Rational Functions Of Best Approximation Research Articles

Joseph L. Walsh: Selected papers, edited by Theodore J. Rivlin and Edward B. Saff. Pp. 682. £86. 2000. ISBN 0 387 98782 7 (Springer Verlag).

1 Zeros and Critical Points of Polynomials and Rational Functions.- [18*] On the location of the roots of the Jacobian of two binary forms and of the derivative of a rational function.- [21-a*] On the location of the roots of the derivative of a polynomial.- [21-b*] On the location of the roots of the Jacobian of two binary forms and of the derivative of a rational function.- [20-c*] On the location of the roots of the derivative of a polynomial.- [33-1*] Note on the location of the roots of the derivative of a polynomial.- [22-g*] On the location of the roots of certain types of polynomials.- [64-e*] A Theorem of Grace on the zeros of polynomials, revisited.- [64-j*] The location of the zeros of the derivative of a rational function, revisited.- [24-h*] An inequality for the roots of an algebraic equation.- Commentary.- Comments on [18*], [21-a*], and [21-b*].- Comments on [20-c*] and [33-i*].- Comments on [22-g*].- Editors' Note.- Comments on [64-e*], [64-j*].- Comments on [24-h*].- 2 Walsh Functions.- [23-b*] A closed set of normal orthogonal functions.- Commentary.- The Walsh System.- The Impact of Walsh Functions on Modern Mathematics.- Probability Theory.- Harmonic Analysis.- Functional Analysis.- Generalizations.- Technical Applications.- Commentary by T. J. Rivlin.- 3 Qualitative Approximation.- [26-b*] Uber die Entwicklung einer analytischen Funktion nach Polynomen.- [26-c*] Uber die Entwicklung einer Funktion einer komiexen Veranderlichen nach Polynomen.- [28-a*] On the expansion of analytic functions in series of polynomials and in series of polynomials and in series of other analytic functions.- [28-d*] Uber die Entwicklung einer harmonischen Funktion nach harmonischen Polynomen.- [29-b*] The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions.- Commentary.- 4 Conformai Mapping.- [37-d*] On the shape of level curves of Green's function.- [38-a*] Note on the curvature of orthogonal trajectories of level curves of Green's functions.- [39-d*] On the circles of curvature of the images of circles under a conformal map.- [40-a*] Note on the curvature of the orthogonal trajectories of level curves of Green's functions.- [70-a*] On the shape of the level loci of harmonic measure.- [55-a*] (With D. Gaier) Zur Methode der variablen Gebiete bei der Randverserrung.- [56-b*] (With L. Rosenfeld) On the boundary behavior of a conformal map.- [56-d*] On the conformal mapping of multiply connected regions.- Commentary Dieter Gaier.- Topic I: Geometry of level curves and related topics.- I.1. Domains convex in one direction.- I.2. Length and area problems.- I.3. On the geometry of lemniscates.- Topic II: Conformal mapping near the boundary.- II.1. Conformal mapping of strip domains.- II.2. Holder continuity of the mapping function.- Topic III: Conformal mapping of multiply connected domains.- III. 1. Walsh's new canonical map.- III. 2. New approaches to Walsh'1 Zeros and Critical Points of Polynomials and Rational Functions.- [18*] On the location of the roots of the Jacobian of two binary forms and of the derivative of a rational function.- [21-a*] On the location of the roots of the derivative of a polynomial.- [21-b*] On the location of the roots of the Jacobian of two binary forms and of the derivative of a rational function.- [20-c*] On the location of the roots of the derivative of a polynomial.- [33-1*] Note on the location of the roots of the derivative of a polynomial.- [22-g*] On the location of the roots of certain types of polynomials.- [64-e*] A Theorem of Grace on the zeros of polynomials, revisited.- [64-j*] The location of the zeros of the derivative of a rational function, revisited.- [24-h*] An inequality for the roots of an algebraic equation.- Commentary.- Comments on [18*], [21-a*], and [21-b*].- Comments on [20-c*] and [33-i*].- Comments on [22-g*].- Editors' Note.- Comments on [64-e*], [64-j*].- Comments on [24-h*].- 2 Walsh Functions.- [23-b*] A closed set of normal orthogonal functions.- Commentary.- The Walsh System.- The Impact of Walsh Functions on Modern Mathematics.- Probability Theory.- Harmonic Analysis.- Functional Analysis.- Generalizations.- Technical Applications.- Commentary by T. J. Rivlin.- 3 Qualitative Approximation.- [26-b*] Uber die Entwicklung einer analytischen Funktion nach Polynomen.- [26-c*] Uber die Entwicklung einer Funktion einer komiexen Veranderlichen nach Polynomen.- [28-a*] On the expansion of analytic functions in series of polynomials and in series of polynomials and in series of other analytic functions.- [28-d*] Uber die Entwicklung einer harmonischen Funktion nach harmonischen Polynomen.- [29-b*] The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions.- Commentary.- 4 Conformai Mapping.- [37-d*] On the shape of level curves of Green's function.- [38-a*] Note on the curvature of orthogonal trajectories of level curves of Green's functions.- [39-d*] On the circles of curvature of the images of circles under a conformal map.- [40-a*] Note on the curvature of the orthogonal trajectories of level curves of Green's functions.- [70-a*] On the shape of the level loci of harmonic measure.- [55-a*] (With D. Gaier) Zur Methode der variablen Gebiete bei der Randverserrung.- [56-b*] (With L. Rosenfeld) On the boundary behavior of a conformal map.- [56-d*] On the conformal mapping of multiply connected regions.- Commentary Dieter Gaier.- Topic I: Geometry of level curves and related topics.- I.1. Domains convex in one direction.- I.2. Length and area problems.- I.3. On the geometry of lemniscates.- Topic II: Conformal mapping near the boundary.- II.1. Conformal mapping of strip domains.- II.2. Holder continuity of the mapping function.- Topic III: Conformal mapping of multiply connected domains.- III. 1. Walsh's new canonical map.- III. 2. New approaches to Walsh's theorem.- III. 3. General canonical domains.- 5 Polynomial Approximation.- [32-c*] On polynomial interpolation to analytic functions with singularities.- [37-g*] (With W.E. Sewell) Note on the relation between continuity and degree of polynomial approximation in the complex domain.- [38-d*] (With W.E. Sewell) Note on degree of trigonometric and polynomial approximation to an analytic function.- [53-c*] (With T.S. Motzkin) On the derivative of a polynomial and Chebyshev approximation.- [73-a*] (With T.S. Motzkin) Equilibrium of inverse-distance forces in three-dimensions.- [34-c*] Note on the orthogonality of Tchebycheff polynomials on confocal ellipses.- [42-a*] Note on the coefficients of overconvergent power series.- [51-c*] Note on approximation by bounded analytic functions.- [68-e*] Approximation by bounded analytic functions: Uniform convergence as implied by mean convergence.- [73-c*] History of the Riemann mapping theorem.- Commentary.- Comments on [32-c*].- Comments on [37-g*] and [38-d*].- Comments on [53-c*] and [73-a*].- Comments on [34-c*].- Comments on [42-a*].- Comments on [51-c*].- Comments on [73-c*].- 6 Rational Approximation 465.- [31-c*] On the overconvergence of certain sequences of rational functions of best approximation.- [34-b*] On approximation to an analytic function by rational functions of best approximation.- [40-b*] On the degree of convergence of sequences of rational functions.- [46-c*] Overconvergence, degree of convergence, and zeros of sequences of analytic functions.- [64-a*] Pade approximants as limits of rational functions of best approximation.- [65-i*] The convergence of sequences of rational functions of best approximation with some free poles.- [67-c*] An extension of the generalized Bernstein lemma.- [68-a*] Degree of approximation by rational functions and polynomials.- [71-b*] (With Dov Aharonov) Some examples in degree of approximation by rational functions.- Commentary.- Comments on [31-c*].- Comments on [34-b*].- Comments on [40-b*].- Comments on [46-c*].- Comments on [64-a*].- Comments on [65-i*].- Comments on [67-c*].- Comments on [68-a*].- Comments on [71-b*].- 7 Spline Functions.- [65-b*] (With J.H. Ahlberg and E.N. Nilson) Fundamental properties of generalized splines.- [67-d*] (With J.H. Ahlberg and E.N. Nilson) Complex cubic splines.- [68-f*] (With J.H. Ahlberg and E.N. Nilson) Cubic splines on the real line.- Commentary by Walter Schempp.

Pade approximants as limits of rational functions of best approximation in Orlicz space

In this paper, we prove that the best rational approximation of a given analytic function in Orlicz space L*(G), where G={⋎z⋎≤∈}, converges to the Pade approximants of the function as the measure of G approaches zero.

Degree of rational functions of best approximation in Lp(Rm)

Degree of rational functions of best approximation in Lp(Rm)

The distribution of the poles of the best approximating rational functions and the analytical properties of the approximated function

The following theorem was proved in our paper in Math. USSR-Sb.9 (1969): Iff(z)∈H 2 and the poles of the rational functions of best approximation tend to infinity sufficiently quickly, thenf(z) is an entire function. In the present article we weaken the restrictions on the distribution of poles by assuming only that these poles have no finite limit point (Theorem 1). Some generalizations of this result are also given.

Padé approximants as limits of rational functions of best approximation, real domain

Padé approximants as limits of rational functions of best approximation, real domain

On the convergence of rational functions of best approximation to a meromorphic function

On the convergence of rational functions of best approximation to a meromorphic function

Note on the Convergence of Sequences of Rational Functions of Best Approximation to a Meromorphic Function

If the function f(z) is meromorphic at every finite point of the plane, analytic on C: [unk]z[unk] </= 1, there are results that can be established concerning the convergence of the sequence R(n)(z) of rational functions of each degree n of best approximation to f(z) on C. However, we need to assume certain additional conditions on the convergence of the sequence R(n)(z) itself. The results generalize to more general sets C and to more general functions.

The Convergence of Rational Functions of Best Approximation to the Exponential Function. II

Let ${W_{m,n}}(z)$ be a rational function of type (m, n) of best uniform approximation to the function ${e^z}$ on the closed unit disk. In this paper we show that any sequence $\{ {W_{m,n}}(z)\}$ for which $m + n \to \infty$ must converge to ${e^z}$ for all values of z. This is the first result which describes completely the regions of convergence of arbitrary sequences formed from a Walsh array.

The convergence of rational functions of best approximation to the exponential function. II

Let ${W_{m,n}}(z)$ be a rational function of type (

Some Examples in Degree of Approximation by Rational Functions

We exhibit examples of (1) series that converge more rapidly than any geometric series where the function represented has a natural boundary, (2) the convergence of a series with maximum geometric degree of convergence yet having limit points of poles of the series everywhere dense on a circumference in the complement of $E$, (3) a Padé table for an entire function whose diagonal has poles every-where dense in the plane and (4) a corresponding example for the table of rational functions of best approximation of prescribed type.

Some examples in degree of approximation by rational functions

We exhibit examples of (1) series that converge more rapidly than any geometric series where the function represented has a natural boundary, (2) the convergence of a series with maximum geometric degree of convergence yet having limit points of poles of the series everywhere dense on a circumference in the complement of E E , (3) a Padé table for an entire function whose diagonal has poles every-where dense in the plane and (4) a corresponding example for the table of rational functions of best approximation of prescribed type.

The Convergence of Rational Functions of Best Approximation to the Exponential Function

The Convergence of Rational Functions of Best Approximation to the Exponential Function

The convergence of rational functions of best approximation to the exponential function

The object of the paper is to establish convergence throughout the entire complex plane of sequences of rational functions of prescribed types which satisfy a certain degree of approximation to the function a e y z a{e^{yz}} on the disk | z | ≦ ρ |z| \leqq \rho . It is assumed that the approximating rational functions have a bounded number of free poles. Estimates are given for the degree of best approximation to the exponential function by rational functions of prescribed types. The results obtained in the paper imply that the successive rows of the Walsh array for a e y z a{e^{yz}} on | z | ≦ ρ |z| \leqq \rho converge uniformly to a e y z a{e^{yz}} on each bounded subset of the plane.

Distribution of poles of rational functions of best approximation

This article describes the construction of an entire function E(z) such that for any sequence $$\{ \mathop r\limits^* _n (z)\} $$ of rational functions of best approximation to E(z) on the unit disc K, the corresponding set of poles {α nk * } is everywhere dense in the complement of K.

The convergence of sequences of rational functions of best approximation. III

The convergence of sequences of rational functions of best approximation. III

The Convergence of Sequences of Rational Functions of Best Approximation. III

The Convergence of Sequences of Rational Functions of Best Approximation. III

On the Convergence of Sequences of Rational Functions

There are numerous kinds of sequences of rational functions, such as sequences of Pad6 approximants and sequences of rational functions of best approximation on a given point set, where the poles of the individual functions are not known even asymptotically, and no effective methods for their determination are at hand. Yet regions of convergence and degree of convergence depend heavily on the location of those poles. If assumptions on the location of poles are made, it may be possible to deduce results on regions and degree of convergence, as has been shown by Baker, Chisholm, Walsh, and others. The present note on these topics seems appropriate because a number of previous results, as set forth in a recent report by Baker [1], can be materially improved, especially with reference to degree of convergence. We emphasize Pad6 approximants and rational functions of best approximation. A rational function is said to be of type (m, n) if it can be expressed in the form

The Convergence of Sequences of Rational Functions of Best Approximation. II

The Convergence of Sequences of Rational Functions of Best Approximation. II

The convergence of sequences of rational functions of best approximation. II

The convergence of sequences of rational functions of best approximation. II

The convergence of sequences of rational functions of best approximation

Unless the poles of a rational function of best approximation are prescribed in advance, the determination of those poles even approximately may be difficult, yet may be necessary for a complete study of approximation or convergence. Certain cases where this determination is possible have recently been considered [1] by the present author; the object of the present note is to enlarge the category of those cases, especially to study approximation on a set having no interior points, and on a set with several components. Montessus de Ballore considered in 1902 an analogous situation related to the Pade table, with analogous results so far as concerns regions of convergence, but his methods relate to the special properties of the Taylor development and do not apply to the hypotheses of the present paper. Here our methods are based primarily on the deeper use of geometric degree of convergence, in various forms, including methods developed [2] in the theory of approximating polynomials. A rational function of the form