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

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 A Similarity Invariant and the Commutant of some Multiplication Operators on the Sobolev Disk Algebra

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Ruifang Zhao
Keyword(s):  
Sobolev Space ◽  
Unit Disk ◽  
Analytic Functions ◽  
Entire Functions ◽  
Rational Functions ◽  
Multiplication Operators ◽  
Disk Algebra ◽  
Strongly Irreducible ◽  
Similarity Invariant ◽  
Sobolev Disk Algebra

LetR(𝔻)be the algebra generated in Sobolev spaceW22(𝔻)by the rational functions with poles outside the unit disk𝔻¯. In this paper, we study the similarity invariant of the multiplication operatorsMginℒ(R(𝔻)), whengis univalent analytic on𝔻orMgis strongly irreducible. And the commutants of multiplication operators whose symbols are composite functions, univalent analytic functions, or entire functions are studied.

scholarly journals Fractional iteration of entire and rational functions

1964 ◽  
Vol 4 (2) ◽  
pp. 129-142 ◽  
Author(s):  
G. Szekeres
Keyword(s):  
Analytic Functions ◽  
Rational Functions ◽  
Image Position

We shall be concerned with the behaviour of the fractional iterates of analytic functions which have a fixpoint ζ with multiplier 11. The general form of such a function is If ζ is finite, if ζ = ∞.

scholarly journals ON A PROBLEM POSED BY MAHLER

2015 ◽  
Vol 100 (1) ◽  
pp. 86-107 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOHANNES SCHLEISCHITZ
Keyword(s):  
Rational Number ◽  
Analytic Functions ◽  
Rational Functions ◽  
Liouville Numbers ◽  
Transcendental Functions

Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that they can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were constructed (though with no information on derivatives). More generally, we study the image of Liouville numbers under analytic functions, with particular attention to$f(z)=z^{q}$, where$q$is a rational number.

scholarly journals Generalization of Padé approximation from rational functions to arbitrary analytic functions — Theory

2015 ◽  
Vol 84 (294) ◽  
pp. 1835-1860 ◽  
Author(s):  
Can Evren Yarman ◽  
Garret M. Flagg
Keyword(s):  
Analytic Functions ◽  
Padé Approximation ◽  
Rational Functions ◽  
Pade Approximation

Expansions of Two Arbitrary Analytic Functions in a Series of Rational Functions

1934 ◽  
Vol 35 (4) ◽  
pp. 759
Author(s):  
P. W. Ketchum
Keyword(s):  
Analytic Functions ◽  
Rational Functions
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