Constructing Least-Squares Polynomial Approximations.

Polynomial approximations constructed using a least-squares approach form a ubiquitous technique in numerical computation. One of the simplest ways to generate data for least-squares problems is with random sampling of a function. We discuss theory and algorithms for stability of the least-squares problem using random samples. The main lesson from our discussion is that the intuitively straightforward ("standard") density for sampling frequently yields suboptimal approximations, whereas sampling from a non-standard density, called the induced distribution, yields near-optimal approximations. We present a recent, theory that demonstrates why sampling from the induced distribution is optimal and provide several numerical experiments that support the theory. Software is also provided that reproduces the figures in this paper.