International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 908104 | https://doi.org/10.1155/2012/908104

Devendra Kumar, "On the Rational Approximation of Analytic Functions Having Generalized Types of Rate of Growth", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 908104, 10 pages, 2012. https://doi.org/10.1155/2012/908104

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

Academic Editor: Narendra Govil
Received21 Mar 2012
Accepted02 Jul 2012
Published02 Aug 2012

Abstract

The present paper is concerned with the rational approximation of functions holomorphic on a domain 𝐺𝐶, 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 𝐸𝑛 be the error in the best uniform approximation of a function 𝑓 (holomorphic on 𝐾) on 𝐾 in the class 𝑅𝑛 of all rational functions of order 𝑛: 𝐸𝑛=𝐸𝑛(𝑓,𝐾)=inf𝑟𝑅𝑛𝑓𝑟𝐾(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 𝐶𝑀, where 𝑀 is a compact set in 𝐶 and 𝑀𝐾=𝜙, then limsup𝑛𝐸𝑛1/𝑛1𝑑,(1.2) where 𝑑=exp(1/𝐶(𝐾,𝑀)) and 𝐶(𝐾,𝑀) is the capacity of the condenser (𝐾,𝑀), (see [24], for the definition and properties of the capacity).

The theory of Hankel operators permits one [57] to estimate the order of decrease of the product 𝐸1𝐸2𝐸𝑛: limsup𝑛𝐸1𝐸2𝐸𝑛1/𝑛21𝑑.(1.3)

The last relation implies Walsh's inequality (1.2) and the following upper estimate for liminf𝑛𝐸𝑛1/𝑛: liminf𝑛𝐸𝑛1/𝑛1𝑑2.(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 [912]. 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(𝐾𝑀) 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(𝐾𝑀) is regular with respect to the Dirichlet problem, this problem is solvable. Let 𝑤(𝑧) be the solution which is extended by continuity to 𝐶𝑤(𝑧)=1 for 𝑧𝑀 and 𝑤(𝑧)=0 for 𝑧𝐾. For 0<𝜀<1, let 𝛾(𝜀)={𝑧𝑤(𝑧)=𝜀}.

Let 𝛼 and 𝛽 be continuous positive functions on [𝑎,) satisfying the following properties:(i)lim𝑥𝛼(𝑥)=+, and lim𝑥𝛽(𝑥)=+;(ii)lim𝑥(𝛽(𝑥+𝑜(𝑥))/𝛽(𝑥))=1;(iii)𝛼1(log(1/𝛽(𝑥)))/𝛼1(log(1/𝛽(𝑥)))=𝑜(𝑥) as 𝑥0 for all >>0.

Let 𝑓 be holomorphic on 𝐺=𝐶𝑀. 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)=limsup𝜀1(𝛼(log𝑓𝛾(𝜀))/𝛽(log(1/(1𝜀)))), (b)𝑇(𝛼,𝛽,𝑓)=limsup𝜀1(𝛼(𝑓𝛾(𝜀))/[(1/(1𝜀))]𝜌(𝛼,𝛽,𝑓)), where 𝑓𝛾(𝜀)=max𝑧𝛾(𝜀)|𝑓(𝑧)|.

It is easy to see that for the functions 𝛼(𝑥)=log𝑝𝑥,𝑝2, and 𝛽(𝑥)=𝑥 properties (i)–(iii) will hold. The following theorem gives the characterization of the rate of decay of product 𝐸0𝐸1𝐸𝑛 for functions 𝑓 having fast rates of growth of the maximum modulus. So to avoid some trivial cases, we will assume that lim𝜀1𝑓𝛾(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, limsup𝑛𝛽exp𝛼(𝑛)log+𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌𝑇(𝛼,𝛽,𝑓),(1.5) where log+𝑥=max(0,log𝑥) for 𝑥0.

Proof. Let us assume that 𝑇(𝛼,𝛽,𝑓)<. Fix arbitrary numbers 𝑇>𝑇>𝑇(𝛼,𝛽,𝑓). For 𝑛=1,2,, we set 𝛿𝑛1=min4,𝛽1𝑇exp(𝛼(𝑛))1/𝜌.(1.6)
We have 𝛿𝑛0 as 𝑛. Using (1.5) for all sufficiently large values of 𝑛,𝑛𝑛1, we set log𝑓𝛾2,𝑛𝛼1𝑇log𝛽𝛿𝑛𝜌=𝛼11𝜌log𝑇1/𝜌𝛽𝛿𝑛.(1.7)
From (1.6), we have 𝑛=𝛼1𝑇log𝛽𝛿𝑛𝜌.(1.8)
In (1.7), 𝛾2,𝑛 defined as subsets of the extended complex plane 𝐶: 𝛾𝑘,𝑛=𝑧𝑤(𝑧)=𝜀𝑘,𝑛,𝐷𝑛=𝑧𝑤(𝑧)>𝜀0,𝑛,(1.9) where 𝜀0,𝑛=𝑘/2𝑛,𝜀1,𝑛=𝑘/𝑛,𝜀2,𝑛=1𝛿𝑛,𝑘=0,1,2, and 𝑛=1,2,. It is given [13] that 𝛾0,𝑛,𝛾1,𝑛, and 𝛾2,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 𝛾0,𝑛 and 𝛾2,𝑛 are positively oriented with respect to 𝐷𝑛 and {𝑧𝑤(𝑧)>𝜀2,𝑛}, respectively.
In view of (1.8) for 𝑛max(𝑛0,𝑛1), we may use the inequality 3.1 of [13] in the form: 𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝐶𝑑𝑛𝑚(𝑛+1)!𝑛8𝑛1/𝑛(𝑛+1)𝛼×exp1𝜌log1/𝑇1/𝜌𝛽𝛿𝑛𝛼1𝜌log1/𝑇1/𝜌𝛽𝛿𝑛+𝛿𝑛.𝐶(𝐾,𝑀)(1.10)
Now, using property (iii), we get log+𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝑑𝛿𝑛𝛿𝐶(𝐾,𝑀)+𝑜𝑛.(1.11)
It gives limsup𝑛𝛽exp𝛼(𝑛)log+𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌𝑇.(1.12)
On letting 𝑇𝑇(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: limsup𝑛𝛽exp(𝛼(𝑛))log+𝐸𝑛1/𝑛𝑑𝜌𝑇(𝛼,𝛽,𝑓),(1.13)limsup𝑛𝛽exp(𝛼(𝑛))log+𝐸𝑛1/𝑛𝑑2𝜌𝑇(𝛼,𝛽,𝑓).(1.14)

Proof. Using the fact 𝐸𝑛𝐸𝑛1𝐸0, we obtain (1.13) immediately from (1.5). To prove (1.14), let us suppose that liminf𝑛𝛽exp𝛼(𝑛)log+𝐸𝑛1/𝑛𝑑2𝜌>𝑇>𝑇(𝛼,𝛽,𝑓).(1.15)
Then, for sufficiently large values of 𝑛, we get 𝛽log+𝐸𝑛1/𝑛𝑑2>𝑇exp𝛼(𝑛)1/𝜌(1.16) or log+𝐸𝑛𝑑2𝑛𝑛𝛽1𝑇exp(𝛼(𝑛))1/𝜌.(1.17)
Since the functions 𝛼 and 𝛽 are increasing, (1.17) gives log+𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝑑𝑛𝑘=0𝑘𝛽1𝑇/exp𝛼(𝑛)1/𝜌+𝑐𝑛(𝑛+1)𝛽1𝑇exp𝛼(𝑛)1/𝜌+𝑐,𝑛(𝑛+1)(1.18) where 𝑐 is a constant. Using (ii), we get 𝛽log+𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌𝑇exp𝛼(𝑛)(1.19) or liminf𝑛𝛽exp𝛼(𝑛)log+𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝜌𝑇>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=limsup𝜀1log𝑓𝛾(𝜀)(1/1𝜀)𝜌.(2.1)

For 𝛼(𝑥)=log𝑥 and 𝛽(𝑥)=𝑥, the property (iii) fails to hold. However, we have the following: 𝛼1(𝑐log(1/𝛽(𝑥)))𝛼1((𝑐+1)log(1/𝛽(𝑥)))=𝑥,(2.2) and we may repeat the arguments involving (1.10), we get 𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝑐𝑑𝑛(𝑛+1)!𝑛8𝑛1/𝑛(𝑛+1)𝛼×exp1log1/𝑇1/𝜌𝛽𝛿𝑛𝜌𝛼1log1/𝑇1/𝜌𝛽𝛿𝑛𝜌+𝛿𝑛.𝐶(𝐾,𝑀)(2.3)

Taking 𝑇=𝑇+1, and 𝑥=𝑇1/𝜌𝛿𝑛 in (2.2), for sufficiently large values of 𝑛 we have 𝑛𝑇+11/𝜌log+𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌𝑇(2.4) or limsup𝑛𝑛log+𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌𝑇.𝑇+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, limsup𝑛𝑛log+𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌𝑇.𝑇+1(2.6)

By the inequality 𝐸𝑛𝐸𝑛1𝐸0, one gets the following.

Corollary 2.2. With the assumption of Theorem 2.1limsup𝑛𝑛log+𝐸𝑛1/𝑛𝑑𝜌𝑇.𝑇+1(2.7)

Theorem 2.1 also gives us the following corollary.

Corollary 2.3. With the assumption of Theorem 2.1, liminf𝑛𝑛log+𝐸𝑛1/𝑛𝑑2𝜌𝑇.𝑇+1(2.8)

Proof. Let liminf𝑛𝑛log+𝐸𝑛1/𝑛𝑑2𝜌>𝑇1>𝑇.𝑇+1(2.9)
Then, from the relation lim𝑛𝑛𝑘=0𝑘11/𝑇1𝑛21/𝑇1=121/𝑇1,(2.10) we obtain liminf𝑛𝑛log+𝐸1𝐸2𝐸𝑛1/𝑛(𝑛+1)𝑑𝜌𝑇1>𝑇,𝑇+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 on [𝑎,+) belongs to the class Λ, if this function satisfies the following.

is strictly increasing on [𝑎,+), lim𝑥(𝑥)=+,(2.12)lim𝑥+(𝑐𝑥)(𝑥)=1,(2.13) for any 𝑐>0.

Let 𝛼Λ. We define 𝛼-order and 𝛼-type of 𝑓 in 𝐺 by the formulae: 𝜌(𝛼,𝑓)=limsup𝜀1𝛼log𝑓𝛾(𝜀),𝛼(log(1/(1𝜀)))(2.14)𝑇(𝛼,𝑓)=limsup𝜀1𝛼log𝑓𝛾(𝜀)[]𝛼(1/(1𝜀))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)=log𝑝𝑥,𝑝1, 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),𝛼Λ. Then, limsup𝑛𝛼𝐸0𝐸1𝐸𝑛1/𝑛(𝑛+1)𝑑𝑛[]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 𝑓𝛾(𝜀) as 𝜀1. Fix 𝑇>𝑇(𝛼,𝑓). Then, for 𝜀 sufficiently close to 1, from (2.15), we have 𝑓𝛾(𝜀)𝛼1𝑇𝛼11𝜀𝜌,𝜌(𝛼,𝑓)𝜌.(2.17)
Define 𝛿𝑛=min(1/4,1/𝑛),𝑛=1,2,. Using [13, Equation (3.1)] with (2.17), for all sufficiently large values of 𝑛,𝑛𝑛0, we have 𝐸0𝐸1𝐸𝑛𝑑𝑛(𝑛+1)(𝑛+1)!𝑐𝑛𝑛8𝑛𝛼exp(𝑛+1)log1𝑇[]𝛼(𝑛)𝜌+1𝐶(𝐾,𝑀).(2.18)
Since 𝛼 is strictly increasing, for 𝑛𝑛0, we get 𝐸0𝐸1𝐸𝑛1/𝑛+1𝑑𝑛𝑐1𝛼1𝑇[]𝛼(𝑛)𝜌.(2.19)
In view of (2.13), (2.19) gives limsup𝑛𝛼𝐸0𝐸1𝐸𝑛1/𝑛+1𝑑𝑛[]𝛼(𝑛)𝜌𝑇.(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, limsup𝑛𝛼𝐸𝑛𝑑𝑛[]𝛼(𝑛)𝜌(𝛼,𝑓)𝑇(𝛼,𝑓).(2.21)

The proof is immediate in view of 𝐸𝑛𝐸𝑛1𝐸0.

For 𝑐>0, let 𝐹[]𝛼𝑥,𝑐,𝜌=log1[](𝑐𝛼(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 𝛼Λ 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 lim𝑥𝑥(𝐹(𝑥,𝑐,𝜌))𝐹(𝑥,𝑐,𝜌)=0.(2.23)
Then, liminf𝑛𝛼𝐸𝑛𝑑2𝑛[]𝛼(𝑛)𝜌(𝛼,𝑓)𝑇(𝛼,𝑓).(2.24)

Proof. We may assume that 𝑇(𝛼,𝑓)<. Let liminf𝑛𝛼𝐸𝑛𝑑2𝑛[]𝛼(𝑛)𝜌>𝑇>𝑇(𝛼,𝑓).(2.25)
For sufficiently large values of 𝑛, 𝛼𝐸0𝐸1𝐸𝑛1/𝑛+1𝑑𝑛[]𝛼(𝑛)𝜌𝛼exp(1/(𝑛+1))𝑛𝑘=1𝐹𝑘,𝑇,𝜌+𝑐[]𝛼(𝑛)𝜌.(2.26)
Since 𝐹[𝑥,𝑇,𝜌] is increasing, we get 𝑛1𝑘=1𝐹𝑘,𝑇,𝜌𝑛1𝐹𝑥,𝑇,𝜌𝑑𝑥𝑛𝑘=2𝐹𝑘,𝑇,,𝜌𝑛1𝐹𝑥,𝑇,𝜌𝑑𝑥=𝑛𝐹𝑛,𝑇,𝜌𝐹1,𝑇,𝜌𝑛1𝑥𝐹𝑥,𝑇,𝜌𝑑𝑥.(2.27)
We see that 1𝑛𝐹𝑛,𝑇,𝜌𝑛1𝑥𝐹𝑥,𝑇,𝜌𝑑𝑥0as𝑛.(2.28)
Thus, (1/(𝑛+1))𝑛𝑘=1𝐹𝑘,𝑇,𝜌𝐹𝑛,𝑇,𝜌1as𝑛.(2.29)
From this and (2.26), we get liminf𝑛𝛼𝐸0𝐸1𝐸𝑛1/𝑛+1𝑑𝑛[]𝛼(𝑛)𝜌𝛼exp𝐹𝑛,𝑇,𝜌[]𝛼(𝑛)𝜌𝑇>𝑇(𝛼,𝐹)(2.30) which contradicts (2.16). Hence the proof is complete.

Remark 2.7. The function 𝛼(𝑥)=log𝑝𝑥,𝑝1, 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.

References

  1. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, American Mathematical Society, Providence, RI, USA, 5th edition, 1965.
  2. N. S. Landkof, Foundations of Modern Potential Theory, Nauka, Morscow, Russia, 1966, English Translation in Springer, Berlin, Germany,1972.
  3. M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, Japan, 1959.
  4. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, vol. 316, Springer, Berlin, Germany, 1997.
  5. O. G. Parfënov, “Estimates for singular numbers of the Carleson embedding operator,” Matematicheskiĭ Sbornik, vol. 131, no. 173, pp. 501–518, 1986, English Translation in Mathematics of the USSR-Sbornik, vol. 59, 1988. View at: Google Scholar
  6. V. A. Prokhorov, “Rational approximation of analytic functions,” Matematicheskiĭ Sbornik, vol. 184, no. 2, pp. 3–32, 1993, English Translation in Russian Academy of Sciences. Sbornik Mathematics, vol. 78, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. V. A. Prokhorov and E. B. Saff, “Rates of best uniform rational approximation of analytic functions by ray sequences of rational functions,” Constructive Approximation, vol. 15, no. 2, pp. 155–173, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. M. N. Šeremeta, “Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion,” Izvestija Vysših Učebnyh Zavedeniĭ Matematika, vol. 2, no. 57, pp. 100–108, 1967, English Translation in American Mathematical Society Translations, vol. 2, no. 88, 1970. View at: Google Scholar
  9. B. J. Aborn and H. Shankar, “Generalized growth for functions analytic in a finite disc,” Pure and Applied Mathematika Sciences, vol. 12, no. 1-2, pp. 83–94, 1980. View at: Google Scholar | Zentralblatt MATH
  10. G. P. Kapoor and A. Nautiyal, “On the coefficients of a function analytic in the unit disc having slow rate of growth,” Annali di Matematica Pura ed Applicata, vol. 131, pp. 281–290, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. A. Nautiyal, “A remark on the generalized order of an analytic function,” Pure and Applied Mathematika Sciences, vol. 17, no. 1-2, pp. 15–17, 1983. View at: Google Scholar | Zentralblatt MATH
  12. M. N. Šeremeta, “The connection between the growth of functions of order zero which are entire or analytic in a disc and their power series coefficients,” Izvestija Vysših Učebnyh Zavedeniĭ Matematika, vol. 6, no. 73, pp. 115–121, 1968 (Russian). View at: Google Scholar
  13. V. A. Prokhorov, “Rational approximation of analytic functions having generalized orders of rate of growth,” Journal of Computational Analysis and Applications, vol. 5, no. 1, pp. 129–146, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. G. P. Kapoor and K. Gopal, “Decomposition theorems for analytic functions having slow rates of growth in a finite disc,” Journal of Mathematical Analysis and Applications, vol. 74, no. 2, pp. 446–455, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2012 Devendra Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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