# Diagonally Square Root Integrable Kernels in System Identification

## TL;DR

This paper investigates diagonally square root integrable kernels within RKHS theory, showing their stability, integrability, and topological properties, with implications for system identification and Gaussian process stability.

## Contribution

It introduces and analyzes the class of DSRI kernels, demonstrating their stability, integrability, and relevance to Gaussian process stability in system identification.

## Key findings

- Various well-known kernels are DSRI.
- DSRI kernels are stable and integrable.
- Stability of Gaussian processes is characterized by DSRI kernels.

## Abstract

In recent years, the reproducing kernel Hilbert space (RKHS) theory has played a crucial role in linear system identification. The core of a RKHS is the associated kernel characterizing its properties. Accordingly, this work studies the class of diagonally square root integrable (DSRI) kernels. We demonstrate that various well-known stable kernels introduced in system identification belong to this category. Moreover, it is shown that any DSRI kernel is also stable and integrable. We look into certain topological features of the RKHSs associated with DSRI kernels, particularly the continuity of linear operators defined on the respective RKHSs. For the stability of a Gaussian process centered at a stable impulse response, we show that the necessary and sufficient condition is the diagonally square root integrability of the corresponding kernel. Furthermore, we elaborate on this result by providing proper interpretations.