an:01295197
Zbl 0924.58005
Agricola, Ilka; Friedrich, Thomas
The Gaussian measure on algebraic varieties
EN
Fundam. Math. 159, No. 1, 91-98 (1999).
00056624
1999
j
58C35 14P99 28A75 58A07
ring of polynomials; real algebraic variety; Hilbert space; Gaussian measure
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 \(L^2(\mathbb{R}^n, e^{-| x|^2}d\mu)\).