Infinite-dimensional measure spaces and frame analysis. (English) Zbl 1405.60006
Summary: 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, 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.
MSC:
| 60B11 | Probability theory on linear topological spaces |
| 42C15 | General harmonic expansions, frames |
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
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