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.