Infinite-Dimensional Measure Spaces and Frame Analysis
Palle E. T. Jorgensen, Myung-Sin Song

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 , we study three cases of measures. We first show that, for infinite dimensional, 1 one must resort to infinite dimensional measure spaces which properly contain . The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures.
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Taxonomy
TopicsMathematical Analysis and Transform Methods · Image and Signal Denoising Methods · Mathematical Dynamics and Fractals
