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Smooth equilibrium measures and approximation. (English) Zbl 1123.41005

Let ${\Sigma }$ be a closed subset on the real line and $w$ a nonnegative continuous function on ${\Sigma }$, such that $w\left(x\right)x\to 0$ as $x\to ±\infty$ if ${\Sigma }$ is unbounded. Let ${A}_{w}$ be the set of functions $f$ for which there exists a sequence of weighted polynomials ${\left\{{w}^{n}{P}_{n}\right\}}_{n=1}^{\infty }$ converging to $f$ uniformly on ${\Sigma }$. Here ${P}_{n}$ is a polynomial of degree at most $n$. ${A}_{w}$ is a subalgebra of ${C}_{0}\left({\Sigma }\right)·$ Let ${Z}_{w}$ be the closed subset of ${\Sigma }$, such that $f\in {A}_{w}$ if and only if $f$ is continuous on ${\Sigma }$ and vanishes on ${Z}_{w}$. The non-trivial approximation of $f$ is possible only on the support ${S}_{w}$ of an extremal measure ${\mu }_{w}$ that solves an associated equilibrium problem and is smooth, i.e. ${\Sigma }\setminus {S}_{w}\subseteq {Z}_{w}·$

In the paper, the following results are shown.

1. If ${x}_{0}\in \phantom{\rule{4.pt}{0ex}}\text{Int}\left({S}_{w}\right)$ does not belong to ${Z}_{w}$ then ${\mu }_{w}$ is smooth on some neighborhood $\left({x}_{0}-\delta ,{x}_{0}+\delta \right)$ of ${x}_{0}·$

2. Suppose that ${\mu }_{w}$ is smooth on $\left({x}_{0}-\delta ,{x}_{0}+\delta \right)·$ Then ${x}_{0}\notin {Z}_{w}$ if one of the following conditions holds.

a) ${S}_{w}$ can be written as the union of finitely many intervals ${J}_{k}$ and the restriction of ${\mu }_{w}$ to each ${J}_{k}$ is a doubling measure on ${J}_{k}$.

b) ${\mu }_{w}$ has a positive lower bound in a neighborhood $\left({x}_{0}-{\delta }_{0},{x}_{0}+{\delta }_{0}\right)·$

As corollaries, the authors obtain all previous results for approximation as well as the solution of a problem of T. Bloom and M. Branker. A connection to level curves of homogeneous polynomials of two variables is also explored.

MSC:
 41A10 Approximation by polynomials 30C10 Polynomials (one complex variable) 31A15 Potentials and capacity, harmonic measure, extremal length (two-dimensional)