Let \(E\) be a closed subset of \(R\) and \(w: E\to (0,\infty)\) a weight function on \(E\). The paper is concerned with the problem of characterization of functions in \(C(E)\) (real-valued continuous functions on \(E\)) which are uniform limits of a sequence of weighted polynomials of the form \(w^ n p_ n\), \(p_ n\) a polynomial of degree \(\leq n\). Considerations from potential theory [see H. H. Mhaskar and E. B. Saff, Constructive Approximation 1, 71-91 (1985; Zbl 0582.41009)] show the existence of a unique smallest compact interval \(S_ w\), such that \(\| w^ n p_ n\|_ E= \| w^ n p_ n\|_{S_ w}\). The second author conjectured that for \(E\) a compact interval and \(w(x)= \exp(-Q(x))\), with \(Q(x)\) convex on \(E\), a function \(f\in C(E)\) is a uniform limit of a sequence \(\{w^ n p_ n\}_{n=1}^ \infty\) iff \(f\) vanishes identically on \(E\setminus S_ w\). The aim of this paper is to prove this conjecture in the special case of Chebyshev polynomials with respect to the system \(\{w^ n1, w^ n x,\dots,w^ n x^ n\}\). The proofs are direct, without appealing to any potential theoretic considerations and are based on the denseness in \(S_ w\) of the alternation points of the Chebyshev polynomials.
For the entire collection see [Zbl 0764.00001].