Reproducing Kernel Hilbert Spaces and Covariance Functions

Let $K : T \times T \to \mathbb{C}$ be a covariance function such that the associated RKHS $\mathcal{H}_K$ is separable where $T \subset \mathbb{R}$. Then there exists a family of vector functions

$\displaystyle \Psi (t, x) = (\psi_n (t, x), n \geq 1) \forall t \in T$

and a Borel measure $\mu$ on $T$ such that $\psi_n (t, x) \in L^2 (T, \mu)$ in terms of which $K$ is representable as:

$\displaystyle K (s, t) = \int_T \sum_{n = 1}^{\infty} \psi_n (s, x) \overline{\psi_n (t, x)} d \mu (x)$

The vector functions $\Psi (s, .), s \in T$ and the measure $\mu$ may not be unique, but all such $(\Psi, .), .)$ determine $K$ and its reproducing kernel Hilbert space (RKHS) $H_K$ uniquely and the cardinality of the components determining $K$ remains the same. [1, ]

Remark 2. 1. If $\Psi (t, .)$ is a scalar, then we have

$\displaystyle K (s, t) = \int_T \Psi (s, x) \overline{\Psi (t, x)} d \mu (x)$

which includes the tri-diagonal triangular covariance with $\mu$ absolutely continuous relative to the Lebesgue measure.

2. The following notational simplification of (25) can be made. Let $n = R \times Z_+ = S \otimes P$, where $P$ is the power set of integers Z, and let P = u @ o where o is the counting measure. Then

$\displaystyle \Psi (t, n) = (\psi_n (t, x), n \in Z)$

Hence

$\displaystyle | \Psi^{\ast} (t) |^2_{L^2} = \int_T | \psi_n (t, x) |^2 d \mu (x)$

Bibliography

[1]

Malempati M. Rao. Stochastic Processes: Inference Theory. Springer Monographs in Mathematics. Springer, 2nd edition, 2014.