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The Gaussian measure on algebraic varieties. (English) Zbl 0924.58005
The aim of the present note is to prove that the ring \(\mathbb{R}[M]\) of all polynomials defined on a real algebraic variety \(M\subset\mathbb{R}^n\) is dense in the Hilbert space \(L^2(M, e^{-| x|^2}d\mu)\), where \(d\mu\) denotes the volume form of \(M\) and \(d\nu= e^{-| x|^2}d\mu\) is the Gaussian measure on \(M\). For \(M=\mathbb{R}^n\), the result is well-known since the Hermite polynomials constitute a complete orthonormal basis of \(L^2(\mathbb{R}^n, e^{-| x|^2}d\mu)\).
58C35 Integration on manifolds; measures on manifolds
14P99 Real algebraic and real-analytic geometry
28A75 Length, area, volume, other geometric measure theory
58A07 Real-analytic and Nash manifolds
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