The logarithmic asymptotic expansions for the norms of evaluation functionals

A. A. Dovgoshey^a, F. G. Abdullaev^b, M. Kuchukaslan<sup>b

a</sup> Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences
^b University of Mersin

Abstract: Let $\mu$ be a compactly supported finite Borel measure in $\mathbb{C}$, and let $\Pi_n$ be the space of holomorphic polynomials of degree at most $n$ furnished with the norm of $L^2(\mu)$. We study the logarithmic asymptotic expansions of the norms of the evaluation functionals that relate to polynomials $p\in\Pi_n$ their values at a point $z\in\mathbb{C}$. The main results demonstrate how the asymptotic behavior depends on regularity of the complement of the support of $\mu$ and the Stahl–Totik regularity of the measure. In particular, we study the cases of pointwise and $\mu$-a.e. convergence as $n\to\infty$.

Keywords: general orthogonal polynomials, logarithmic asymptotic expansion, evaluation functionals, Green?s function, irregularity points for the Dirichlet problem.

UDC: 517.538.3

Received: 13.08.2003
Revised: 28.01.2005

 English version: Siberian Mathematical Journal, 2005, 46:4, 613–622

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