scholarly journals Uniform Approximation by Rational Functions Which All Satisfy the Same Algebraic Differential Equation

1996 ◽  
Vol 84 (2) ◽  
pp. 123-128 ◽  
Author(s):  
Lee A. Rubel
Keyword(s):  
Differential Equation ◽  
Uniform Approximation ◽  
Rational Functions ◽  
Algebraic Differential Equation ◽  
Approximation By Rational Functions

scholarly journals On Uniform Approximation by Rational Functions with an Application to Chordal Cluster Sets

1969 ◽  
Vol 34 ◽  
pp. 143-148 ◽  
Author(s):  
J.T. Gresser
Keyword(s):  
Meromorphic Function ◽  
Uniform Approximation ◽  
Boundary Behavior ◽  
Rational Functions ◽  
Approximation Techniques ◽  
Bounded Set ◽  
Perfect Set ◽  
Approximation By Rational Functions ◽  
Bounded Sets ◽  
Geometric Nature

For a closed and bounded set E in the complex plane, let A(E) denote the collection of all functions continuous on E and analytic on E°, its interior; let R(E) denote the collection of all functions which are uniform limits on E of rational functions with poles outside E. Then let A denote the collection of all closed, bounded sets for which A(E) = R(E). The purpose of this paper is to formulate a condition on a set, which is essentially of a geometric nature, in order that the set belong to A. Then using approximation techniques, we shall construct a meromorphic function having a certain boundary behavior on a perfect set; this answers a question raised in [1].

On The Factorization of Partial Differential Equations

1987 ◽  
Vol 39 (4) ◽  
pp. 825-834 ◽  
Author(s):  
W. Dale Brownawell
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nevanlinna Theory ◽  
Algebraic Differential Equation ◽  
List Type ◽  
Variable Result ◽  
Several Variables ◽  
Function Of Several Variables ◽  
Degenerate Cases

In [4] N. Steinmetz used Nevanlinna theory to establish remarkably versatile theorems on the factorization of ordinary differential equations which implied numerous previous results of various authors. (Here factorization is taken in the sense of function composition as introduced by F. Gross in [2].) The thrust of Steinmetz’ central results on factorization is that if g(z) is entire and f(z) is meromorphic in C such that the composite fog satisfies an algebraic differential equation, then so do f(z) and, degenerate cases aside, g(z). In addition, the more one knows about the equation for fog (e.g. degree, weight, autonomy), the more one can conclude about the equations for f and g.In this note we generalize Steinmetz’ work to show the following:a) Steinmetz’ two basic results, Satz 1 and Korollar 1 of [4] can be seen as one-variable specializations of a single two variable result, andb) the function g(z) can itself be allowed to be a function of several variables.

scholarly journals An application of Ritt’s low power theorem

1977 ◽  
Vol 68 ◽  
pp. 17-19 ◽  
Author(s):  
Michihiko Matsuda
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Low Power ◽  
Singular Solution ◽  
Algebraic Differential Equation ◽  
Classical Concept ◽  
First Order ◽  
Particular Solutions ◽  
Rigorous Definition ◽  
Particular Solution

AbstractConsider an algebraic differential equation F = 0 of the first order. A rigorous definition will be given to the classical concept of “particular solutions” of F = 0. By Ritt’s low power theorem we shall prove that a singular solution of F = 0 belongs to the general solution of F if and only if it is a particular solution of F = 0.

Approximation by Rational Functions

1977 ◽  
Vol s2-15 (2) ◽  
pp. 319-328 ◽  
Author(s):  
P. Erdős ◽  
D. J. Newman ◽  
A. R. Reddy
Keyword(s):  
Rational Functions ◽  
Approximation By Rational Functions
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