Karhunen–Loève decomposition of Gaussian measures on Banach spaces

Abstract

The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen–Ločve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. In the special case of the standardWiener measure, this decomposition matches with Lévy–Ciesielski construction of Brownian motion.

Maximum a posteriori estimators in $\ell^p$ are well-defined for diagonal Gaussian priors

“…We estimate that especially the step towards non-diagonal Gaussian priors will require deeper insight into the compatibility between the ambient space's structure and the Gaussian measure's inherent sequential nature, for example using ideas as in (Bay and Croix, 2019).…”