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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:
41A10Approximation by polynomials
30C10Polynomials (one complex variable)
31A15Potentials and capacity, harmonic measure, extremal length (two-dimensional)
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References:
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