Rational Approximation Of Analytic Functions Research Articles

Generalized Hypergeometric Function 3F2 Ratios and Branched Continued Fraction Expansions

The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction expansions. We used combinations of three- and four-term recurrence relations of the generalized hypergeometric function 3F2 to construct the branched continued fraction expansions of the ratios of this function. We also used the concept of correspondence and the research method to extend convergence, already known for a small region, to a larger region. As a result, we have established some convergence criteria for the expansions mentioned above. It is proved that the branched continued fraction expansions converges to the functions that are an analytic continuation of the ratios mentioned above in some region. The constructed expansions can approximate the solutions of certain differential equations and analytic functions, which are represented by generalized hypergeometric function 3F2. To illustrate this, we have given a few numerical experiments at the end.

The Gonchar-Stahl -theorem and associated directions in the theory of rational approximations of analytic functions

The Gonchar-Stahl -theorem characterizes the rate of convergence of best uniform (Chebyshev) rational approximations (with free poles) for one basic class of analytic functions. The theorem itself, modifications and generalizations of it, methods involved in its proof and other related details constitute an important subfield in the theory of rational approximations of analytic functions and complex analysis. This paper briefly outlines the essentials of the subfield. The fundamental contributions of A. A. Gonchar and H. Stahl are at the heart of the exposition. Bibliography: 70 titles.

The Gonchar-Stahl $\rho^2$-theorem and associated directions in the theory of rational approximations of analytic functions

Теорема Гончара-Шталя о $\rho^2$ характеризует скорость сходимости наилучших (чебышeвских) равномерных рациональных приближений (со свободными полюсами) для одного из важнейших классов аналитических функций. Сама эта теорема, ее варианты и обобщения, методы, используемые в доказательстве, и прочие моменты составляют важную подобласть теории рациональных приближений аналитических функций и комплексного анализа. В статье вкратце очертены контуры этой подобласти. В центре изложения находится фундаментальный вклад А. А. Гончара и Г. Шталя в эту теорию. Библиография: 70 названий.

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

The present paper is concerned with the rational approximation of functions holomorphic on a domainG⊂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 functionsfhaving fast and slow rates of growth of the maximum modulus.

Rational approximation of analytic functions

The paper provides an overview of the author’s contribution to the theory of constructive rational approximations of analytic functions. The results presented are related to the convergence theory of Pade approximants and of more general rational interpolation processes, which significantly expand the classical theory’s framework of continuous fractions, to inverse problems in the theory of Pade approximants, to the application of multipoint Pade approximants (solutions of Cauchy-Jacobi interpolation problem) in explorations connected with the rate of Chebyshev rational approximation of analytic functions and to the asymptotic properties of Pade-Hermite approximation for systems of Markov type functions.

On estimates for the ratio of errors in best rational approximation of analytic functions

Let E E be an arbitrary compact subset of the extended complex plane C ¯ \overline {\mathbb C} with nonempty interior. For a function f f continuous on E E and analytic in the interior of E E denote by ρ n ( f ; E ) \rho _n(f; E) the least uniform deviation of f f on E E from the class of all rational functions of order at most n n . In this paper we show that if f f is not a rational function and if K K is an arbitrary compact subset of the interior of E , E, then ∏ k = 0 n ( ρ k ( f ; K ) / ρ k ( f ; E ) ) , \prod _{k=0}^n (\rho _k(f; K) /\rho _k(f; E) ), the ratio of the errors in best rational approximation, converges to zero geometrically as n → ∞ n \to \infty and the rate of convergence is determined by the capacity of the condenser ( ∂ E , K ) (\partial E, K) . In addition, we obtain results regarding meromorphic approximation and sharp estimates of the Hadamard type determinants.

On best rational approximation of analytic functions

This paper contains some theorems related to the best approximation ρ n ( f ; E ) to a function f in the uniform metric on a compact set E ⊂ C ¯ by rational functions of degree at most n. We obtain results characterizing the relationship between ρ n ( f ; K ) and ρ n ( f ; E ) in the case when complements of compact sets K and E are connected, K is a subset of the interior Ω of E, and f is analytic in Ω and continuous on E.

Rational Approximation of Analytic Functions Having Generalized Orders of Rate of Growth

Let f be holomorphic on a domain G ⊂ C¯ and ρn be the error in best approximation of f in the supremum norm on a compact set E ⊂ G by rational functions of order n. We obtain results characterizing the degree of decrease of the best approximation ρn in terms connected with the condenser (E,F), F=C¯ \ G¯, and the rate of growth of the maximum modulus of f(z). In particular, if f has a generalized order σ(α, β, f) in the domain G, then

Sharp constants for rational approximations of analytic functions

Theorems describing the sharp constants for the approximation of a general class of analytic functions by rational functions are proved. Magnus's conjecture on the sharp constant for the approximation of on is established as a consequence. For the proof of the theorems new formulae expressing the strong asymptotics of polynomials orthogonal with respect to a varying complex weight are obtained.

A study of the rate of rational approximation of a class of analytic functions

This paper is devoted to results belonging to the theory of rational approximation of analytic functions. We study the rate of decrease of the best approximations ρn of a functionf holomorphic in a disk ¦z ∶ ¦z¦ 1> by rational functions of order at most n in the uniform metric on the unit disk with center at z=0. We prove theorems that connect the rate of decrease of the quantities ρn with the order σ≥0 and type Τ≥0 of the functionf. The proofs of these results are based on the methods of the theory of Hankel operators.

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class \( {\cal R}_{n,m} \) of all rational functions of order (n,m) are considered. Let ρn,m = ρn,m(f;E) be the distance of f in the uniform metric on E from the class \( {\cal R}_{n,m} \) . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρn,m(n) }n=0∞ , m(n)/n→θ∈ (0,1] as n→∞ . In particular, we give an upper estimate for the liminfn →∞ρn,m(n)1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators.

On the rate of convergence of the partial sums of dirichlet series and rational approximation of analytic functions

We study the asymptotic behavior of the rate of convergence of Dirichlet series that are absolutely convergent in a half-plane; the results obtained are applicable to rational approximation of functions analytic in the unit disk with nonnegative Taylor coefficients.

Inequalities for Rational Functions

A new hyperbolic area estimate for the level sets of finite Blaschke products is presented. The following inversion of the usual Sobolev embedding theorem is proved:‖r‖W1q(D)⩽Cn‖r‖Lp(D),p>2,1/q=1/p+1/2.Hereris a rational function of degreenwith poles outside D. This estimate implies a new inverse theorem for rational approximation of analytic functions with respect to areaLpnorm.

ON THE DEGREE OF RATIONAL APPROXIMATION OF MEROMORPHIC FUNCTIONS

Questions concerning the theory of rational approximation of analytic functions are considered. One of the main results is Theorem 1, which characterizes the degree of rational approximation of meromorphic functions of finite order. The proof of the results obtained are based on the methods of the theory of Hankel operators. Bibliography: 29 titles.

RATIONAL APPROXIMATION OF ANALYTIC FUNCTIONS

This article is devoted to results relating to the theory of rational approximation of analytic functions. An analysis is made of the rate of decrease of the best approximations ρn of an analytic functions of order at most n in the uniform metric in terms connected with the properties of the function to be approximated, more precisely, connected with the distribution and structure of its set of singularities and with the character of its analytic continuation. Bibliography: 31 titles.

Some Computational Aspects of a Method for Rational Approximation

Numerical aspects of a method for rational approximation of analytic functions on regions in the complex plane are considered. The approximation method divides the problem of computing a rational approximant into three subproblems. First, one chooses a. space of rational functions, then one selects a basis for this space, and finally one determines an element of the space by interpolation. For approximation on regions with bounded simply connected complement, we discuss the choice of space and basis from a numerical point of view. We illustrate with computed examples.

5.6. Rational approximation of analytic functions

5.6. Rational approximation of analytic functions

Numerical Chebyshev Approximation in the Complex Plane

The paper describes a numerical method for computing rational approximations of analytic functions on arbitrary simply-connected closed regions of the complex plane. The method is based on approximation on finite point sets and can be regarded as a generalization of the exchange algorithm for real rational approximation. Numerical examples show that for practical purposes, the approximations may be regarded as optimal in the sense of Chebyshev.

Rational approximation of analytic functions with finite number of singularities on the real axis

Rational approximation of analytic functions with finite number of singularities on the real axis