Geodesic
The simultaneous approximation by polynomials on the circle and in the interior of the disc
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 60-84 
A. L. Vol'berg. The simultaneous approximation by polynomials on the circle and in the interior of the disc. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 60-84. http://geodesic.mathdoc.fr/item/ZNSL_1979_92_a3/
@article{ZNSL_1979_92_a3,
     author = {A. L. Vol'berg},
     title = {The simultaneous approximation by polynomials on the circle and in the interior of the disc},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {60--84},
     year = {1979},
     volume = {92},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_92_a3/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.