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Zbl 1123.41005
Totik, Vilmos; Varjú, Péter P.
Smooth equilibrium measures and approximation.
(English)
[J] Adv. Math. 212, No. 2, 571-616 (2007). ISSN 0001-8708

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. $\Sigma \backslash S_w \subseteq Z_w.$ \par 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 $J_k$. \par b) $\mu_w$ has a positive lower bound in a neighborhood $(x_0 - \delta_0 , x_0 + \delta_0).$ \par 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.
[Gerlind Plonka (Duisburg)]
MSC 2000:
*41A10 Approximation by polynomials
30C10 Polynomials (one complex variable)
31A15 Potentials, etc. (two-dimensional)

Keywords: smooth and doubling measures; polynomials; approximation; equilibrium measures; homogeneous polynomials; level curves

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