On the Rational Approximation of Analytic Functions Having Generalized Types of Rate of Growth

Abstract

The present paper is concerned with the rational approximation of functions holomorphic on a domain 𝐺⊂𝐶, having generalized types of rates of growth. Moreover, we obtain the characterization of the rate of decay of product of the best approximation errors for functions f having fast and slow rates of growth of the maximum modulus.

1. Introduction

Let 𝐾 be a compact subset of the extended complex plane 𝐶 and let 𝐸_𝑛 be the error in the best uniform approximation of a function 𝑓 (holomorphic on 𝐾) on 𝐾 in the class 𝑅_𝑛 of all rational functions of order 𝑛: 𝐸_𝑛=𝐸_𝑛(𝑓,𝐾)=inf_{𝑟∈𝑅𝑛}‖𝑓−𝑟‖_𝐾(1.1) for each nonnegative integer 𝑛, where ‖⋅‖_𝐾 is the supremum norm on 𝐾.

In view of Walsh's inequality [1], if 𝑓 is holomorphic on 𝐶⧵𝑀, where 𝑀 is a compact set in 𝐶 and 𝑀∩𝐾=𝜙, then limsup_𝑛→∞𝐸^𝑛1/𝑛≤1𝑑,(1.2) where 𝑑=exp(1/𝐶(𝐾,𝑀)) and 𝐶(𝐾,𝑀) is the capacity of the condenser (𝐾,𝑀), (see [2–4], for the definition and properties of the capacity).

The theory of Hankel operators permits one [5–7] to estimate the order of decrease of the product 𝐸₁𝐸₂⋯𝐸_𝑛: limsup_𝑛→∞𝐸₁𝐸₂⋯𝐸_𝑛^1/𝑛2≤1𝑑.(1.3)

The last relation implies Walsh's inequality (1.2) and the following upper estimate for liminf_𝑛→∞𝐸^𝑛1/𝑛: liminf_𝑛→∞𝐸^𝑛1/𝑛≤1𝑑².(1.4)

The present paper is concerned to results that make the inequalities (1.2), (1.3) and (1.4) more precise for analytic functions having generalized types of the rate of growth of the maximum modulus in the domain of analyticity of 𝑓.

The generalized order 𝜌(𝛼,𝛽,𝑓) of the rate of growth of entire functions 𝑓 was introduced by Šeremeta [8], who obtained a characterization of 𝜌(𝛼,𝛽,𝑓) in terms of the coefficients of the power series of 𝑓. In [8], the relationship between the generalized order of entire functions 𝑓 and the degree of polynomial approximation of 𝑓 was studied. The coefficient characterization of a generalized order of the rate of growth of functions analytic in a disk has been discussed in several papers [9–12]. The degree of rational approximation of entire functions of a finite generalized order is investigated in [6].

Now let us consider the Dirichlet problem in the domain 𝐶⧵(𝐾∪𝑀) with boundary function equal to 1 on 𝜕𝑀 and to 0 on 𝜕𝐾. Here, 𝐾 and 𝑀 be disjoint compact sets with connected complements in the extended complex plane 𝐶 such that their boundaries consist of finitely many closed analytic Jordan curves. Since the domain 𝐶⧵(𝐾∪𝑀) is regular with respect to the Dirichlet problem, this problem is solvable. Let 𝑤(𝑧) be the solution which is extended by continuity to 𝐶∶𝑤(𝑧)=1 for 𝑧∈𝑀 and 𝑤(𝑧)=0 for 𝑧∈𝐾. For 0<𝜀<1, let 𝛾(𝜀)={𝑧∶𝑤(𝑧)=𝜀}.

Let 𝛼 and 𝛽 be continuous positive functions on [𝑎,∞) satisfying the following properties:(i)lim_𝑥→∞𝛼(𝑥)=+∞, and lim_𝑥→∞𝛽(𝑥)=+∞;(ii)lim_𝑥→∞(𝛽(𝑥+𝑜(𝑥))/𝛽(𝑥))=1;(iii)𝛼⁻¹(log(1/℧𝛽(𝑥)))/𝛼⁻¹(log(1/℧^𝛽(𝑥)))=𝑜(𝑥) as 𝑥→0 for all ℧^>℧>0.

Let 𝑓 be holomorphic on 𝐺=𝐶⧵𝑀. We define the generalized order 𝜌(𝛼,𝛽,𝑓) and generalized type 𝑇(𝛼,𝛽,𝑓) of 𝑓 in the domain 𝐺 by the formulae:(a)𝜌(𝛼,𝛽,𝑓)=limsup_𝜀→1(𝛼(log‖𝑓‖_𝛾(𝜀))/𝛽(log(1/(1−𝜀)))), (b)𝑇(𝛼,𝛽,𝑓)=limsup_𝜀→1(𝛼(‖𝑓‖_𝛾(𝜀))/[(1/(1−𝜀))]^{𝜌(𝛼,𝛽,𝑓)}), where ‖𝑓‖_𝛾(𝜀)=max_{𝑧∈𝛾(𝜀)}|𝑓(𝑧)|.

It is easy to see that for the functions 𝛼(𝑥)=log_𝑝𝑥,𝑝≥2, and 𝛽(𝑥)=𝑥 properties (i)–(iii) will hold. The following theorem gives the characterization of the rate of decay of product 𝐸₀𝐸₁⋯𝐸_𝑛 for functions 𝑓 having fast rates of growth of the maximum modulus. So to avoid some trivial cases, we will assume that lim_𝜀→1‖𝑓‖_𝛾(𝜀)=∞.

Theorem 1.1. Suppose that 𝑓 is holomorphic on 𝐺,𝛼 and 𝛽 satisfy conditions (i)–(iii), and 𝑓 has generalized order 𝜌(𝛼,𝛽,𝑓)>0 and generalized type 𝑇(𝛼,𝛽,𝑓) in the domain 𝐺. Then, limsup_𝑛→∞exp𝛼(𝑛)𝛽log⁺𝐸⁰𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑^𝜌≤𝑇(𝛼,𝛽,𝑓),(1.5) where log⁺𝑥=max(0,log𝑥) for 𝑥≥0.

Proof. Let us assume that 𝑇(𝛼,𝛽,𝑓)<∞. Fix arbitrary numbers 𝑇^>𝑇^>𝑇(𝛼,𝛽,𝑓). For 𝑛=1,2,…, we set 𝛿_𝑛=min14,𝛽⁻¹𝑇^exp(−𝛼(𝑛))^1/𝜌.(1.6)
We have 𝛿_𝑛→0 as 𝑛→∞. Using (1.5) for all sufficiently large values of 𝑛,𝑛≥𝑛₁, we set log‖𝑓‖_𝛾2,𝑛≤=𝛼𝛼⁻¹₋₁log𝑇^𝛽𝛿_𝑛^−𝜌𝜌log𝑇^1/𝜌𝛽1𝛿_𝑛.(1.7)
From (1.6), we have 𝑛=𝛼⁻¹log𝑇^𝛽𝛿_𝑛^−𝜌.(1.8)
In (1.7), 𝛾_2,𝑛 defined as subsets of the extended complex plane 𝐶: 𝛾𝐷_{𝑘,𝑛𝑛}=𝑧∶𝑤(𝑧)=𝜀_𝑘,𝑛,=𝑧∶𝑤(𝑧)>𝜀_0,𝑛,(1.9) where 𝜀_0,𝑛=𝑘/2𝑛,𝜀_1,𝑛=𝑘/𝑛,𝜀_2,𝑛=1−𝛿_𝑛,𝑘=0,1,2, and 𝑛=1,2,…. It is given [13] that 𝛾_0,𝑛,𝛾_1,𝑛, and 𝛾_2,𝑛,𝑛=1,2,…, consist of finitely many closed analytic curves whose lengths are bounded from above by a positive quantity not depending on 𝑛. It is assumed that 𝛾_0,𝑛 and 𝛾_2,𝑛 are positively oriented with respect to 𝐷_𝑛 and {𝑧∶𝑤(𝑧)>𝜀_2,𝑛}, respectively.
In view of (1.8) for 𝑛≥max(𝑛₀,𝑛₁), we may use the inequality 3.1 of [13] in the form: 𝐸₀𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑≤𝐶^𝑛𝑚(𝑛+1)!𝑛^8𝑛^{1/𝑛(𝑛+1)}×exp𝛼⁻¹𝜌log1/𝑇^1/𝜌𝛽𝛿_𝑛𝛼⁻¹𝜌log1/𝑇^1/𝜌𝛽𝛿_𝑛+𝐶(𝐾𝛿^𝑛,𝑀).(1.10)
Now, using property (iii), we get log⁺𝐸⁰𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑≤𝛿^𝑛𝐶(𝐾,𝑀)+𝑜𝛿^𝑛.(1.11)
It gives limsup_𝑛→∞exp𝛼(𝑛)𝛽log⁺𝐸₀𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑^𝜌≤𝑇^.(1.12)
On letting 𝑇^→𝑇(𝛼,𝛽,𝑓), the proof is complete.

In the consequence of Theorem 1.1, we have the following.

Corollary 1.2. With the assumption of Theorem 1.1, the following inequalities are valid: lilmimsusup_𝑛→∞p_𝑛→∞exp(𝛼(𝑛))𝛽log⁺𝐸^𝑛1/𝑛𝑑^𝜌≤𝑇(𝛼,𝛽,𝑓),(1.13)exp(𝛼(𝑛))𝛽log⁺𝐸^𝑛1/𝑛𝑑²^𝜌≤𝑇(𝛼,𝛽,𝑓).(1.14)

Proof. Using the fact 𝐸_𝑛≤𝐸_𝑛−1≤⋯≤𝐸₀, we obtain (1.13) immediately from (1.5). To prove (1.14), let us suppose that liminf_𝑛→∞exp𝛼(𝑛)𝛽log⁺𝐸^𝑛1/𝑛𝑑²^𝜌>𝑇^>𝑇(𝛼,𝛽,𝑓).(1.15)
Then, for sufficiently large values of 𝑛, we get 𝛽log⁺𝐸^𝑛1/𝑛𝑑²>𝑇^exp𝛼(𝑛)^1/𝜌(1.16) or log⁺𝐸_𝑛𝑑^2𝑛≥𝑛𝛽⁻¹exp𝑇(𝛼^(𝑛))^1/𝜌.(1.17)
Since the functions 𝛼 and 𝛽 are increasing, (1.17) gives log⁺𝐸⁰𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑≥∑^𝑛𝑘=0𝑘𝛽⁻¹𝑇^/exp𝛼(𝑛)^1/𝜌+𝑐≥𝛽₋₁𝑛(𝑛+1)exp𝑇𝛼^(𝑛)^1/𝜌+𝑐𝑛(𝑛+1),(1.18) where 𝑐 is a constant. Using (ii), we get 𝛽log⁺𝐸₀𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑^𝜌≥exp𝑇𝛼^(𝑛)(1.19) or liminf_𝑛→∞exp𝛼(𝑛)𝛽log⁺𝐸₀𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}^𝜌≥𝑇^>𝑇(𝛼,𝛽,𝑓)(1.20) which contradicts (1.5). Thus, (1.14) is valid.

2. Rational Approximation of Analytic Functions Having Slow Rates of Growth

For a function 𝑓 analytic in a domain 𝐺, the type of 𝑓 in 𝐺 can be defined by (b) for 𝛼(𝑥)=log𝑥 and 𝛽(𝑥)=𝑥: 𝑇=limsup_𝜀→1log‖𝑓‖_𝛾(𝜀)(1/1−𝜀)^𝜌.(2.1)

For 𝛼(𝑥)=log𝑥 and 𝛽(𝑥)=𝑥, the property (iii) fails to hold. However, we have the following: 𝛼⁻¹(𝑐log(1/𝛽(𝑥)))𝛼⁻¹((𝑐+1)log(1/𝛽(𝑥)))=𝑥,(2.2) and we may repeat the arguments involving (1.10), we get 𝐸₀𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑≤𝑐^𝑛(𝑛+1)!𝑛^8𝑛^{1/𝑛(𝑛+1)}×exp𝛼⁻¹log1/𝑇^1/𝜌𝛽𝛿_𝑛^𝜌𝛼⁻¹log1/𝑇^1/𝜌𝛽𝛿_𝑛^𝜌+𝐶(𝐾𝛿^𝑛,𝑀).(2.3)

Taking 𝑇^=𝑇^+1, and 𝑥=𝑇^1/𝜌𝛿_𝑛 in (2.2), for sufficiently large values of 𝑛 we have 𝑛𝑇^+1^1/𝜌log⁺𝐸₀𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑𝜌≤𝑇^(2.4) or limsup_𝑛→∞𝑛log⁺𝐸₀𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑^𝜌≤𝑇𝑇+1.(2.5)

We summarize the above facts in the following.

Theorem 2.1. Let 𝑓 have an order 𝜌>0 and generalized type 𝑇 in the domain 𝐺. Then, limsup_𝑛→∞𝑛log⁺𝐸₀𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑^𝜌≤𝑇𝑇+1.(2.6)

By the inequality 𝐸_𝑛≤𝐸_𝑛−1≤⋯≤𝐸₀, one gets the following.

Corollary 2.2. With the assumption of Theorem 2.1limsup_𝑛→∞𝑛log⁺𝐸^𝑛1/𝑛𝑑^𝜌≤𝑇𝑇+1.(2.7)

Theorem 2.1 also gives us the following corollary.

Corollary 2.3. With the assumption of Theorem 2.1, liminf_𝑛→∞𝑛log⁺𝐸^𝑛1/𝑛𝑑²^𝜌≤𝑇𝑇+1.(2.8)

Proof. Let liminf_𝑛→∞𝑛log⁺𝐸^𝑛1/𝑛𝑑²^𝜌>𝑇₁>𝑇𝑇+1.(2.9)
Then, from the relation lim_𝑛→∞∑^𝑛𝑘=0𝑘^1−1/𝑇1𝑛^2−1/𝑇1=12−1/𝑇₁,(2.10) we obtain liminf_𝑛→∞𝑛log⁺𝐸¹𝐸₂⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑^𝜌≥𝑇₁>𝑇𝑇+1,(2.11) which contradicts the inequality (2.6).

Now, we define 𝛼-type of 𝑓 to classify functions having slow rates of growth.

A continuous positive function ℎ on [𝑎,+∞) belongs to the class Λ, if this function satisfies the following.

ℎ is strictly increasing on [𝑎,+∞), limli_𝑥→+∞m_𝑥→∞ℎ(𝑥)=+∞,(2.12)ℎ(𝑐𝑥)ℎ(𝑥)=1,(2.13) for any 𝑐>0.

Let 𝛼∈Λ. We define 𝛼-order and 𝛼-type of 𝑓 in 𝐺 by the formulae: 𝜌(𝛼,𝑓)=limsup_𝜀→1𝛼log‖𝑓‖_𝛾(𝜀)𝑇(𝛼,𝑓)=limsup_𝜀→1𝛼(log(1/(1−𝜀))),(2.14)𝛼log‖𝑓‖_𝛾(𝜀)[𝛼(1/(1−𝜀))]^𝜌.(2.15)

The following results are concerned with the degree of rational approximation of functions having 𝛼-type 𝑇(𝛼,𝑓). The functions 𝛼(𝑥)=log_𝑝𝑥,𝑝≥1, and 𝛼(𝑥)=exp(log𝑥)^𝛿,0<𝛿<1, satisfy the condition 𝛼𝜖Λ. For 𝛼(𝑥)=log𝑥, the parameter 𝑇(𝛼,𝑓) is called the logarithmic type of 𝑓 in 𝐺 [14].

Theorem 2.4. Let 𝑓, analytic in 𝐺, be of 𝛼-order 𝜌(𝛼,𝑓)≥1, and 𝛼-type 𝑇(𝛼,𝑓),𝛼∈Λ. Then, limsup_𝑛→∞𝛼𝐸₀𝐸₁⋯𝐸_𝑛^{1/𝑛(𝑛+1)}𝑑^𝑛[𝛼(𝑛)]^{𝜌(𝛼,𝑓)}≤𝑇(𝛼,𝑓).(2.16)

Proof. The inequality (2.16) holds for 𝑇(𝛼,𝑓)=∞ obviously. Now, let 𝑇(𝛼,𝑓)<∞ and ‖𝑓‖_𝛾(𝜀)→∞ as 𝜀→1. Fix 𝑇^>𝑇(𝛼,𝑓). Then, for 𝜀 sufficiently close to 1, from (2.15), we have ‖𝑓‖_𝛾(𝜀)≤𝛼⁻¹𝑇^𝛼11−𝜀^𝜌,𝜌(𝛼,𝑓)≡𝜌.(2.17)
Define 𝛿_𝑛=min(1/4,1/𝑛),𝑛=1,2,…. Using [13, Equation (3.1)] with (2.17), for all sufficiently large values of 𝑛,𝑛≥𝑛₀, we have 𝐸₀𝐸₁⋯𝐸_𝑛𝑑^𝑛(𝑛+1)≤(𝑛+1)!𝑐^𝑛𝑛^8𝑛exp(𝑛+1)log𝛼⁻¹𝑇^[𝛼(𝑛)]^𝜌+1𝐶(𝐾,𝑀).(2.18)
Since 𝛼 is strictly increasing, for 𝑛≥𝑛₀, we get 𝐸₀𝐸₁⋯𝐸_𝑛^1/𝑛+1𝑑^𝑛≤𝑐₁𝛼⁻¹𝑇^[𝛼(𝑛)]^𝜌.(2.19)
In view of (2.13), (2.19) gives limsup_𝑛→∞𝛼𝐸₀𝐸₁⋯𝐸_𝑛^1/𝑛+1𝑑^𝑛[𝛼(𝑛)]^𝜌≤𝑇^.(2.20)
In order to complete the proof, it remains to let 𝑇^ tend to 𝑇(𝛼,𝑓).

Now, we have the following corollaries.

Corollary 2.5. With assumption of Theorem 2.4, limsup_𝑛→∞𝛼𝐸_𝑛𝑑^𝑛[𝛼(𝑛)]^{𝜌(𝛼,𝑓)}≤𝑇(𝛼,𝑓).(2.21)

The proof is immediate in view of 𝐸_𝑛≤𝐸_𝑛−1≤⋯𝐸₀.

For 𝑐>0, let 𝐹[𝑥,𝑐,𝜌]=log𝛼⁻¹(𝑐[𝛼(𝑥)]^𝜌).(2.22)

Corollary 2.6. Let a function 𝑓, analytic in 𝐺, be of 𝛼-order 𝜌(𝛼,𝑓)≥1, and 𝛼-type 𝑇(𝛼,𝑓) where 𝛼∈Λ is continuously differentiable on [𝑎,+∞) and for all 1<𝑐<∞ the function 𝑥(𝐹(𝑥,𝑐,𝜌))^=𝑂(1) as 𝑥→∞ or is increasing and lim_𝑥→∞𝑥(𝐹(𝑥,𝑐,𝜌))^𝐹(𝑥,𝑐,𝜌)=0.(2.23)
Then, liminf_𝑛→∞𝛼𝐸_𝑛𝑑^2𝑛[𝛼(𝑛)]^{𝜌(𝛼,𝑓)}≤𝑇(𝛼,𝑓).(2.24)

Proof. We may assume that 𝑇(𝛼,𝑓)<∞. Let liminf_𝑛→∞𝛼𝐸_𝑛𝑑^2𝑛[𝛼(𝑛)]^𝜌>𝑇^>𝑇(𝛼,𝑓).(2.25)
For sufficiently large values of 𝑛, 𝛼𝐸₀𝐸₁⋯𝐸_𝑛^1/𝑛+1𝑑^𝑛[𝛼(𝑛)]^𝜌≥𝛼exp(1/(𝑛+1))∑^𝑛𝑘=1𝐹𝑘,𝑇^,𝜌+𝑐[𝛼(𝑛)]^𝜌.(2.26)
Since 𝐹[𝑥,𝑇^,𝜌] is increasing, we get ^𝑛−1_𝑘=1𝐹𝑘,𝑇^,𝜌≤^𝑛1𝐹𝑥,𝑇^,𝜌𝑑𝑥≤^𝑛_𝑘=2𝐹𝑘,𝑇^,𝜌_𝑛1,𝐹𝑥,𝑇^,𝜌𝑑𝑥=𝑛𝐹𝑛,𝑇^,𝜌−𝐹1,𝑇^,𝜌−^𝑛1𝑥𝐹𝑥,𝑇^,𝜌^𝑑𝑥.(2.27)
We see that 1𝑛𝐹𝑛,𝑇^,𝜌^𝑛1𝑥𝐹𝑥,𝑇^,𝜌^𝑑𝑥⟶0as𝑛⟶∞.(2.28)
Thus, (1/(𝑛+1))∑^𝑛𝑘=1𝐹𝑘,𝑇^,𝜌𝐹𝑛,𝑇^,𝜌⟶1as𝑛⟶∞.(2.29)
From this and (2.26), we get liminf_𝑛→∞𝛼𝐸₀𝐸₁⋯𝐸_𝑛^1/𝑛+1𝑑^𝑛[𝛼(𝑛)]^𝜌≥𝛼exp𝐹𝑛,𝑇^,𝜌[𝛼(𝑛)]^𝜌≥𝑇^>𝑇(𝛼,𝐹)(2.30) which contradicts (2.16). Hence the proof is complete.

Remark 2.7. The function 𝛼(𝑥)=log_𝑝𝑥,𝑝≥1, and 𝛼(𝑥)=exp(log𝑥)^𝜌,0<𝛿<1, satisfy the assumptions of Corollary 2.6.

Acknowledgment

The author is extremely thankful to the reviewers for giving fruitful comments to improve the paper.