Interpolatory properties of best \(L_ 2\)-approximants (Q757770)

Let f be a continuous function and \(S_ n\) be the polynomial of degree at most n of best \(L_ 2(\mu)\)-approximation to f on [-1,1]. Let \(Z_ n(f)=\{x\in [-1,1]\); \(f(x)-S_ n(x)=0\}\), i.e. \(S_ n(x)\) interpolates f(x) at the points of \(Z_ n(f)\). The main result of this paper is that \(\cup_{n\in N}Z_ n(f)\) is dense in [-1,1] under mild conditions on the measure \(\mu\). This answers a question posed independently by A. Kroo and V. M. Tikhomiroff. Some related results for the space \(L_ p(\mu)\) (1\(\leq p\leq \infty)\) are also proved.