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Smooth equilibrium measures and approximation. (English) Zbl 1123.41005

Let Σ be a closed subset on the real line and w a nonnegative continuous function on Σ, such that w(x)x0 as x± if Σ is unbounded. Let A w be the set of functions f for which there exists a sequence of weighted polynomials {w n P n } n=1 converging to f uniformly on Σ. Here P n is a polynomial of degree at most n. A w is a subalgebra of C 0 (Σ)· Let Z w be the closed subset of Σ, such that fA w if and only if f is continuous on Σ and vanishes on Z w . The non-trivial approximation of f is possible only on the support S w of an extremal measure μ w that solves an associated equilibrium problem and is smooth, i.e. ΣS w Z w ·

In the paper, the following results are shown.

1. If x 0 Int(S w ) does not belong to Z w then μ w is smooth on some neighborhood (x 0 -δ,x 0 +δ) of x 0 ·

2. Suppose that μ w is smooth on (x 0 -δ,x 0 +δ)· Then x 0 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 μ w to each J k is a doubling measure on J k .

b) μ w has a positive lower bound in a neighborhood (x 0 -δ 0 ,x 0 +δ 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.

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