Geodesic
Density of Polynomials in the L^2 Space on the Real and the Imaginary Axes and in a Sobolev Space
Serdica Mathematical Journal, Tome 35 (2009) no. 2, pp. 147-168.

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In this paper we consider an L^2 type space of scalar functions L^2 M, A (R u iR) which can be, in particular, the usual L^2 space of scalar functions on R u iR. We find conditions for density of polynomials in this space using a connection with the L^2 space of square-integrable matrix-valued functions on R with respect to a non-negative Hermitian matrix measure. The completness of L^2 M, A (R u iR ) is also established.
Keywords: Density of Polynomials, Moment Problem, Measure 
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Klotz, Lutz; Zagorodnyuk, Sergey M. Density of Polynomials in the L^2 Space on the Real and the Imaginary Axes and in a Sobolev Space. Serdica Mathematical Journal, Tome 35 (2009) no. 2, pp. 147-168. http://geodesic.mathdoc.fr/item/SMJ2_2009_35_2_a2/