The Paley-Wiener Functional of a Vector Relative to a Variance Sequence

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinates $x_{k}=\langle x,e_{k}\rangle$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, let $c$ be a variance sequence and let $h\in X$, with coordinates $h_{k}$.

1. (Partial sums)¶ For $n\in\mathbb{N}$, the $n$-th Paley-Wiener partial sum of $h$ relative to $c$ is the function

$$\ell_{h,n}:X\to\mathbb{R},\qquad\ell_{h,n}(x)=\sum_{k=1}^{n}\frac{h_{k}x_{k}}{c_{k}}.$$

2. (Paley-Wiener functional)¶ The Paley-Wiener functional of $h$ relative to $c$ is the function $\ell_{h}:X\to\mathbb{R}$ given by $\ell_{h}(x)=\lim_{n\to\infty}\ell_{h,n}(x)$ for those $x\in X$ for which the real sequence $(\ell_{h,n}(x))_{n\in\mathbb{N}}$ converges, and by $\ell_{h}(x)=0$ for all other $x\in X$.