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Abstract

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.

Details

Title
ASYMPTOTIC DISTRIBUTION OF ZEROS OF APPROXIMATING POLYNOMIALS
Author
SIMKANI, MEHRDAD
Year
1987
Publisher
ProQuest Dissertations & Theses
ISBN
979-8-206-98651-8
Source type
Dissertation or Thesis
Language of publication
English
ProQuest document ID
303632247
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.