TY - JOUR

T1 - Spectral Theory for Gaussian Processes

T2 - Reproducing Kernels, Boundaries, and L2-Wavelet Generators with Fractional Scales

AU - Alpay, Daniel

AU - Jorgensen, Palle

N1 - Funding Information:
D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. The research of the authors was supported in part by the Binational Science Foundation Grant 2010117.
Publisher Copyright:
© 2015 Copyright Taylor & Francis Group, LLC.

PY - 2015/10/3

Y1 - 2015/10/3

N2 - A recurrent theme in functional analysis is the interplay between the theory of positive definite functions and their reproducing kernels on the one hand, and Gaussian stochastic processes on the other. This central theme is motivated by a host of applications in mathematical physics. In this article, we show that it is possible to obtain explicit formulas amenable to computations of the respective Gaussian stochastic processes. To achieve this, we first develop two functional analytic tools. These are the identification of a universal sample space Ω where we realize the particular Gaussian processes in the correspondence, a procedure for discretizing computations in Ω. Our processes are as follows: Processes associated with arbitrarily given sigma finite regular measures on a fixed Borel measure space, with Hilbert spaces of sigma-functions, and with systems of self-similar measures arising in the theory of iterated function systems. In our last theorem, starting with a non-degenerate positive definite function K on some fixed set T, we show that there is a choice of a universal sample space Ω which can be realized as a boundary of (T, K). Its boundary-theoretic properties are analyzed, and we point out their relevance to the study of electrical networks on countable infinite graphs.

AB - A recurrent theme in functional analysis is the interplay between the theory of positive definite functions and their reproducing kernels on the one hand, and Gaussian stochastic processes on the other. This central theme is motivated by a host of applications in mathematical physics. In this article, we show that it is possible to obtain explicit formulas amenable to computations of the respective Gaussian stochastic processes. To achieve this, we first develop two functional analytic tools. These are the identification of a universal sample space Ω where we realize the particular Gaussian processes in the correspondence, a procedure for discretizing computations in Ω. Our processes are as follows: Processes associated with arbitrarily given sigma finite regular measures on a fixed Borel measure space, with Hilbert spaces of sigma-functions, and with systems of self-similar measures arising in the theory of iterated function systems. In our last theorem, starting with a non-degenerate positive definite function K on some fixed set T, we show that there is a choice of a universal sample space Ω which can be realized as a boundary of (T, K). Its boundary-theoretic properties are analyzed, and we point out their relevance to the study of electrical networks on countable infinite graphs.

KW - Bernoulli measures

KW - Boundary-representations

KW - Covariance

KW - Cuntz-relations

KW - Direct integral decompositions

KW - Gaussian processes

KW - Independence

KW - Iterated function systems

KW - Positive definite functions

KW - Reproducing kernel-Hilbert spaces

KW - Reproducing kernels

KW - Sigma-functions

KW - Wavelets

UR - http://www.scopus.com/inward/record.url?scp=84942881234&partnerID=8YFLogxK

U2 - 10.1080/01630563.2015.1062777

DO - 10.1080/01630563.2015.1062777

M3 - Article

AN - SCOPUS:84942881234

VL - 36

SP - 1239

EP - 1285

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 10

ER -