On a spectral representation for correlation measures in configuration space analysis
| dc.creator | Berezansky, Yu. M. | |
| dc.creator | Kondratiev, Yu. G. | |
| dc.creator | Kuna, T. | |
| dc.creator | Lytvynov, E. | |
| dc.date | 2006-08-14 | |
| dc.date.accessioned | 2026-07-07T07:21:45Z | |
| dc.date.available | 2026-07-07T07:21:45Z | |
| dc.description | The paper is devoted to the study of configuration space analysis by using the projective spectral theorem. For a manifold $X$, let $Γ_X$, resp.\ $Γ_{X,0}$ denote the space of all, resp. finite configurations in $X$. The so-called $K$-transform, introduced by A. Lenard, maps functions on $Γ_{X,0}$ into functions on $Γ_{X}$ and its adjoint $K^*$ maps probability measures on $Γ_X$ into $σ$-finite measures on $Γ_{X,0}$. For a probability measure $μ$ on $Γ_X$, $ρ_μ:=K^*μ$ is called the correlation measure of $μ$. We consider the inverse problem of existence of a probability measure $μ$ whose correlation measure $ρ_μ$ is equal to a given measure $ρ$. We introduce an operation of $\star$-convolution of two functions on $Γ_{X,0}$ and suppose that the measure $ρ$ is $\star$-positive definite, which enables us to introduce the Hilbert space ${\cal H}_ρ$ of functions on $Γ_{X,0}$ with the scalar product $(G^{(1)},G^{(2)})_{{\cal H}_ρ}= \int_{Γ_{X,0}}(G^{(1)}\star\bar G{}^{(2)})(η) ρ(dη)$. Under a condition on the growth of the measure $ρ$ on the $n$-point configuration spaces, we construct the Fourier transform in generalized joint eigenvectors of some special family $A=(A_ϕ)_{ϕ\in\D}$, $\D:=C_0^\infty(X)$, of commuting selfadjoint operators in ${\cal H}_ρ$. We show that this Fourier transform is a unitary between ${\cal H}_ρ$ and the $L^2$-space $L^2(Γ_X,dμ)$, where $μ$ is the spectral measure of $A$. Moreover, this unitary coincides with the $K$-transform, while the measure $ρ$ is the correlation measure of $μ$. | |
| dc.identifier | https://arxiv.org/abs/math/0608343 | |
| dc.identifier | http://arxiv.org/abs/math/0608343 | |
| dc.identifier | Meth. Funct. Anal. Topol. 5 (1999), no.4, 87-100 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/115413 | |
| dc.subject | Probability | |
| dc.subject | 60G57, 47A75; 60K35 | |
| dc.title | On a spectral representation for correlation measures in configuration space analysis | |
| dc.type | text |