Infinite-Dimensional Measure Spaces and Frame Analysis

TL;DR

This paper explores the properties of infinite-dimensional probability measures related to frame analysis, highlighting key differences from finite-dimensional cases and examining Gaussian, Markov, and determinantal measures in infinite-dimensional Hilbert spaces.

Contribution

It introduces a framework for analyzing infinite-dimensional measure spaces in the context of frame analysis, extending prior finite-dimensional work to infinite-dimensional settings.

Findings

Infinite-dimensional measure spaces must properly contain the Hilbert space.

Three types of measures are studied: Gaussian, Markov path-space, and determinantal.

Distinct properties of these measures are identified in infinite-dimensional contexts.

Abstract

We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs infinite dimensions. For the case of infinite-dimensional Hilbert space $\mathcal{H}$, we study three cases of measures. We first show that, for $\mathcal{H}$ infinite dimensional, 1 one must resort to infinite dimensional measure spaces which properly contain $\mathcal{H}$. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures.