The role of the endpoint in weighted polynomial approximation with varying weights
Approximation by weighted polynomials of the form \(w^n\cdot p_n\) on an interval \([a,b]\) is considered, where \(w\) is a positive weight function and \(p_n\in\Pi_n\). The problem is to describe those functions \(f\), which can be approximated arbitrarily close by weighted polynomials of this type. Special emphasis is laid on the role of the interval endpoints and on that of internal zeros of the density function \(v\) of the extremal measure associated with \(w\). As a main result, it is shown that every function \(f\) which is approximable in the above sense must vanish in \(a\) if \(v(t)\) behaves like \((t-a)^\beta\) as \(t\to a\), with \(\beta>-1/2\). Furthermore, this result is applied to three types of illustrative examples, characterized by the Freud weights and two other types of weight functions, which lead to the incomplete polynomials resp. some fast decreasing polynomials.
scientific article; zbMATH DE number 922397
scientific article