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

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.