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The Gaussian measure on algebraic varieties

100%

Agricola I. , Friedrich T.

Fundamenta Mathematicae

1999

tom 159

nr 1

91-98

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.

On measure concentration of vector-valued maps

80%

Ledoux M. , Oleszkiewicz K.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

tom Vol. 55, no 3

261-278

We study concentration properties for vector-valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in R^k. To this end, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions with earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite-dimensional setting.

The higher order Riesz transform for Gaussian measure need not be of weak type (1,1)

80%

Forzani L. , Scotto R.

Studia Mathematica

1998

tom 131

nr 3

205-214

The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator $L:= d^2/dx^2 - 2xd/dx$, x ∈ ℝ, need not be of weak type (1,1). A function in $L^1(dγ)$, where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.

Karhunen-Loève Decomposition of Gaussian Measures on Banach Spaces

80%

Bay X. , Croix J.-C.

Probability and Mathematical Statistics

2019

tom Vol. 39, Fasc. 2

279--297

The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectra theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen-Loève expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. In the special case of the standardWiener measure, this decomposition matches with Lévy-Ciesielski construction of Brownian motion.

On an invariant Borel measure in Hilbert space

51%

Pantsulaia G.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

tom Vol. 52, nr 1

47-51

An example of a nonzero [sigma]-finite Borel measure [my] with everywhere dense linear manifold I[my] of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space l[2] such that [my] and any shift [my]^[alpha] of [my] by a vector [alpha] is an element of l[2] \ I[my] are neither equivalent nor orthogonal. This extends a result established in [7].