛 similarly.

Throughout this paper 𝐶,𝐶₁,𝐶₂,… denote positive constants independent of 𝑛,𝑥,𝑡, or polynomials 𝑃_𝑛(𝑥), and the same symbol does not necessarily denote the same constant in different 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 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)lim_𝑥→∞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 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 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 Λ_𝑖, 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)=𝑄_𝑟,𝛼(𝑥)∶=exp_𝑟(|𝑥|^𝛼)−exp_𝑟(0),(2.5) where, for 𝑟⩾1, exp_𝑟(𝑥)∶=exp(exp(exp(⋯exp𝑥)⋯))(𝑟-times)(2.6) and exp₀(𝑥)∶=𝑥. (3)(See [10]) For 𝑚⩾0,𝛼⩾0 with 𝛼+𝑚>1, we put 𝑄(𝑥)=𝑄_{𝑟,𝛼,𝑚}(𝑥)∶=|𝑥|^𝑚exp_𝑟(|𝑥|^𝛼)−𝛼^∗exp_𝑟(0),(2.7) 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 𝑄(𝑥)=𝑄_𝛼(𝑥)∶=(1+|𝑥|)^|𝑥|𝛼−1.(2.8) For 𝑥>0, we define the Mhaskar-Rakhmanov-Saff number 𝑎_𝑥 by the equation𝑥=2𝜋¹⁰𝑎_𝑥𝑢𝑄^𝑎_𝑥𝑢1−𝑢²^1/2𝑑𝑢.(2.9) We have the following estimates for the coefficients 𝑒_𝑠,𝑖(𝜈,𝑘,𝑛)(𝑒_𝑖(𝜈,𝑘,𝑛)∶=𝑒_0,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,…,𝑛 and 𝑠=0,1,…,𝜈−1, we have 𝑒_0,0(𝜈,𝑘,𝑛)=𝑒₀(𝜈,𝑘,𝑛)=1, ||𝑒_𝑠,𝑖(𝜈,𝑘,𝑛)||⩽𝐶⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑘,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 𝐿_𝑛(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 𝐿_𝑛(1,𝑓;𝑥) and the ordinary Hermite-Fejér interpolation polynomial 𝐿_𝑛(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. Let 𝑤(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 constants 𝐶_𝑖>0 such that||𝑄^(𝑖+1)(𝑥)||⩽𝐶_𝑖||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 ℱ_𝜈(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 ℱ_𝜈(𝐶²+).

Example 2.5 (cf. [10, Theorem 3.1]). Let 𝜈 be a positive integer, and let 𝑄_{𝑟,𝛼,𝑚} be defined in (2.7).(1)Let 𝑚 and 𝛼 be nonnegative even integers with 𝑚+𝛼>1. Then 𝑤(𝑥)=exp(−𝑄_{𝑟,𝛼,𝑚})∈ℱ_𝜈(𝐶²+). (a)If 𝑟>0, then we see that 𝑄^{𝑟,𝛼,𝑚}(𝑥)/𝑄_{𝑟,𝛼,𝑚}(𝑥) is quasi-increasing on a certain positive interval (𝑐₁,∞) and 𝑄_𝑟,0,𝑚(𝑥) is nondecreasing on (0,∞). (b)If 𝑟=0, then we see that 𝑄_0,0,𝑚(𝑥),𝑚⩾2, is nondecreasing on (0,∞). Hence, 𝑤(𝑥)=exp(−𝑄_{𝑟,𝛼,𝑚})∈ℱ_𝜈(𝐶²+). (2)Let 𝑚+𝛼−𝜈>0. Then 𝑤(𝑥)=exp(−𝑄_{𝑟,𝛼,𝑚})∈ℱ_𝜈(𝐶²+), and one has the following. (c)If 𝑟⩾2 and 𝛼>0, then there exists a constant 𝑐₁>0 such that 𝑄^{𝑟,𝛼,𝑚}(𝑥)/𝑄_{𝑟,𝛼,𝑚}(𝑥) is quasi-increasing on (𝑐₁,∞). (d)Let 𝑟=1. If 𝛼⩾1, then there exists a constant 𝑐₂>0 such that 𝑄^{1,𝛼,𝑚}(𝑥)/𝑄_1,𝛼,𝑚(𝑥) is quasi-increasing on (𝑐₂,∞), and, if 0<𝛼<1, then 𝑄^{1,𝛼,𝑚}(𝑥)/𝑄_1,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 positive interval on (𝑐₂,∞).Hence, 𝑤(𝑥)=exp(−𝑄_{𝑟,𝛼,𝑚})∈ℱ_𝜈(𝐶²+).

Definition 2.6. One uses the following notation: 𝜑_𝑢(𝑥)=⎧⎪⎨⎪⎩𝑎𝑢^𝑢√11−−|𝑥|𝑥||/𝑎/𝑎_𝑢+^2𝑢𝛿_𝑢𝜑_𝑢,|𝑥|⩽𝑎_𝑢;𝑎_𝑢,𝑎_𝑢<|𝑥|,𝛿^𝑢=𝑢𝑇𝑎_𝑢^−2/3,𝑢>0.(2.13)

Lemma 2.7 (see [18, Corollary 4.5]). Let 𝑤_𝜌(𝑥)=|𝑥|^𝜌exp(−𝑄(𝑥)), exp(−𝑄)∈ℱ_𝜈(𝐶²+). If 𝑥_𝑘,𝑛≠0 and |𝑥_𝑘,𝑛|⩽𝑎_𝑛(1+𝛿_𝑛), then 𝑒₀(𝜈,𝑘,𝑛)=1 and, for 𝑖=1,2,…,𝜈−1, ||𝑒_𝑖(𝜈,𝑘,𝑛)||⩽𝐶𝑇𝑎^𝑛𝑎_𝑛+||𝑄^𝑥_𝑘,𝑛||+1||𝑥_𝑘,𝑛||^⟨𝑖⟩𝑎_2𝑛−𝑛||𝑥_𝑘,𝑛||+𝑇𝑎^𝑛𝑎_𝑛^{𝑖−⟨𝑖⟩},(2.14) where ⟨𝑖⟩=10,,𝑖𝑖::oevdedn,.(2.15) For 𝑥_𝑘,𝑛=0 one has 𝑒₀(𝜈,𝑘,𝑛)=1,||𝑒_𝑖(𝜈,𝑘,𝑛)||⩽𝐶𝑛𝑎_𝑛^𝑖,𝑖=1,2,…,𝜈−1.(2.16)

Remark 2.8. In [19, Theorem 2.2] we see that 𝑥_1,𝑛<𝑎_𝑛 if 𝑛 is large enough. Therefore Lemma 2.7 holds for all 𝑥_𝑘,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, 𝑇𝑎_𝑡⩽𝐶𝑡^2−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,𝑎_𝑡⩽𝐶(𝜂)𝑡^𝜂.(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𝑎_𝑡⩽𝐶(𝜆,𝜂)𝑡^{2(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𝑎_𝑡⩽𝐶(𝜆,𝜂)𝑡^𝜂,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) is unbounded, then 𝑤 is called an Erdős-type weight.
(2) In (2.20) and (2.21), we set 0<𝜂<2 and 𝜀=⎧⎪⎨⎪⎩22−(2𝜂−,𝜂)0<𝜆⩽1,(𝜆+1),𝜆>1.(2.22)
Then (2.17) holds.
(3) If limsup_𝑥→∞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𝑎_𝑡=𝑚,𝑎_𝑡∼𝑡^1/𝑚.(2.24)
(5) The inequality (2.18) implies 0<𝐶⩽𝑛𝑎_𝑛𝑇𝑎^𝑛𝑎_𝑛^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 weight 𝑤_𝜌(𝑥)=|𝑥|^𝜌exp(−𝑄(𝑥)), exp(−𝑄(𝑥))∈ℱ_𝜈(𝐶²+), 𝜌⩾0. (a)(cf. [20, Theorem 1.4]) If 𝑇(𝑥) is bounded, then we suppose, for 𝛿 in (2.12), 𝑎_𝑛⩽𝐶𝑛^{1/(1+𝜈−𝛿)}.(3.1)(b)There exist 0⩽𝛾<1 and 𝐶(𝛾)>0 such that 𝑇𝑎_𝑛⩽𝐶(𝛾)𝑛^𝛾;(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 𝜀_𝑛∶=⎧⎪⎪⎨⎪⎪⎩𝑎^𝑛𝑛,𝑇𝑎^𝑛𝑎_𝑛1<1,𝑛^1−𝛾,𝑇𝑎^𝑛𝑎_𝑛⩾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 𝜀_𝑛log𝑛→0 as 𝑛→∞.

We shall state our theorems. Put 𝑋_𝑛(𝜈,𝑓;𝑥)∶=^𝑛_𝑗=1𝑓𝑥_{𝑗,𝑛,𝜌}𝑙^{𝜈𝑗,𝑛,𝜌}(𝑥)^𝜈−2_𝑖=0𝑌_𝑛𝑒_𝑖(𝜈,𝑗,𝑛)𝑥−𝑥_{𝑗,𝑛,𝜌}^𝑖,(𝜈,𝑓;𝑥)∶=^𝑛_𝑗=1𝑍_𝑛𝑓𝑥_{𝑗,𝑛,𝜌}𝑙^{𝜈𝑗,𝑛,𝜌}(𝑥)𝑒_𝜈−1(𝜈,𝑗,𝑛)𝑥−𝑥_{𝑗,𝑛,𝜌}^𝜈−1,(𝑙,𝜈,𝑓;𝑥)∶=^𝑛_𝑙𝑗=1_𝑠=1𝑓^(𝑠)𝑥_{𝑗,𝑛,𝜌}𝑙^{𝜈𝑗,𝑛,𝜌}(𝑥)^𝜈−1_𝑖=𝑠𝑒_𝑠𝑖(𝜈,𝑗,𝑛)𝑥−𝑥_{𝑗,𝑛,𝜌}^𝑖.(3.5) Furthermore, we consider the class 𝐺={𝑔_𝑠∈𝐶(ℝ), 𝑠=𝑙+1,𝑙+2,…,𝜈−1} and construct the following interpolation polynomial:𝑊_𝑛(𝑙,𝜈,𝐺;𝑥)∶=^𝑛_{𝑗=1𝜈−1𝑠=𝑙+1}𝑔_𝑠𝑥_{𝑗,𝑛,𝜌}𝑙^{𝜈𝑗,𝑛,𝜌}(𝑥)^𝜈−1_𝑖=𝑠𝑒_𝑠𝑖(𝜈,𝑗,𝑛)𝑥−𝑥_{𝑗,𝑛,𝜌}^𝑖.(3.6)

Then we have 𝐿𝐿_𝑛(𝜈,𝑓;𝑥)=𝑋_𝑛(𝜈,𝑓;𝑥)+𝑌_𝑛(𝜈,𝑓;𝑥),_𝑛𝐿(𝑙,𝜈,𝑓;𝑥)=𝐿_𝑛(𝜈,𝑓;𝑥)+𝑍_𝑛(𝑙,𝜈,𝑓;𝑥),_𝑛(𝑙,𝜈,𝑓⊕𝐺;𝑥)∶=𝐿_𝑛(𝜈,𝑓;𝑥)+𝑍_𝑛(𝑙,𝜈,𝑓;𝑥)+𝑊_𝑛(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)^1/3𝑥𝑄^(𝑥),|𝑥|⩾𝑑>0.(3.9)

Moreover, we define Φ_𝑛(𝑥)∶=max𝛿_𝑛,1−|𝑥|𝑎_𝑛,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,𝑎_𝐿𝑡∼𝑎_𝑡.(3.11)(b)(See [9, Lemma 3.5(b)]) Uniformly for 𝑡>0,𝑄^(𝑗)𝑎_𝐿𝑡∼𝑄^(𝑗)𝑎_𝑡,𝑗=0,1.(3.12)Moreover, 𝑇𝑎_𝐿𝑡∼𝑇𝑎_𝑡.(3.13)(c)(See [9, Lemma 3.11 (3.52)]) Uniformly for 𝑡>0, ||||1−𝑎_𝐿𝑡𝑎_𝑡||||∼1𝑇𝑎_𝑡.(3.14)(d)(See [9, Lemma 3.4 (3.17), (3.18)]) Uniformly for 𝑡>0, one has𝑄𝑎_𝑡∼𝑡𝑇𝑎_𝑡,𝑄^𝑎_𝑡∼𝑡𝑇𝑎_𝑡𝑎_𝑡.(3.15)

Lemma 3.4. For 𝑥∈ℝ, we have Φ(𝑥)⩽𝐶Φ_𝑛(𝑥),𝑛⩾1.(3.16)

Proof. Let 𝑥=𝑎_𝑢,𝑢⩾1. By Lemma 3.3(d) we have 𝑢∼𝑄𝑎_𝑢𝑇𝑎_𝑢.(3.17) So, we have 𝛿^𝑢−1∼𝑄𝑎_𝑢^2/3𝑇𝑎_𝑢=𝑎^𝑢𝑄^𝑎_𝑢𝑄𝑎_𝑢^1/3=𝑥𝑄^(𝑥)𝑄^1/3(𝑥)∼Φ⁻¹𝑎_𝑢.(3.18) Now, if 𝑢⩽𝑛/2, then 1−𝑎^𝑢𝑎_𝑛⩾1−𝑎^𝑛/2𝑎_𝑛∼1⩾1𝑇𝑎_𝑛(byLemma3.2(c))𝑛𝑇𝑎_𝑛^2/3=𝛿_𝑛(by(2.17)).(3.19) So, we have Φ_𝑛(𝑥)=1−𝑎^𝑢𝑎_𝑛⩾1−𝑎^𝑢𝑎_2𝑢∼1⩾1𝑇𝑎_𝑢(byLemma3.2(c))𝑢𝑇𝑎_𝑢^2/3=𝛿_𝑢∼Φ(𝑥)(by(2.17)and(3.18)).(3.20) Let 𝑛/2<𝑢<𝑛. Then we have Φ_𝑛(𝑥)⩾𝛿_𝑛∼𝛿_𝑢∼Φ(𝑥)(byLemmas3.2(a),(2.17),and(3.18)).(3.21)

Let 𝐶(𝑓) denote a positive constant depending only on 𝑓.

Assumption 3.5. Let 𝑤_𝜌(𝑥)=|𝑥|^𝜌𝑤(𝑥) and 𝑤(𝑥)=exp(−𝑄(𝑥))∈ℱ_𝜈(𝐶²+), 𝜌⩾0. Suppose the following.(A-1)Let 𝑓∈𝐶(ℝ) satisfy that, for a given 0<𝛿<1,||𝑓(𝑥)||𝑤^𝜈−𝛿(𝑥)||𝑄^(𝑥)||+1|𝑥|⩽𝐶(𝑓),𝑥∈ℝ,(3.22)where we suppose limsup_|𝑥|→0|𝑓(𝑥)/𝑥|⩽𝐶(𝑓). (A-2)Let 𝑓∈𝐶^𝑙(ℝ) for a certain 0<𝑙⩽𝜈−1. Then we suppose that 𝑓 satisfies ||𝑓^(𝑠)(𝑥)||Φ^−3/4(𝑥)𝑤_𝜌(𝑥)^𝜈⩽𝐶(𝑓),𝑥∈ℝ,𝑠=1,2,…,𝑙.(3.23)(A-3) Let 𝐺={𝑔_𝑠∈𝐶(ℝ),𝑠=𝑙+1,𝑙+2,…,𝜈−1}. Then we suppose that there exists a constant 𝐶>0 such that||𝑔_𝑠(𝑥)||Φ^−3/4(𝑥)𝑤_𝜌(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)𝑤_𝜌(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), it follows that ||𝑓(𝑥)||Φ^−3/4(𝑥)𝑤_𝜌(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 ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑋_𝑛(𝜈,𝑓;𝑥)‖‖‖_𝐿∞(ℝ)⩽𝐶(𝑓).(3.27)

Proposition 3.8. Let 𝜈=2,4,6,…. For 𝑓(𝑥) satisfying (3.25), one has ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑌_𝑛(𝜈,𝑓;𝑥)‖‖‖_𝐿∞(ℝ)⩽𝐶(𝑓)𝜀_𝑛log𝑛,(3.28) where 𝜀_𝑛 is defined by (3.3).

Proposition 3.9. Let 𝜈=1,2,3,…. For 𝑓(𝑥) satisfying (3.23), one has ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑍_𝑛(𝑙,𝜈,𝑓;𝑥)‖‖‖_𝐿∞(ℝ)⩽𝐶(𝑓)𝑎^𝑛log𝑛𝑛,(3.29) and for 𝑓(𝑥) satisfying (3.24) one has ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑊_𝑛(𝑙,𝜈,𝐺;𝑥)‖‖‖_𝐿∞(ℝ)⩽𝐶(𝐺)𝑎^𝑛log𝑛𝑛.(3.30)

Proposition 3.10. Let 𝜈=1,2,3,…. Let 𝑃∈𝒫_𝑚 be fixed. Then one has ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿_𝑛(𝜈,𝑃,𝑥)−𝑃(𝑥)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(3.31)

Proposition 3.11. Let 𝜈=2,4,6,…. Let 𝑃∈𝒫_𝑚 with 𝑃(0)=0 be fixed. Then one has ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑋_𝑛(𝜈,𝑃;𝑥)−𝑃(𝑥)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(3.32)

Theorem 3.12. Let 𝜈=2,4,6,…. For 𝑓(𝑥) satisfying (3.22), one has ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿_𝑛(𝜈,𝑓;𝑥)−𝑓(𝑥)‖‖‖_𝐿∞(ℝ)⟶0𝑎𝑠𝑛⟶∞.(3.33)

Theorem 3.13. Let 𝜈=2,4,6,…. For 𝑓(𝑥) satisfying (3.22) and (3.23), one has ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿_𝑛(𝑙,𝜈,𝑓;𝑥)−𝑓(𝑥)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(3.34)

Corollary 3.14. Let 𝜈=2,4,6,…. For 𝑓(𝑥) satisfying (3.22), one has ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑋_𝑛(𝜈,𝑓;𝑥)−𝑓(𝑥)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(3.35)

Theorem 3.15. Let 𝜈=2,4,6,…. For 𝑓(𝑥) satisfying (3.22), (3.23) and (3.24), one has ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿^𝑛(𝑙,𝜈,𝑓⊕𝐺;𝑥)−𝑓(𝑥)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(3.36)

Define 𝐼_𝑛𝑘,𝐻_𝑛(𝑓)∶=^∞−∞𝑘(𝑥)𝐻_𝑛(𝑓;𝑥)𝑑𝑥,(3.37) where 𝐻_𝑛(𝑓) equals to one of the following: 𝐿_𝑛(𝜈,𝑓),𝐿_𝑛(𝑙,𝜈,𝑓),𝐿_𝑛(𝑙,𝜈,𝑓⊕𝐺),𝑋_𝑛(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)Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^−𝜈𝑑𝑥<∞,𝑛=1,2,3,…,(3.40) then we have lim_𝑛→∞𝐼_𝑛𝑘,𝐻_𝑛(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|)^−ΔΦ^3𝜈/4(𝑥)𝑑𝑥⩽^∞1(1+|𝑥|)^−Δ𝑇(𝑥)𝑄1(𝑥)^2/3^3𝜈/4𝑑𝑥<∞(3.42) (see [10, Lemma 2.1(b)]). Then we set 𝑘(𝑥)∶=(1+|𝑥|)^{𝜌𝜈−Δ}Φ^3/2(𝑥)𝑤(𝑥)^𝜈.(3.43) 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 constants 𝐶,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 sup_𝑥∈ℝ||𝑝_𝑛,𝜌(𝑥)𝑤(𝑥)|||𝑥|+𝑎^𝑛𝑛^𝜌||𝑥²−𝑎^2𝑛||^1/4∼1,(4.1) and for 𝑤(𝑥)=exp(−𝑄(𝑥))∈ℱ(𝐶²+), sup_𝑥∈ℝ||𝑝_𝑛,𝜌(𝑥)𝑤(𝑥)|||𝑥|+𝑎^𝑛𝑛^𝜌∼𝑎^𝑛−1/2𝑛𝑇𝑎_𝑛^1/6.(4.2)
(2) (See [19, Theorem 2.5(d)]) Let 𝑤∈ℱ(𝐶²+) and 𝜌>−1/2. Let 1⩽𝑗⩽𝑛−1 and 𝑥∈[𝑥_𝑗+1,𝑛,𝑥_𝑗,𝑛]. Then ||𝑝_𝑛(𝑥)𝑤(𝑥)|||𝑥|+𝑎^𝑛𝑛^𝜌∼min||𝑥−𝑥_𝑗,𝑛||,||𝑥−𝑥_𝑗+1,𝑛||𝜑_𝑛𝑥_𝑗,𝑛⁻¹||𝑎^2𝑛−𝑥^2𝑗,𝑛||^−1/4.(4.3)
(3) (See [19, Theorem 2.5(c)]) Let 𝑤∈ℱ(𝐶²+) and 𝜌>−1/2. Then one has max_𝑥∈ℝ|||||𝑝_𝑛(𝑥)𝑤(𝑥)|𝑥|+𝑎_𝑛/𝑛^𝜌𝑥−𝑥_𝑗,𝑛𝑝^𝑛𝑥_𝑗,𝑛𝑤𝑥_𝑗,𝑛||𝑥_𝑗,𝑛||+𝑎_𝑛/𝑛^𝜌=max_𝑥∈ℝ||||||||𝑙_𝑗𝑛(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌|||𝑤₋₁𝑥_𝑗,𝑛||𝑥_𝑗,𝑛||+𝑎𝑛^𝑛^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||𝑥_𝑗,𝑛||𝑥_𝑗,𝑛||+𝑎^𝑛𝑛^𝜌∼𝜑^𝑛−1𝑥_𝑗,𝑛||𝑎^2𝑛−𝑥^2𝑗,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 the zeros 𝑥_𝑗,𝑛=𝑥_{𝑗,𝑛,𝜌}, one has the following: (1)For 𝑛⩾1 and 1⩽𝑗⩽𝑛−1,𝜑_𝑛𝑥_𝑗,𝑛−𝑥_𝑗+1,𝑛∼𝜑_𝑛𝑥_𝑗,𝑛,𝑥_𝑗,𝑛∼𝜑_𝑛𝑥_𝑗+1,𝑛(see([19,LemmaA.1(𝐴.3)])).(4.6)(2)For the minimum positive zero 𝑥_{[𝑛/2],𝑛} ([𝑛/2] is the largest integer ⩽𝑛/2), one has𝑥_{[𝑛/2],𝑛}∼𝑎_𝑛𝑛⁻¹,(4.7)and for large enough 𝑛,1−𝑥^1,𝑛𝑎_𝑛∼𝛿_𝑛.(4.8)(3) (See [19, Lemma 4.7]) 𝑏_𝑛=𝛾_𝑛−1/𝛾_𝑛∼𝑎_𝑛∼𝑥_1,𝑛, where 𝛾_𝑛 is defined by (1.3).

Lemma 4.3. Let 𝑤∈ℱ(𝐶²+). Then there exist 𝐶₁,𝐶₂>0 such that sup_𝑥∈ℝ||𝑝_𝑛,𝜌(𝑥)𝑤(𝑥)|||𝑥|+𝑎^𝑛𝑛^𝜌Φ^1/4(𝑥)⩽𝐶₁sup_𝑥∈ℝ||𝑝_𝑛,𝜌(𝑥)𝑤(𝑥)|||𝑥|+𝑎^𝑛𝑛^𝜌⩽𝐶₂Φ^𝑛1/4(𝑥)𝑎^𝑛−1/2.(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}||𝑝_𝑛,𝜌(𝑥)𝑤(𝑥)|||𝑥|+𝑎^𝑛𝑛^𝜌||𝑥²−𝑎^2𝑛||^1/4∼sup_{|𝑥|⩽𝑥1,𝑛}||𝑝_𝑛,𝜌(𝑥)𝑤(𝑥)|||𝑥|+𝑎^𝑛𝑛^𝜌𝑎^𝑛1/2Φ^𝑛1/4(𝑥).(4.10) From (4.2) we see that 𝑎^𝑛−1/2⩾𝐶sup_{|𝑥|>𝑥1,𝑛}||𝑝_𝑛,𝜌(𝑥)𝑤(𝑥)|||𝑥|+𝑎^𝑛𝑛^𝜌𝛿^𝑛1/4∼𝐶sup_{|𝑥|>𝑥1,𝑛}||𝑝_𝑛,𝜌(𝑥)𝑤(𝑥)|||𝑥|+𝑎^𝑛𝑛^𝜌Φ^𝑛1/4(𝑥).(4.11) Therefore we have the result.

Proof of Proposition 3.7. We recall the definition of 𝑋_𝑛(𝜈,𝑓;𝑥): 𝑋_𝑛(𝜈,𝑓;𝑥)=^𝑛_𝑗=1𝑓𝑥_𝑗,𝑛𝑙^𝜈𝑗𝑛(𝑥)^𝜈−2_𝑖=0𝑒_𝑖(𝜈,𝑗,𝑛)𝑥−𝑥_𝑗,𝑛^𝑖=^𝜈−2_𝑛𝑖=0_𝑗=1𝑓𝑥_𝑗,𝑛𝑙^𝜈𝑗𝑛(𝑥)𝑥−𝑥_𝑗,𝑛^𝑖𝑒_𝑖(𝜈,𝑗,𝑛).(4.12) Using Lemma 2.3, we may estimate, for 𝑖=0,1,…,𝜈−2 and 1⩽𝑗⩽𝑛, 𝐴_𝑖,𝑗(𝑥)∶=Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈||𝑓𝑥_𝑗,𝑛||||𝑙_𝑗𝑛(𝑥)||^𝜈||𝑥−𝑥_𝑗,𝑛||^𝑖⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑗,𝑛⎞⎟⎟⎟⎠^𝑖.(4.13) Let (𝑥_𝑚+1,𝑛+𝑥_𝑚,𝑛)/2<𝑥⩽𝑥_𝑚,𝑛 or 𝑥_𝑚,𝑛⩽𝑥<(𝑥_{𝑚−1,𝑛}+𝑥_𝑚,𝑛)/2. For simplicity, we assume 𝑥>0, and let 𝑥_0,𝑛∶=𝑥_1,𝑛+𝛿𝜑_𝑛(𝑥_1,𝑛) for a fixed 𝛿>0 small enough. Then we can assume that there exists a constant 𝑐>0 such that 𝑥_0,𝑛∶=𝑥_1,𝑛+𝛿𝜑_𝑛𝑥_1,𝑛<𝑎_𝑛−𝑐𝛿_𝑛.(4.14) Assume that 𝑥<𝑥_0,𝑛. Then we first estimate 𝐴_𝑖,𝑚(𝑥). By Lemma 4.1(3) and the definition of 𝜑_𝑛(𝑥), we have Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎𝑛^𝑛^𝜌^𝜈||𝑙_𝑚𝑛(𝑥)||^𝜈⩽𝐶Φ^3𝜈/4(𝑥)𝑤(𝑥_𝑚,𝑛)||𝑥_𝑚,𝑛||+𝑎^𝑛𝑛^𝜌^𝜈||𝑥−𝑥_𝑚,𝑛,(4.15)||^𝑖⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑚,𝑛⎞⎟⎟⎟⎠^𝑖⩽𝐶⎛⎜⎜⎜⎝𝑛𝜑^𝑛𝑥_𝑚,𝑛𝑎^22𝑛−𝑥^2𝑚,𝑛⎞⎟⎟⎟⎠^𝑖⩽𝐶⎛⎜⎜⎜⎝1−||𝑥_𝑚,𝑛||/𝑎_2𝑛1−||𝑥_𝑚,𝑛||/𝑎_𝑛⎞⎟⎟⎟⎠^𝑖,(4.16) and by Remark 3.6||𝑓𝑥_𝑚,𝑛||⩽𝐶(𝑓)Φ^3𝜈/4𝑥_𝑚,𝑛𝑤𝑥_𝑚,𝑛||𝑥_𝑚,𝑛||+𝑎^𝑛𝑛^𝜌^−𝜈.(4.17) Therefore, we have 𝐴_𝑖,𝑚(𝑥)⩽𝐶(𝑓)Φ^3𝜈/4(𝑥)Φ^3𝜈/4𝑥_𝑚,𝑛⎛⎜⎜⎜⎝1−||𝑥_𝑚,𝑛||/𝑎_2𝑛1−||𝑥_𝑚,𝑛||/𝑎_𝑛⎞⎟⎟⎟⎠^𝑖=𝐶(𝑓)Φ^3𝜈/4(𝑥)max1−||𝑥^𝑚,𝑛||𝑎_𝑛,𝛿_𝑛^3𝜈/4√1−|𝑥_𝑚,𝑛|/𝑎_2𝑛√1−|𝑥_𝑚,𝑛|/𝑎_𝑛^𝑖=𝐶(𝑓)Φ^3𝜈/4(𝑥)1−||𝑥^𝑚,𝑛||𝑎_𝑛^{3𝜈/4−𝑖/2}⎛⎜⎜⎝1−||𝑥^𝑚,𝑛||𝑎_2𝑛⎞⎟⎟⎠^𝑖⩽𝐶(𝑓).(4.18) Next, we estimate ∑_{1⩽𝑗⩽𝑛,𝑗≠𝑚}𝐴_𝑖,𝑗(𝑥). For 1⩽𝑗⩽𝑛, 𝑗≠𝑚, we have, by Lemma 4.3 and Lemma 4.1(4), Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎_𝑛/𝑛^𝜌^𝜈||𝑙_𝑗𝑛(𝑥)||^𝜈||𝑥−𝑥_𝑗,𝑛||^𝑖=Φ^𝜈/2(𝑥)|||||𝑝^𝑛(𝑥)Φ^1/4(𝑥)𝑤(𝑥)|𝑥|+𝑎_𝑛/𝑛^𝜌𝑝^𝑛𝑥_𝑗,𝑛𝑤𝑥_𝑗,𝑛||𝑥_𝑗,𝑛||+𝑎_𝑛/𝑛^𝜌|||||^𝜈𝑤𝑥_𝑗,𝑛||𝑥_𝑗,𝑛||+𝑎_𝑛/𝑛^𝜌^𝜈||𝑥−𝑥_𝑗,𝑛||^𝜈−𝑖=Φ^𝜈/2(𝑥)𝑎^{𝑛−𝜈/2}𝜑^𝜈𝑛𝑥_𝑗,𝑛||𝑎^2𝑛−𝑥^2𝑗,𝑛||^𝜈/4𝑤𝑥_𝑗,𝑛||𝑥_𝑗,𝑛||+𝑎_𝑛/𝑛^𝜌^𝜈||𝑥−𝑥_𝑗,𝑛||^𝜈−𝑖.(4.19) Then, since we know from Remark 3.6 that ||𝑓𝑥_𝑗,𝑛||⩽𝐶(𝑓)Φ^3𝜈/4𝑥_𝑗,𝑛𝑤(𝑥_𝑗,𝑛)||𝑥_𝑗,𝑛||+𝑎^𝑛𝑛^𝜌^−𝜈,(4.20) we have, by Lemma 3.4, _𝑗_≠𝑚𝐴_𝑖,𝑗(𝑥)⩽×𝐶(𝑓1)−||𝑥_𝑗≠𝑚Φ^𝑛𝜈/2_𝑗,𝑛(𝑥)Φ^𝑛3𝜈/4𝑥_𝑗,𝑛𝜑^𝜈𝑛𝑥_𝑗,𝑛||𝑎_𝑛^𝜈/4||𝑥−1𝑥_𝑗,𝑛||^𝜈−𝑖⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑗,𝑛⎞⎟⎟⎟⎠^𝑖.(4.21) By the definition of 𝜑_𝑛(𝑥) and by Lemma 3.3(a) 𝑎_2𝑛∼𝑎_𝑛, we have 𝜑^𝜈𝑛𝑥_𝑗,𝑛⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑗,𝑛⎞⎟⎟⎟⎠^𝑖⩽𝐶𝑎^𝑛𝑛^𝜈−𝑖1−||𝑥^𝑗,𝑛||𝑎_2𝑛^{𝜈−𝑖/2}1−||𝑥^𝑗,𝑛||𝑎_𝑛^−𝜈/2.(4.22) By Lemma 4.2(1), we have 1||𝑥−𝑥_𝑗,𝑛||⩽⩽𝐶𝐶∑1_{𝑗⩽𝑠⩽𝑚+1or𝑚−1⩽𝑠⩽𝑗}𝜑_𝑛𝑥𝑛_𝑠,𝑛/𝑎_𝑛∑_{𝑗⩽𝑠⩽𝑚+1or𝑚−1⩽𝑠⩽𝑗}1−||𝑥_𝑠,𝑛||/𝑎_2𝑛/1−|𝑥_𝑠,𝑛|/𝑎_𝑛+𝛿_𝑛^1/2⩽𝐶𝑛/𝑎^𝑛∑_{𝑗⩽𝑠⩽𝑚+1or𝑚−1⩽𝑠⩽𝑗}1−||𝑥_𝑠,𝑛||/𝑎_𝑛/1−||𝑥_𝑠,𝑛||/𝑎_𝑛+𝛿_𝑛^1/2⩽𝐶𝑛/𝑎^𝑛||𝑚−𝑗||1−max|𝑥|,||𝑥_𝑗,𝑛||/𝑎_𝑛^1/2.(4.23) Here, we note from (4.14) and (4.8) that Φ_𝑛(𝑥)∼1−|𝑥|𝑎_𝑛,Φ^𝑛𝑥_𝑗,𝑛∼1−||𝑥^𝑗,𝑛||𝑎_𝑛.(4.24) Thus, we have, for 𝑗≠𝑚, Φ^𝑛𝜈/2(𝑥)Φ^𝑛3𝜈/4𝑥_𝑗,𝑛𝜑^𝜈𝑛𝑥_𝑗,𝑛1−||𝑥^𝑗,𝑛||𝑎_𝑛^𝜈/4|𝑥−1𝑥_𝑗,𝑛|^𝜈−𝑖⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑗,𝑛⎞⎟⎟⎟⎠^𝑖⩽𝐶1−|𝑥|𝑎_𝑛^𝜈/21−|𝑥^𝑗,𝑛|𝑎_𝑛^𝜈/21−|𝑥^𝑗,𝑛|𝑎_2𝑛^{𝜈−𝑖/2}×1|𝑚−𝑗|1−max|𝑥|,||𝑥_𝑗,𝑛||/𝑎_𝑛^1/2^𝜈−𝑖⩽𝐶1|𝑚−𝑗|^𝜈−𝑖.(4.25) Therefore, we have _1⩽𝑗_{⩽𝑛,𝑗≠𝑚}𝐴_𝑖,𝑗(𝑥)⩽𝐶(𝑓)_𝑗≠𝑚||𝑚1−𝑗||^𝜈−𝑖⩽𝐶(𝑓),(4.26) because of 𝜈−𝑖⩾2.Remark 4.4. If we consider the estimate of ∑^𝑛𝑖=1𝐴_{𝑖,𝜈−1}(𝑥) with (4.13), then we obtain ^𝑛_𝑖=1𝐴_{𝑖,𝜈−1}(𝑥)⩽𝐶(𝑓)log𝑛.(4.27)
We continue the proof Proposition 3.7. We need to estimate ∑^𝑛𝑗=1𝐴_𝑖,𝑗(𝑥) for |𝑥|>𝑥_0,𝑛. Now, suppose 𝑥>𝑥_0,𝑛. Then similarly to (4.23), we have1||𝑥_0,𝑛−𝑥_𝑗,𝑛||⩽𝐶∑_{1⩽𝑠⩽𝑗}𝜑1_𝑛𝑥_𝑠,𝑛⩽𝐶𝑗1−𝑛𝑥/𝑎^𝑛_0,𝑛/𝑎_𝑛^1/2⩽𝐶1𝑗𝛿^𝑛1/2𝑛𝑎_𝑛.(4.28) Similarly to (4.21), we have using (4.22) 𝐴^𝑖,𝑗(𝑥)⩽×𝐶(𝑓1)−_𝑗||𝑥Φ^𝜈/2_𝑗,𝑛(𝑥)Φ^𝑛3𝜈/4𝑥_𝑗,𝑛𝜑^𝜈𝑛𝑥_𝑗,𝑛||𝑎_𝑛^𝜈/4|𝑥_0,𝑛−1𝑥_𝑗,𝑛|^𝜈−𝑖⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑗,𝑛⎞⎟⎟⎟⎠^𝑖⩽𝐶(𝑓)_𝑗Φ^𝑛𝜈/2(𝑥)1−||𝑥^𝑗,𝑛||𝑎_𝑛^𝜈/21−|𝑥^𝑗,𝑛|𝑎_2𝑛^{𝜈−𝑖/2}𝑗𝛿1^𝑛1/2^𝜈−𝑖⩽𝐶(𝑓)_𝑗1𝑗^𝜈−𝑖𝐶(𝑓),(4.29) because of 𝜈−𝑖⩾2 and Φ^𝜈/2(𝑥)𝛿^{𝑛−(𝜈−𝑖)/2}⩽𝛿^𝑛𝜈/2𝛿^{𝑛−(𝜈−𝑖)/2}⩽𝐶. Consequently we achieve the result.

Remark 4.5. The above proof implies the following: there exists a constant 𝐶>0 such that ^𝑛_𝑗=1||ℎ_{𝑗,𝑛,𝜌}(𝜈;𝑥)||⩽^𝑛_𝑗=1||𝑙_𝑗𝑛(𝑥)||^𝜈𝜈−2_𝑖=0||𝑒_𝑖(𝜈,𝑘,𝑛)||||𝑥−𝑥_𝑗,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 (𝑥_𝑚+1,𝑛+𝑥_𝑚,𝑛)/2<𝑥⩽𝑥_𝑚,𝑛 or 𝑥_𝑚,𝑛⩽𝑥<(𝑥_{𝑚−1,𝑛}+𝑥_𝑚,𝑛)/2. For simplicity, we assume 𝑥>0, and let 𝑥_0,𝑛∶=𝑥_1,𝑛+𝛿𝜑_𝑛(𝑥_1,𝑛) for a fixed 𝛿>0 small enough and there exists a constant 𝛿₁>0 such that 𝑥_0,𝑛∶=𝑥_1,𝑛+𝛿𝜑_𝑛(𝑥_1,𝑛)<𝑎_𝑛−𝛿₁𝛿_𝑛. Assume that 𝑥<𝑥_0,𝑛. Since 𝑓(0)=0, we may leave out the term with 𝑥_𝑗,𝑛=0. Hence we consider only the term of 𝑥_𝑗,𝑛≠0. Noting Assumption 3.1, (3.2), and Lemma 2.7, we estimate ∑^𝑛𝑗=1𝐵_{𝜈−1,𝑗}(𝑥), where 𝐵_{𝜈−1,𝑗}(𝑥)∶=Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈×||𝑓𝑥_𝑗,𝑛||||𝑙_𝑗𝑛(𝑥)||^𝜈||𝑥−𝑥_𝑗,𝑛||^𝜈−1𝑇𝑎^𝑛𝑎_𝑛+||𝑄^𝑥_𝑗,𝑛||+1||𝑥_𝑗,𝑛||𝑎_2𝑛−𝑛||𝑥_𝑗,𝑛||^𝜈−2.(4.31) First we estimate 𝐵_{𝜈−1,𝑚}(𝑥). By Lemma 4.2, (4.16), and the definition of 𝜑_𝑛(𝑥), we have ||𝑥−𝑥_𝑚,𝑛||^𝜈−1⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑚,𝑛⎞⎟⎟⎟⎠^𝜈−2⩽𝐶𝜑_𝑛𝑥_𝑚,𝑛√1−|𝑥_𝑚,𝑛|/𝑎_2𝑛√1−|𝑥_𝑚,𝑛|/𝑎_𝑛^𝜈−2⩽𝐶𝑎^𝑛𝑛1−||𝑥^𝑚,𝑛||𝑎_2𝑛^1/2√1−|𝑥_𝑚,𝑛|/𝑎_2𝑛√1−|𝑥_𝑚,𝑛|/𝑎_𝑛^𝜈−1.(4.32) Since here |𝑄′(𝑥_𝑚,𝑛)|+1/|𝑥_𝑚,𝑛|⩾𝐶 for some positive constant 𝐶, we know that 𝑇𝑎^𝑛𝑎_𝑛+||𝑄^𝑥_𝑚,𝑛||+1||𝑥_𝑚,𝑛||⩽𝐶max1,𝑇𝑎^𝑛𝑎_𝑛||𝑄^𝑥_𝑚,𝑛||+||𝑥1_𝑚,𝑛||,(4.33) and by (3.25) ||𝑓𝑥_𝑚,𝑛||⩽𝐶(𝑓)Φ^3𝜈/4𝑥_𝑚,𝑛𝑤𝑥_𝑚,𝑛||𝑥_𝑚,𝑛||+𝑎^𝑛𝑛^𝜌^−𝜈||𝑄^𝑥_𝑚,𝑛||+||𝑥1_𝑚,𝑛||⁻¹.(4.34) Therefore, using (3.3) and (4.15), we have 𝐵_{𝜈−1,𝑚}(𝑥)⩽𝐶(𝑓)max1,𝑇𝑎^𝑛𝑎_𝑛Φ^𝑛3𝜈/4(𝑥)Φ^𝑛3𝜈/4𝑥_𝑚,𝑛𝑎^𝑛𝑛1−||𝑥^𝑚,𝑛||𝑎_2𝑛^1/2×⎛⎜⎜⎜⎝1−||𝑥_𝑚,𝑛||/𝑎_2𝑛1−||𝑥_𝑚,𝑛||/𝑎_𝑛⎞⎟⎟⎟⎠^𝜈−1⩽𝐶(𝑓)𝜀_𝑛Φ^𝑛3𝜈/4(𝑥)Φ^𝑛3𝜈/4𝑥_𝑚,𝑛1−||𝑥^𝑚,𝑛||𝑎_2𝑛^1/2⎛⎜⎜⎜⎝1−||𝑥_𝑚,𝑛||/𝑎_2𝑛1−||𝑥_𝑚,𝑛||/𝑎_𝑛⎞⎟⎟⎟⎠^𝜈−1.(4.35) Noting (4.24), we have 𝐵_{𝜈−1,𝑚}(𝑥)⩽𝐶(𝑓)𝜀_𝑛Φ^𝑛3𝜈/4(𝑥)1−||𝑥^𝑚,𝑛||𝑎_𝑛^(𝜈+2)/41−||𝑥^𝑚,𝑛||𝑎_2𝑛^𝜈/2⩽𝐶(𝑓)𝜀_𝑛.(4.36) Next, we estimate ∑_𝑗≠𝑚𝐵_{𝜈−1,𝑗}(𝑥). Noting (4.19) and (4.22), we have Φ^𝑛3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎𝑛^𝑛^𝜌^𝜈||𝑙_𝑗𝑛(𝑥)||^𝜈||𝑥−𝑥_𝑗,𝑛||^𝜈−1=Φ^𝑛𝜈/2(𝑥)𝑎^{𝑛−𝜈/2}𝜑^𝜈𝑛𝑥_𝑗,𝑛||𝑎^2𝑛−𝑥^2𝑗,𝑛||^𝜈/4𝑤𝑥_𝑗,𝑛||𝑥_𝑗,𝑛||+𝑎_𝑛/𝑛^𝜌^𝜈𝜑_𝜈𝑛||𝑥−𝑥_𝑗,𝑛||,𝑥_𝑗,𝑛⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑗,𝑛⎞⎟⎟⎟⎠^𝜈−2⩽𝐶𝑎^𝑛𝑛²1−|𝑥^𝑗,𝑛|𝑎_𝑛^−𝜈/21−|𝑥^𝑗,𝑛|𝑎_2𝑛^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 𝐵_{𝜈−1,𝑗}(𝑥)⩽𝐶(𝑓)max1,𝑇𝑎^𝑛𝑎_𝑛𝑎𝑛^𝑛²Φ^𝑛𝜈/2(𝑥)Φ^𝑛3𝜈/4𝑥_𝑗,𝑛𝑎^{𝑛−𝜈/2}||𝑎^2𝑛−𝑥^2𝑗,𝑛||^𝜈/4×1||𝑥−𝑥_𝑗,𝑛||1−||𝑥^𝑗,𝑛||𝑎_𝑛^−𝜈/21−||𝑥^𝑗,𝑛||𝑎_2𝑛^𝜈/2+1⩽𝐶(𝑓)𝜀_𝑛𝑎_𝑛𝑛Φ^𝑛𝜈/2(𝑥)Φ^𝑛3𝜈/4𝑥_𝑗,𝑛1−||𝑥^𝑗,𝑛||𝑎_𝑛^−𝜈/41−||𝑥^𝑗,𝑛||𝑎_2𝑛^𝜈/2+11⩽𝐶(𝑓)𝜀_𝑛||𝑥−𝑥_𝑗,𝑛||𝑎_𝑛𝑛1−|𝑎𝑥_𝑛|^𝜈/21−||𝑥^𝑗,𝑛||𝑎_𝑛^𝜈/21−|𝑥^𝑗,𝑛|𝑎_2𝑛^𝜈/2+11||𝑥−𝑥_𝑗,𝑛||.(4.38) Therefore, since we know from (4.23) that 1||𝑥−𝑥_𝑗,𝑛||⩽𝐶||𝑚−𝑗||1−max𝑛/𝑎^𝑛|𝑥|,||𝑥_𝑗,𝑛||/𝑎_𝑛^1/2,(4.39) noting (4.24), we have _𝑗_≠𝑚𝐵_{𝜈−1,𝑗}(𝑥)⩽𝐶(𝑓)𝜀_𝑛1−|𝑎𝑥_𝑛|^𝜈/2_𝑗≠𝑚1−||𝑥_𝑗,𝑛||/𝑎_𝑛^𝜈/21−||𝑥_𝑗,𝑛||/𝑎_2𝑛^𝜈/2+1||𝑚−𝑗||1−max|𝑥|,||𝑥_𝑗,𝑛||/𝑎_𝑛^1/2⩽𝐶(𝑓)𝜀_𝑛1−|𝑎𝑥_𝑛|^(𝜈−1)/2_𝑗≠𝑚1−|𝑥_𝑗,𝑛|/𝑎_𝑛^(𝜈−1)/21−||𝑥_𝑗,𝑛||/𝑎_2𝑛^𝜈/2+1⩽𝐶(𝑓)𝜀_𝑛||𝑚−𝑗||_𝑗_≠𝑚1||𝑚−𝑗||⩽𝐶(𝑓)𝜀^𝑛log𝑛.(4.40) Finally, we estimate ∑^𝑛𝑗=1𝐵_{𝜈−1,𝑗}(𝑥) for 𝑥>𝑥_0,𝑛. Suppose 𝑥>𝑥_0,𝑛. Then similar to the above computations, we have 𝐵_{𝜈−1,𝑗}(𝑥)⩽𝐶(𝑓)𝜀_𝑛𝑎_𝑛𝑛Φ^𝜈/2(𝑥)Φ^𝑛3𝜈/4𝑥_𝑗,𝑛1−||𝑥^𝑗,𝑛||𝑎_𝑛^−𝜈/41−||𝑥^𝑗,𝑛||𝑎_2𝑛^𝜈/2+11⩽𝐶(𝑓)𝜀_𝑛||𝑥_0,𝑛−𝑥_𝑗,𝑛||𝑎_𝑛𝑛Φ^𝜈/2(𝑥)1−||𝑥^𝑗,𝑛||𝑎_𝑛^𝜈/21−||𝑥^𝑗,𝑛||𝑎_2𝑛^𝜈/2+11||𝑥_0,𝑛−𝑥_𝑗,𝑛||.(4.41) Then, since we know by (4.28) that 1||𝑥_0,𝑛−𝑥_𝑗,𝑛||⩽𝐶𝑗𝛿1^𝑛1/2𝑛𝑎_𝑛,(4.42) we have 𝐵_{𝜈−1,𝑗}(𝑥)⩽𝐶(𝑓)𝜀_𝑛Φ^𝜈/2(𝑥)1−||𝑥^𝑗,𝑛||𝑎_𝑛^𝜈/21−||𝑥^𝑗,𝑛||𝑎_2𝑛^𝜈/2+11𝑗𝛿^𝑛1/2.(4.43) Then, since Φ^𝜈/2(𝑥)𝛿^𝑛−1/2<𝐶 for 𝑥>𝑥_0,𝑛, we have ^𝑛_𝑗=1𝐵_{𝜈−1,𝑗}(𝑥)⩽𝐶(𝑓)𝜀_𝑛𝑛_𝑗=11𝑗⩽𝐶(𝑓)𝜀^𝑛log𝑛.(4.44) Therefore, the result is proved.

Proof of Proposition 3.9. By Lemma 2.3 we see that, for a constant 𝐶₁>0, ||𝑍_𝑛(𝑙,𝜈,𝑓;𝑥)||⩽𝐶_1𝑛_𝑙𝑗=1_𝑠=1||𝑓^(𝑠)𝑥_𝑗,𝑛||||𝑙_𝑗𝑛(𝑥)||^𝜈𝜈−1_𝑖=𝑠⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑗,𝑛⎞⎟⎟⎟⎠^𝑖−𝑠||𝑥−𝑥_𝑗,𝑛||^𝑖⩽𝐶₁𝑎_𝑛𝑛^𝑛_𝑙𝑗=1_𝑠=1||𝑓^(𝑠)𝑥_𝑗,𝑛||||𝑙_𝑗𝑛(𝑥)||^𝜈𝜈−1_𝑖=𝑠⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑗,𝑛⎞⎟⎟⎟⎠^𝑖||𝑥−𝑥_𝑗,𝑛||^𝑖⩽𝐶₁𝑎_𝑛𝑛^𝑙_{𝑠=1𝜈−1𝑛𝑖=0}_𝑗=1||𝑓^(𝑠)𝑥_𝑗,𝑛||||𝑙_𝑗𝑛(𝑥)||^𝜈||𝑥−𝑥_𝑗,𝑛||^𝑖⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑗,𝑛⎞⎟⎟⎟⎠^𝑖=∶𝑎^𝑛𝑛^𝑙_{𝑠=1𝜈−1𝑖=0}𝐶_𝑖,𝑠(𝑥),(4.45) where, 𝐶_𝑖,𝑠(𝑥)∶=^𝑛_𝑗=1||𝑓^(𝑠)𝑥_𝑗,𝑛||||𝑙_𝑗𝑛(𝑥)||^𝜈||𝑥−𝑥_𝑗,𝑛||^𝑖⎛⎜⎜⎜⎝𝑛𝑎^22𝑛−𝑥^2𝑗,𝑛⎞⎟⎟⎟⎠^𝑖.(∗) We set 𝑋_{𝑖,𝑠,𝑛}(𝜈,𝑓;𝑥)∶=^𝑛_𝑗=1𝑓^(𝑠)𝑥_𝑗,𝑛𝑙^𝜈𝑗𝑛(𝑥)𝑒_𝑖(𝜈,𝑗,𝑛)𝑥−𝑥_𝑗,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 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 Φ^𝑛3/4(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 𝑙_𝑚𝑛(𝑥) in the proof of Proposition 3.7; furthermore we use Remark 4.4. Then we have Φ^𝑛3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎𝑛^𝑛^𝜈||𝑋||_{𝜈−1,𝑠,𝑛}(𝜈,𝑓;𝑥)⩽𝐶₁𝐶(𝑓)log𝑛.(4.48) Consequently, we have Φ^𝑛(𝑥)^3/4𝑤(𝑥)|𝑥|+𝑎𝑛^𝑛^𝜈||𝑍_𝑛(𝑙,𝜈,𝑓;𝑥)||⩽𝐶₁𝐶(𝑓)𝑎^𝑛log𝑛𝑛.(4.49) Similarly, we have Φ(𝑥)^3/4𝑤(𝑥)|𝑥|+𝑎𝑛^𝑛^𝜈||𝑊_𝑛(𝑙,𝜈,𝐺;𝑥)||⩽𝐶₁𝐶(𝐺)𝑎^𝑛log𝑛𝑛.(4.50)

Proof of Proposition 3.10. Let 𝑃∈𝒫_𝑚 be fixed. From Proposition 3.9 we see ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿_𝑛(𝜈,𝑃)−𝑃‖‖‖_𝐿∞(ℝ)=‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿_𝑛(𝜈,𝑃)−𝐿_𝑛(𝜈−1,𝜈,𝑃)‖‖‖_𝐿∞(ℝ)=‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑍_𝑛(𝜈−1,𝜈,𝑃)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(4.51)

Proof of Proposition 3.11. Let 𝑃∈𝒫_𝑚 with 𝑃(0)=0. Then 𝑃 satisfies the condition (A-1). By Proposition 3.8, we see ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑌_𝑛(𝜈,𝑃)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(4.52) So, from Propositions 3.10 and 3.8 (noting (3.3)) we have ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑋_𝑛(𝜈,𝑃)−𝑃‖‖‖_𝐿∞(ℝ)⩽𝐶₁‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿_𝑛(𝜈,𝑃)−𝑃‖‖‖_𝐿∞(ℝ)+‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑌_𝑛(𝜈,𝑃)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(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∈𝒫_𝑚 with 𝑃(0)=0 such that sup_𝑥∈ℝ||𝑓(𝑥)−𝑃(𝑥)||Φ^−3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈<𝜀.(4.55) In fact, by [21, Theorem 1.4], there exists a polynomial 𝑅∈𝒫_𝑚 such that sup_𝑥∈ℝ||𝑓(𝑥)−𝑅(𝑥)||𝑤^𝜈−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⩽𝐶𝑤^𝜈−𝛿(𝑥),𝑥∈ℝforsome𝐶>0(4.57) and 𝑓(0)=0, we have sup_𝑥∈ℝ||𝑓(𝑥)−𝑃(𝑥)||Φ^−3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈⩽⩽ssuupp_{𝑥∈ℝ𝑥∈ℝ}||𝑓(𝑥)−𝑃(𝑥)||𝑤^𝜈−𝛿(𝑥)||𝑓(𝑥)−𝑅(𝑥)||𝑤^𝜈−𝛿(𝑥)+||𝑓(0)−𝑅(0)||𝑤^𝜈−𝛿(0)<𝜀;(4.58) that is, we have (4.55). Here, we know that 𝐿_𝑛(𝜈,𝑓)(𝑥)−𝑓(𝑥)=𝑋_𝑛(𝜈,𝑓−𝑃)+𝑋_𝑛(𝜈,𝑃)−𝑃+(𝑃−𝑓)+𝑌_𝑛(𝜈,𝑓).(4.59) Therefore, by Propositions 3.7, 3.11, and 3.8, we have for 𝑛 large enough ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿_𝑛(𝜈,𝑓)−𝑓‖‖‖_𝐿∞(ℝ)⩽𝐶‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑋_𝑛(𝜈,𝑓−𝑃)‖‖‖_𝐿∞(ℝ)+‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈(𝑋_𝑛(𝜈,𝑃)−𝑃)‖‖‖_𝐿∞(ℝ)+‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈(𝑃−𝑓)‖‖‖_𝐿∞(ℝ)+‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑌_𝑛(𝜈,𝑓)‖‖‖_𝐿∞(ℝ)=∶𝐼_𝑛+𝐼𝐼_𝑛+𝐼𝐼𝐼_𝑛+𝐼𝑉_𝑛.(4.60) Here we see that, by Proposition 3.7 with 𝜀∶=𝐶(𝑓−𝑃) (constant depending only on 𝑓−𝑃) and (4.55), 𝐼_𝑛⩽𝐶𝜀,𝐼𝐼𝐼_𝑛⩽𝜀,(4.61) and for 𝑛⩾𝑛₀ large enough, we have 𝐼𝐼𝑉𝐼_𝑛𝑛⩽𝐶𝜀,(byProposition3.9),⩽𝐶𝜀,(byProposition3.6).(4.62) Consequently, noting (3.3), we have lim_𝑛→∞𝐼_𝑛+𝐼𝐼_𝑛+𝐼𝐼𝐼_𝑛+𝐼𝑉_𝑛=0.(4.63) Then we have Theorem 3.12.

Proof of Theorem 3.13. From Theorem 3.12 and Proposition 3.9, we have ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿_𝑛(𝑙,𝜈,𝑓)−𝑓‖‖‖_𝐿∞(ℝ)⩽𝐶₁‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿_𝑛(𝜈,𝑓)−𝑓‖‖‖_𝐿∞(ℝ)+‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑍_𝑛(𝑙,𝜈,𝑓)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(4.64)

Proof of Corollary 3.14. From Theorem 3.12 and Proposition 3.8, we have ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑋_𝑛(𝜈,𝑓)−𝑓‖‖‖_𝐿∞(ℝ)⩽𝐶₁‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈(𝐿_𝑛(𝜈,𝑓)−𝑓)‖‖‖_𝐿∞(ℝ)+‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑌_𝑛(𝜈,𝑓)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(4.65)

Proof of Theorem 3.15. From Theorem 3.13 and Proposition 3.10 we have ‖‖‖Φ_3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿^𝑛(𝑙,𝜈,𝑓⊕𝐺)−𝑓‖‖‖_𝐿∞(ℝ)⩽𝐶‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐿_𝑛(𝑙,𝜈,𝑓)−𝑓‖‖‖_𝐿∞(ℝ)+‖‖‖Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑊_𝑛(𝑙,𝜈,𝐺)‖‖‖_𝐿∞(ℝ)⟶0as𝑛⟶∞.(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||𝐼_𝑛=𝑘,𝐻_𝑛(𝑓);𝑓−𝐼[𝑘,𝑓_∞−∞]||⩽𝐶‖‖‖𝑘Φ(𝑥_3/4)𝐻_𝑛(𝑓)(𝑥)−𝑓(𝑥)𝑑𝑥(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝐻_𝑛(𝑓)(𝑥)−𝑓(𝑥)‖‖‖_𝐿∞(ℝ)×^∞−∞Φ^3/4(𝑥)𝑤(𝑥)|𝑥|+𝑎^𝑛𝑛^𝜌^𝜈𝑘(𝑥)𝑑𝑥⟶0as𝑛⟶∞.(4.67)

5. Divergence Theorem

If 𝜈 is a positive odd integer, then we obtain the unboundedness of 𝐿_𝑛(𝜈,𝑓;𝑥). We define Λ_𝑛(𝜈,ℝ)=max_{𝑛𝑥∈ℝ}_𝑘=1||ℎ_𝑘𝑛(𝑥)||.(5.1)

Theorem 5.1 (cf. [1, Theorem 2]). Let 𝜈>0 be an odd integer. Then there exists a constant 𝐶>0 and 𝑛₀>0 such that for 𝑛⩾𝑛₀Λ_𝑛(𝜈,ℝ)⩾𝐶log𝑛.(5.2)

Let −∞⩽𝑎<𝑏⩽∞, and let us define Λ_𝑛(𝜈,(𝑎,𝑏))=max_{𝑛𝑎⩽𝑥⩽𝑏}_𝑘=1||ℎ_𝑘𝑛(𝑥)||.(5.3) Then we will show that for 𝑛⩾𝑛₀Λ_𝑛(𝜈,(𝑎,𝑏))⩾𝐶log𝑛.(5.4)

Remark 5.2. For the interpolation polynomial 𝐿_𝑛(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 ‖‖𝐿_𝑛𝜈,𝑓^∗;𝑥−𝑓(𝑥)‖‖_{𝐿∞([𝑎,𝑏])}⟶0as𝑛⟶∞.(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 ‖‖𝐿_𝑛𝜈,𝑓^∗;𝑥−𝑓^∗(𝑥)‖‖_{𝐿∞([𝑎,𝑏])}↛0as𝑛⟶∞;(5.6) that is, for 𝑓(𝑥)∶=𝑓^∗(𝑥), 𝑎⩽𝑥⩽𝑏 we see that the interpolation polynomials 𝐿_𝑛(𝜈,𝑓^∗;𝑥) do not converge to 𝑓. We also remark that the polynomial 𝐿_𝑛(𝜈,𝑓^∗;𝑥) interpolates 𝑓∈𝐶[𝑎,𝑏] at only {𝑥_𝑗,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 Ψ_𝑗(𝑥) of degree 𝑗 such that (−1)^𝑗Ψ_𝑗(−𝜈)>0 for 𝜈=1,3,5,…, and the following relation holds. Let 0<𝜀<1. Then one has an expression for (1/𝜀)(𝑎_𝑛/𝑛)⩽|𝑥_𝑘,𝑛|⩽𝜀𝑎_𝑛 and 𝑠=0,1,…,(𝜈−1)/2: 𝑒_2𝑠(𝜈,𝑘,𝑛)=(−1)^𝑠(21𝑠)!Ψ^𝑠(−𝜈)𝛽^𝑠(𝑛,𝑘)𝑛𝑎_𝑛^2𝑠1+𝜂_𝑘𝑛(𝜈,𝑠),(5.7) where for 𝛽^𝑠(𝑛,𝑘), 𝑛=1,2,3,…, 𝑘=1,2,…,𝑛, 𝑠=0,1,…,(𝜈−1)/2, there exist the constants 𝐶₁,𝐶₂ such that 0<𝐶₁⩽𝛽^𝑠(𝑛,𝑘)⩽𝐶₂,(5.8) and 𝜂_𝑘𝑛(𝜈,𝑠) satisfies ||𝜂_𝑘𝑛(𝜈,𝑠)||⩽𝜀^∗𝑛,𝜀^∗𝑛=𝜀^∗𝑛(𝜀)⟶0as𝜀⟶0,(5.9) for 𝑘 with (1/𝜀)(𝑎_𝑛/𝑛)⩽|𝑥_𝑘,𝑛|⩽𝜀𝑎_𝑛, and for 𝑠=0,1,2,….

Proof of Theorem 5.1. To get a lower bound of Λ_𝑛(𝜈,ℝ), it suffices to consider a lower bound (5.4) of Λ_𝑛(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)(𝑎_𝑛/𝑛)⩽𝑐<2𝑏−𝑎⩽𝜀𝑎_𝑛. See the expression ||ℎ_𝑘𝑛(𝑥)||⩾|||𝑙^𝜈𝑘𝑛(𝑥)𝑒_𝜈−1(𝜈,𝑘,𝑛)𝑥−𝑥_𝑘,𝑛^𝜈−1|||−|||||𝑙_𝜈𝑘𝑛(𝑥)^𝜈−2_𝑖=0𝑒_𝑖(𝜈,𝑘,𝑛)𝑥−𝑥_𝑘,𝑛^𝑖|||||.(5.10) Then we have Λ_𝑛(𝜈,[𝑎,𝑏])⩾max_{𝑥∈𝐼𝑥}_{𝑘𝑛∈𝐽}|||𝑙_𝜈𝑘𝑛(𝑥)𝑒_𝜈−1(𝜈,𝑘,𝑛)𝑥−𝑥_𝑘,𝑛^𝜈−1−max_𝑥∈𝐼|||_𝑥_{𝑘𝑛∈𝐽}|||||𝑙_𝜈𝑘𝑛(𝑥)^𝜈−2_𝑖=0𝑒_𝑖(𝜈,𝑘,𝑛)𝑥−𝑥_𝑘,𝑛^𝑖=∶max_𝑥∈𝐼|||||𝐹_𝑛(𝑥)−max_𝑥∈𝐼𝐺_𝑛(𝑥).(5.11) It follows from Remark 4.5 that max_{𝑎⩽𝑥⩽𝑏}𝐺_𝑛(𝑥)⩽𝐶. Hence, it is enough to show that max_𝑥∈𝐼𝐹_𝑛(𝑥)⩾𝐶log𝑛. Let 𝑥_𝑘+1,𝑛,𝑥_𝑘,𝑛∈𝐽; then by Lemma 4.2 and the definition of 𝜑_𝑛(𝑥_𝑘,𝑛) there exists 0<𝛼⩽𝛽<∞ such that 𝛼𝑎^𝑛𝑛⩽||𝑥^𝑘,𝑛−𝑥_𝑘+1,𝑛||⩽𝛽𝑎^𝑛𝑛.(5.12) For 𝑥∈𝐼 we consider only 𝑥_𝑘,𝑛∈𝐽 such that, for some positive integer 𝑗_𝑘, 𝑗_𝑘−1𝛼𝑎𝑛^𝑛⩽||𝑥^𝑘,𝑛−𝑥||⩽𝑗_𝑘𝛽𝑎𝑛^𝑛.(5.13) Then for each 𝑥∈𝐼 we define Γ(𝑥)∶=𝑗_𝑘;𝑥_𝑘,𝑛∈𝐽,𝑗_𝑘−1𝛼𝑎^𝑛𝑛⩽||𝑥^𝑘,𝑛−𝑥||⩽𝑗_𝑘𝛽𝑎^𝑛𝑛.(5.14) Here, we will see that 𝑗;1⩽𝑗⩽𝑑2−𝛽𝑐𝑎𝑛_𝑛−1⊂Γ(𝑥).(5.15) Let 𝑥𝑥_{𝑚+1,𝑛𝑘(𝑐)+1,𝑛}<𝑥⩽𝑥_𝑚,𝑛,<𝑐⩽𝑥_{𝑘(𝑐),𝑛},𝑥_{𝑘(𝑑),𝑛}⩽𝑑<𝑥_{𝑘(𝑑)−1,𝑛}.(5.16) Then we have 𝑥−𝑥_𝑚+1,𝑛⩽𝑥_𝑚,𝑛−𝑥_𝑚+1,𝑛⩽𝛽𝑎^𝑛𝑥−𝑥_𝑚+2,𝑛𝑛,⩽𝑥_𝑚,𝑛−𝑥_𝑚+2,𝑛⩽2𝛽𝑎^𝑛𝑥−𝑥_{𝑘(𝑐),𝑛}⋮𝑛,⩽𝑥_𝑚,𝑛−𝑥_{𝑘(𝑐),𝑛}⩽𝑗(𝑐)𝛽𝑎^𝑛𝑥_𝑚,𝑛𝑛,−𝑥⩽𝑥_𝑚,𝑛−𝑥_𝑚+1,𝑛⩽𝛽𝑎^𝑛𝑥_{𝑚−1,𝑛}𝑛,−𝑥⩽𝑥_{𝑚−1,𝑛}−𝑥_𝑚+1,𝑛⩽2𝛽𝑎^𝑛𝑥_{𝑘(𝑑),𝑛}⋮𝑛,−𝑥⩽𝑥_{𝑘(𝑑),𝑛}−𝑥_𝑚+1,𝑛⩽𝑗(𝑑)𝛽𝑎^𝑛𝑛,(5.17) where 𝑗(𝑐) and 𝑗(𝑑) are integers. On the other hand, 𝑥_{𝑘(𝑑),𝑛}−𝑥_{𝑘(𝑐),𝑛}𝛽𝑎_𝑛/𝑛⩾𝑑−𝑐−2𝛽𝑎^𝑛/𝑛𝛽𝑎_𝑛/𝑛=𝑑−𝛽𝑐𝑎𝑛_𝑛−2.(5.18) Therefore, we have max(𝑗(𝑐),𝑗(𝑑))⩾12𝑑−𝛽𝑐𝑎𝑛_𝑛−2=𝑑−𝑐2𝛽𝑎𝑛_𝑛−1.(5.19) Now, we take a positive integer 𝑁(𝑛) such that 𝑁(𝑛)⩽𝑑−𝑐2𝛽𝑎𝑛_𝑛−1<𝑁(𝑛)+1.(5.20) Consequently, we have the following. By Lemmas 4.2(1), Lemma 4.1(1), (2), (4), and Lemma 2.3, we have 𝐹_𝑛(𝑥)=_{𝑥𝑘,𝑛∈𝐽}||𝑙_𝑘𝑛(𝑥)||^𝜈||𝑒_𝜈−1(𝜈,𝑘,𝑛)||||𝑥_𝑘,𝑛−𝑥||^𝜈−1=_{𝑥𝑘,𝑛∈𝐽}||𝑝^𝑛(𝑥)||𝑤_𝜌(𝑥)||𝑥−𝑥_𝑘,𝑛||||𝑝^𝑛𝑥_𝑘,𝑛||𝑤_𝜌𝑥_𝑘,𝑛𝑤^𝜌𝑥_𝑘,𝑛𝑤_𝜌(𝑥)^𝜈||𝑒_𝜈−1(𝜈,𝑘,𝑛)||||𝑥−𝑥_𝑘,𝑛||^𝜈−1=_{𝑥𝑘,𝑛∈𝐽}||𝑝^𝑛(𝑥)||𝑤_𝜌(𝑥)||𝑝^𝑛𝑥_𝑘,𝑛||𝑤_𝜌𝑥_𝑘,𝑛𝑤^𝜌𝑥_𝑘,𝑛𝑤_𝜌(𝑥)^𝜈||𝑒_𝜈−1(𝜈,𝑘,𝑛)||1⩾𝐶_{𝑗𝑘∈Γ(𝑥)}||𝑥−𝑥_𝑘,𝑛||𝑛/𝑎𝑎_𝑛𝑎^{𝑛−1/2𝑛−1/2}𝑤_𝜌(𝑑)𝑤_𝜌(𝑎)^𝜈||𝑒_𝜈−1(𝜈,𝑘,𝑛)||𝑛𝑗_𝑘𝛽𝑎_𝑛⩾𝐶_{𝑗𝑘∈Γ(𝑥)}1𝑗_𝑘𝑎𝑛^𝑛^𝜈−1||𝑒_𝜈−1(𝜈,𝑘,𝑛)||.(5.21) Here, using Lemma 5.3 and (5.15), we have 𝐹_𝑛(𝑥)⩾𝐶_{𝑗𝑘∈Γ(𝑥)}1𝑗_𝑘𝑎𝑛^𝑛^𝜈−1𝑎𝑛_𝑛^𝜈−1⩾𝐶_{1⩽𝑗⩽𝑁(𝑛)}1𝑗.(5.22) Here, there exists 0<𝜂<1 such that 𝑛^𝜂⩽𝑁(𝑛). Therefore we see 𝐹_𝑛(𝑥)⩾𝐶_{1⩽𝑗⩽𝑛𝜂}1𝑗⩾𝐶log𝑛.(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.