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

1999 ◽  
Vol 15 (2) ◽  
pp. 155-173 ◽  
Author(s):  
V. A. Prokhorov ◽  
E. B. Saff
Keyword(s):  
Rational Approximation ◽  
Analytic Functions ◽  
Rational Functions ◽  
Ray Sequences

scholarly journals ABOUT THE RATE OF RATIONAL APPROXIMATION OF SOME ANALYTIC FUNCTIONS

1998 ◽  
Vol 3 (1) ◽  
pp. 168-176
Author(s):  
A. YA. Radyno
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Rational Approximation ◽  
Analytic Functions ◽  
Hankel Operator ◽  
Rational Functions ◽  
Best Approximations ◽  
Singular Numbers

The article is devoted to results relating to the theory of rational approximation of an analytic function. Let ƒ be an analytic function on the disk {z : |z| < ñ), ñ > 1. The rate of decrease of the best approximations ñn of a function ƒ by the rational functions of order at most n in the uniform metric on the unit disk E with the center z = 0 is investigated. The theorem connecting the rate of decrease of ñn with the order ó > 0 of ƒ in the disk {z : |z| < ñ} is proved. The proof of this results is based on an analysis of behavior of the singular numbers of the Hankel operator constructed from the function ƒ.

Conditions for the representability of analytic functions by series of rational functions

1974 ◽  
Vol 15 (2) ◽  
pp. 112-115 ◽  
Author(s):  
T. A. Leont'eva
Keyword(s):  
Analytic Functions ◽  
Rational Functions

Possibility of Uniform Rational Approximation in the Spherical Metric

1976 ◽  
Vol 28 (1) ◽  
pp. 112-115 ◽  
Author(s):  
P. M. Gauthier ◽  
A. Roth ◽  
J. L. Walsh
Keyword(s):  
Compact Subset ◽  
Rational Approximation ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Finite Complex ◽  
Finite Plane ◽  
Extended Plane ◽  
Chordal Metric

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.

scholarly journals Free Function Theory Through Matrix Invariants

2017 ◽  
Vol 69 (02) ◽  
pp. 408-433 ◽  
Author(s):  
Igor Klep ◽  
Špela Špenko
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Function Theory ◽  
Rational Functions ◽  
Generalized Polynomials ◽  
Direct Sums ◽  
Real Analytic Functions ◽  
Power Series Expansions ◽  
Noncommutative Polynomials ◽  
Free Function

Abstract This paper concerns free function theory. Freemaps are free analogs of analytic functions in several complex variables and are defined in terms of freely noncommuting variables. A function of g noncommuting variables is a function on g-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions, and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps with involution, free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invarianttheoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involutionfree counterparts.

scholarly journals Erratum: Optimal ray sequences of rational functions connected with the zolotarev problem

1996 ◽  
Vol 12 (3) ◽  
pp. 437-438 ◽  
Author(s):  
A. L. Levin ◽  
E. B. Saff
Keyword(s):  
Rational Functions ◽  
Zolotarev Problem ◽  
Ray Sequences

On Padé and Best Rational Approximation

1983 ◽  
Vol 26 (1) ◽  
pp. 50-57 ◽  
Author(s):  
Peter B. Borwein
Keyword(s):  
Rational Approximation ◽  
Analytic Functions ◽  
Unit Disc ◽  
Padé Approximants ◽  
Main Diagonal ◽  
Rational Approximations ◽  
Pade Approximants ◽  
Best Rational Approximation

AbstractIt is reasonable to expect that, under suitable conditions, Padé approximants should provide nearly optimal rational approximations to analytic functions in the unit disc. This is shown to be the case for ez in the sense that main diagonal Padé approximants are shown to converge as expeditiously as best uniform approximants. Some more general but less precise related results are discussed.

scholarly journals On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

2016 ◽  
Vol 206 ◽  
pp. 48-67 ◽  
Author(s):  
Viktor I. Buslaev ◽  
Sergey P. Suetin
Keyword(s):  
Rational Approximation ◽  
Analytic Functions
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