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Density of polynomials in the \(L^2\) space on the real and the imaginary axes and in a Sobolev space. (English) Zbl 1224.41018

Summary: We consider an \(L^2\) type space of scalar functions \(L^2_{M,A}({\mathbb R}\cup i{\mathbb R})\) which, in particular, can be the usual \(L^2\) space of scalar functions on \({\mathbb R}\cup i{\mathbb R}\). 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 \({\mathbb R}\) with respect to a non-negative Hermitian matrix measure. The completeness of \(L^2_{M,A}({\mathbb R}\cup i{\mathbb R})\) is also established.

MSC:

41A10 Approximation by polynomials
30E10 Approximation in the complex plane
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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