Weighted approximation with varying weights: The case of a power-type singularity
This paper is a continuation of recent work by the author himself [Constructive Approximation 12, No. 2, 287-301 (1996; Zbl 0864.41006) and ibid. 87, No. 1, 112-115 (1996; Zbl 0862.41006)] and by \textit{L. Totik} [Weighted Approximation with varying weight (1994; Zbl 0808.41001)]. The following type of generalized Weierstraß problem is considered: Characterize the class of functions \(f\) that can be uniformly approximated by weighted polynomials of the form \(w^n p_n\), \(p_n\in\Pi_n\). Denoting by \(\mu_w\) the extremal measure associated with \(w\) the central result of the paper can be stated as follows: If in a neighborhood of a point \(t_0\mu_w\) has a density with a power-type singularity at \(t_0\), then every function \(f\) of the above type vanishes at \(t_0\).
scientific article; zbMATH DE number 1005643
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