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 Padé 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 Padé approximants, to the application of multipoint Padé 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 Padé-Hermite approximation for systems of Markov type functions.     

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 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 functions of best uniform approximation and holomorphic continuation of functions

Let f be a function, continuous and real valued on the segment Δ, Δ ⊂ (−∞, ∞) and {R_n} be the sequence of the rational functions of best uniform approximation to f on Δ of order (n, n). In the present work, the convergence of {R_n} in the complex plane is considered for the special caseswhen the poles (or the zeros, respectively) of {R_n} accumulate in the terms of weak convergence of measures to acompact set of zero capacity. As a consequence, sufficient conditions for the holomorphic and the meromorphic continuability of f are given. The work is supported by Project 69 with Ministry of Science and Education, Bulgaria.     

The best polynomial approximation of functions analytic in the unit circle

In the Banach space (p,q,), studied by M. I. Gvaradze, we establish a relation between the best polynomial approximation of an entire transcendental function and such important characteristics as its order of growth and its type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 838–843, June, 1990.     

Rational interpolation and best rational approximation

A relation between the degree of convergence (in capacity) of Padé approximants and the degree of best rational approximation is derived for functions in Gon?ar's class R₀. The results contain previous convergence results for Padé approximants proved by Nuttall, Pommerenke, and Gammel and Nuttall.     

On the best linear approximation methods and the widths of certain classes of analytic functions

We discuss the best linear approximation methods in the Hardy spaceH _q q≥1, for classes of analytic functions studied by N. Ainulloev; these are generalizations (in a certain sense) of function sets introduced by L. V. Taikov. The exact values of their linear and Gelfandn-widths are obtained. The exact values of the Kolmogorov and Bernsteinn-widths of classes of analytic (in |z|<1) functions whose boundaryK-functionals are majorized by a prescribed functions are also obtained. Translated fromMatermaticheskie Zametki, Vol. 65, No. 2, pp. 186–193, February, 1999.     

On the best approximation by trigonometric polynomials on convolution classes of analytic periodic functions

For a continuous 2π-periodic real-valued function K(t), whose amplitudes decrease as a geometric progression with a denominator q ∈ (0, 1) starting from a given number n ∈ ?, we find sharp upper bounds for q ensuring that K(t) satisfies the Nagy condition N* _n .     

Rational approximation of decay-type functions

Numerically given functions with exponential asymptotic behavior are approximated by rational expressions. The numerators are obtained from the nodes of the functions, and the denominators are powers of polynomials, whose coefficients are obtained by expansion in orthogonal polynomials. Different weights are discussed for uniform absolute or relative errors.An iterative procedure is further described to improve the least-squares approximation to one of Chebyshev-type best fit.The research reported in this paper was sponsored in part by the King Gustaf VI Adolf's 70-Years Fund for Swedish Culture, Knut and Alice Wallenberg's Foundation, The Swedish Natural Science Research Council, The Swedish Technical Research Council, and in part by the Aeronautical Research Laboratory, OAR, through the European Office, Aerospace Research, United States Air Force.Part of the work on this paper was carried out while the author was at the Physics Department, University of Florida, Gainesville, Florida, U.S.A.     

Rational approximation of empirical functions

A method is developed for finding the best rational approximationp(x)/q(x) in the least-squares sense of a functionf(x) known at discrete points of an interval. Especially for functions with a steep tangent the method yields far better results than polynomial approximation. The scheme can easily be extended to functions known at all points of an interval.Part of this work was carried out at SAAB, Linköping. The present form was developed at the Department of Numerical Analysis, Lund.     

Rational approximation to harmonic functions

Padé-type approximation is the rational function analogue of Taylor’s polynomial approximation to a power series. A general method for obtaining Padé-type approximants to Fourier series expansions of harmonic functions is defined. This method is based on the Newton-Cotes and Gauss quadrature formulas. Several concrete examples are given and the convergence behavior of a sequence of such approximants is studied. This revised version was published online in June 2006 with corrections to the Cover Date.