Harmonic analysis of a class of reproducing kernel Hilbert spaces arising from groups

Abstract

We study two extension problems, and their interconnections: (i) extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; and (ii) (in case of Lie groups $G$) representations of the associated Lie algebras $La\left(G\right)$, i.e., representations of $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space ${\mathcal{H}}_F$ (RKHS). Our analysis is non-trivial even if $G={\mathbb{R}}^n$, and even if $n=1$. If $G={\mathbb{R}}^n$, (ii), we are concerned with finding systems of strongly commuting selfadjoint operators $\left\{T_i\right\}$ extending a system of commuting Hermitian operators with common dense domain in ${\mathcal{H}}_F$. Specifically, we consider partially defined positive definite (p.d.) continuous functions $F$ on a fixed group. From $F$ we then build a reproducing kernel Hilbert space ${\mathcal{H}}_F$, and the operator extension problem is concerned with operators acting in ${\mathcal{H}}_F$, and with unitary representations of $G$ acting on ${\mathcal{H}}_F$. Our emphasis is on the interplay between the two problems, and on the harmonic analysis of our RKHSs ${\mathcal{H}}_F$.

139 pages, 26 figures

References in corpus (1)

• Quantum Probability and Quantum Information Theory