Smooth equilibrium measures and approximation
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.
scientific article; zbMATH DE number 5161825
scientific article