Abstract/Details

ASYMPTOTIC DISTRIBUTION OF ZEROS OF APPROXIMATING POLYNOMIALS

SIMKANI, MEHRDAD.   University of South Florida ProQuest Dissertations Publishing,  1987. 8728332.

Abstract (summary)

Consider a sequence of monic polynomials $\{$P$\sb{\rm n}\}\sb{\rm n\in I}$ of respective degrees precisely n. Each P$\sb{\rm n}$ may be identified by a unit discrete measure $\nu\sb{\rm n}$ = $\nu$(P$\sb{\rm n})$ having mass 1/n at every zero of P$\sb{\rm n}$ (counting multiplicities). For a compact set of the complex plane with logarithmic capacity $\gamma$(K) $>$ 0 and equilibrium: measure $\mu\sp*$, we impose the following conditions on the sequence $\{$P$\sb{\rm n}\}\sb{\rm n\in I}$:(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\lim\sb{\scriptstyle \rm n \to \infty\atop\scriptstyle\rm n \in I}\Vert {\rm P\sb{n}\Vert\sbsp{S(\mu\sp\*)}{1/n} = \gamma(K)},\leqno(1)$$(TABLE/EQUATION ENDS)where $\Vert\cdot\Vert\sb{\rm S(\mu\sp\*)}$ denotes the sup-norm (Chebyshev norm) on S($\mu\sp*$), the support of $\mu\sp*$, and(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\lim\sb{\scriptstyle\rm n \to \infty\atop\scriptstyle\rm n \in I} \nu\sb{\rm n}{\rm (A)} = 0,\leqno(2)$$(TABLE/EQUATION ENDS)for every closed set A contained in the union of the bounded components of the complement of S($\mu\sp*$). Then, we conclude that the sequence of measures $\nu\sb{\rm n}$ converges weakly to the equilibrium measure $\mu\sp*$, i.e. for every continuous function $\phi$(z) with compact support, we have(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\lim\sb{\scriptstyle\rm n \to \infty\atop\scriptstyle\rm n \in I}\int\phi{\rm (z)d\nu\sb{n}(z)} = \int\phi{\rm (z)d\mu\sp\*(z).}$$(TABLE/EQUATION ENDS)

We shall apply the above result to the sequences of polynomials approximating non-entire functions, in arbitrary semi-Chebyshev normed linear spaces. In so doing we obtain extensions of the classical results of Szego concerning the zeros of partial sums of power series.

Indexing (details)

Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences
Title
ASYMPTOTIC DISTRIBUTION OF ZEROS OF APPROXIMATING POLYNOMIALS
Author
Number of pages
95
Degree date
1987
School code
0206
Source
DAI-B 48/10, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
979-8-206-98651-8
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
8728332
ProQuest document ID
303632247