Orthogonal expansions of vectors in a Hilbert space for non-Gaussian measures

Abstract:

Let $\mathcal {H}$ be a separable Hilbert space and $\mu$ a probability Radon measure on $\mathcal {H}$ of second order. Then there exist $({a_n}) \in {l^2}$, an O.N.S. $({x_n}) \subset \mathcal {H}$ and an O.N.S. $({\xi _n}) \subset H$ such that the orthogonal series $\sum \nolimits _{n = 1}^\infty {{a_n}{\xi _n}(x){x_n}}$ converges in $\mathcal {H}$ $\mu$-almost everywhere and it holds that $x = \sum \nolimits _{n = 1}^\infty {{a_n}{\xi _n}(x){x_n}}$, $\mu$-almost everywhere, where $H$ is the generating Hubert space of $\mu$. In the case where $\mu$ is a Gaussian measure, a similar result was proved by Kuelbs [2] in general Banach spaces.