Geodesic
Weighted density of polynomials on the line for strongly non-symmetric weights
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 47-54

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For a wide class of non-symmetric weights $w$ on the line $\mathbb R$ necessary and sufficient conditions for the density of polynomials in $L^2(\mathbb R, w(t)\,dt)$ are found. 
@article{ZNSL_1983_126_a5,
     author = {A. L. Vol'berg},
     title = {Weighted density of polynomials on the line for strongly non-symmetric weights},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {47--54},
     publisher = {mathdoc},
     volume = {126},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_126_a5/}
}
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A. L. Vol'berg. Weighted density of polynomials on the line for strongly non-symmetric weights. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 47-54. http://geodesic.mathdoc.fr/item/ZNSL_1983_126_a5/