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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 $w (x) x \to 0$ as $x \to \pm \infty$ if $\Sigma$ is unbounded. Let $A_w$ be the set of functions $f$ for which there exists a sequence of weighted polynomials $\{ w^n P_n \}^\infty_{n=1}$ converging to $f$ uniformly on $\Sigma$. Here $P_n$ is a polynomial of degree at most $n$. $A_w$ is a subalgebra of $C_0 (\Sigma ).$ 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. $\Sigma \backslash S_w \subseteq Z_w.$ In the paper, the following results are shown. {1.} If $x_0 \in \text{ Int}(S_w)$ does not belong to $Z_w$ then $\mu_w$ is smooth on some neighborhood $(x_0 - \delta, x_0 + \delta)$ of $x_0 .$ {2.} Suppose that $\mu_w$ is smooth on $(x_0 - \delta , x_0 + \delta ).$ Then $x_0 \not\in Z_w$ 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 $(x_0 - \delta_0 , x_0 + \delta_0).$ 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)
Full Text:
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