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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 $$L^2(\mathbb{R}^n, e^{-| x|^2}d\mu)$$.
MSC:
 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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