The simultaneous approximation by polynomials on the circle and in the interior of the disc

A. L. Vol'berg

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 60-84

Résumé

The subject of this paper is the investigation of the question whether the polynomials form a dense set in the space $L^2(h)\oplus L^2(\mu_{\mathbb D})$ where $h$ is a weight on the unit circle $\mathbb T$ and $\mu_{\mathbb D}$ is a measure in the unit disc $\mathbb D$. In the case $\operatorname{supp}\mu_{\mathbb D}\subset[0,1]$ some necessary and some (close) sufficient conditions for the answer to be positive are obtained (these conditions say, roughly speaking, thet the functions $\mu_{\mathbb D}(1-\delta,1)$ and $h(e^{i\Theta})$ tend to zero sufficiently rapidly as $\delta\to0$ and $\Theta\to0$). In the general case only sufficient conditions are given.