Approximation by weighted polynomials

Abstract

In 1998 V. Totik settled a basic conjecture in the theory of approximation by weighted polynomials of the form wⁿP_n; the Borwein-Saff conjecture. The dissertation extends this result to a larger class of functions.

Let w(x) = exp(−Q( x)) be an admissible weight on the real line. The question is which functions can be uniformly approximated by weighted polynomials of the form w(x)ⁿP_n(x) where P_n is a polynomial of degree n.

Let S_w denote the support of the extremal measure associated with w; this is a compact subset of the real line. We prove that if w(x) is an admissible weight defined on (0, +∞) and xQ^′(x) is increasing, then a continuous function f can be uniformly approximated by weighted polynomials of the form wⁿP _n if and only if f vanishes outside the support S_w.

In fact a new criterion is introduced which implies that the support of the extremal measure is an interval. This criterion is a common generalization of two often used criteria in logarithmic potential theory, namely the “ Q is convex” and the “xQ^′ (x) is increasing” assumptions.

The above approximation problem is stated and proved under the hypothesis of this more general criterion, thus the previous statement as well as the one where Q is assumed to be convex are special cases of the main theorem of the dissertation.