Weighted polynomial approximation

Summary

Weighted polynomial approximation measures the distance between a function and polynomials with a weight $w$, as in $\sup_x \abs{f(x) - p(x)} w(x)$ or in a weighted $L^p$ norm. On unbounded sets such as $\R$ the weight must decay fast enough to make polynomials have finite norm. A classical question, Bernstein's approximation problem, asks for which weights on $\R$ polynomials are dense in the corresponding space of continuous functions. For the weights $e^{-\abs{x}^\alpha}$ the answer is yes exactly when $\alpha \ge 1$, and orthogonal polynomials for such weights are a main tool.