The Gaussian measure on algebraic varieties

Ilka Agricola; Thomas Friedrich

Fundamenta Mathematicae (1999)

  • Volume: 159, Issue: 1, page 91-98
  • ISSN: 0016-2736

Abstract

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We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety M ⊂ ℝ n is dense in the Hilbert space L 2 ( M , e - | x | 2 d μ ) , where dμ denotes the volume form of M and d ν = e - | x | 2 d μ the Gaussian measure on M.

How to cite

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Agricola, Ilka, and Friedrich, Thomas. "The Gaussian measure on algebraic varieties." Fundamenta Mathematicae 159.1 (1999): 91-98. <http://eudml.org/doc/212322>.

@article{Agricola1999,
abstract = {We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety $M⊂ℝ^n$ is dense in the Hilbert space $L^2(M,e^\{-|x|^2\}dμ)$, where dμ denotes the volume form of M and $dν = e^\{-|x|^2\}dμ$ the Gaussian measure on M.},
author = {Agricola, Ilka, Friedrich, Thomas},
journal = {Fundamenta Mathematicae},
keywords = {Gaussian measure; algebraic variety; ring of polynomials; real algebraic variety; Hilbert space},
language = {eng},
number = {1},
pages = {91-98},
title = {The Gaussian measure on algebraic varieties},
url = {http://eudml.org/doc/212322},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Agricola, Ilka
AU - Friedrich, Thomas
TI - The Gaussian measure on algebraic varieties
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 1
SP - 91
EP - 98
AB - We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety $M⊂ℝ^n$ is dense in the Hilbert space $L^2(M,e^{-|x|^2}dμ)$, where dμ denotes the volume form of M and $dν = e^{-|x|^2}dμ$ the Gaussian measure on M.
LA - eng
KW - Gaussian measure; algebraic variety; ring of polynomials; real algebraic variety; Hilbert space
UR - http://eudml.org/doc/212322
ER -

References

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  1. [Agr] I. Agricola, Dissertation am Institut für Reine Mathematik der Humboldt-Universität zu Berlin, in preparation. 
  2. [Brö] L. Bröcker, Semialgebraische Geometrie, Jahresber. Deutsch. Math.-Verein. 97 (1995), 130-156. 
  3. [Mau] K. Maurin, Analysis, Vol. 2, Reidel and PWN-Polish Sci. Publ., Dordrecht and Warszawa, 1980. 
  4. [Mi1] J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275-280. Zbl0123.38302
  5. [Mi2] J. Milnor, Euler characteristics and finitely additive Steiner measures, in: Collected Papers, Vol. 1, Publish or Perish, 1994, 213-234. 
  6. [Rud] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1966. 
  7. [Sto1] W. Stoll, The growth of the area of a transcendental analytic set. I, Math. Ann. 156 (1964), 47-78. Zbl0126.09502
  8. [Sto2] W. Stoll, The growth of the area of a transcendental analytic set. II, ibid. 156 (1964), 144-170. 

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