ASYMPTOTIC DISTRIBUTION OF ZEROS OF APPROXIMATING POLYNOMIALS

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.