Impossibility theorems for stable approximation

Alexei Shadrin, Rodrigo Platte and I have just finished a paper on stability and instability in approximating analytic functions from nonequispaced points:

Optimal sampling rates for approximating analytic functions from pointwise samples

In it, we generalize Rodrigo’s previous work (with Arno Kuijlaars and Nick Trefethen) from equispaced nodes to arbitrary nonequispaced nodes.  In particular, our result quantifies the tradeoff between convergence rates and ill-conditioning for nodes distributed according to modified Jacobi weight functions.  We also determine a necessary and sufficient sampling rate for stable approximation with polynomial least-squares fitting.