Abstract/Details

Convergence and zero distribution of Laurent-type rational functions

Pritsker, Igor Efimovich. 
 University of South Florida ProQuest Dissertations Publishing,  1995. 9542084.

Abstract (summary)

We study convergence and asymptotic zero distribution of sequences of rational functions with fixed location of poles that approximate an analytic function in a multiply connected domain. The questions of convergence and limiting zero distributions are closely connected, because zeros are known to frame the largest possible regions of convergence. The asymptotic zero distribution of sequences of polynomials is a classical subject which essentially originated with the fundamental results of R. Jentzsch (1914) and G. Szego (1922) on the zero distributions of partial sums of a power series. It is rather surprising that although the study of zero distributions of polynomials has a long history, analogous results for truncations of Laurent series have been obtained only recently by A. Edrei (1982). We obtain extensions of Edrei's results for more general sequences of Laurent-type rational functions. It turns out that the limiting measure describing zero distributions is a linear convex combination of the harmonic measures at the poles of rational functions, which arises as the solution to a minimum weighted energy problem for a special weight. Applications of these results include the asymptotic zero distribution of the best approximants to analytic functions in multiply connected domains, Faber-Laurent polynomials, Laurent-Pade approximants, trigonometric polynomials, orthogonal Laurent polynomials, etc.

We also investigate optimal ray sequences of Laurent-type rational functions that provide the best rates of approximation to the functions analytic on the closure of a multiply connected domain. This gives rise to the adaptive version of the orthonormalization method for the approximation of conformal mapping of annular regions. We also provide the theoretical basis for the numerical algorithms, estimating the rates of the uniform convergence of approximating rational functions to the conformal mapping in the case of domains with piecewise analytic boundaries without cusps. The rates in approximation of the conformal module are studied in both cases.

The main tools in this research are the theories of weighted potentials, weighted polynomial zero distributions and complex function theory.

Indexing (details)


Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences
Title
Convergence and zero distribution of Laurent-type rational functions
Author
Pritsker, Igor Efimovich
Number of pages
126
Degree date
1995
School code
0206
Source
DAI-B 56/08, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
979-8-209-37331-5
Advisor
Saff, Edward B.
University/institution
University of South Florida
University location
United States -- Florida
Degree
Ph.D.
Source type
Dissertation or Thesis
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
9542084
ProQuest document ID
304235680
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
https://www.proquest.com/docview/304235680