RKHS Representation of Algebraic Convolutional Filters with Integral Operators

TL;DR

This paper develops a theory connecting integral operators in signal processing to reproducing kernel Hilbert spaces, enabling new representations and analysis of filters in continuous and graphon models.

Contribution

It introduces a systematic RKHS framework for integral operators, linking spectral properties to kernel models and enabling learnable filter design in neural architectures.

Findings

RKHS convolutional models induced by integral operators

Polynomial filtering corresponds to iterated box products

Finite-dimensional RKHS representations for optimal filters

Abstract

Integral operators play a central role in signal processing, underpinning classical convolution, and filtering on continuous network models such as graphons. While these operators are traditionally analyzed through spectral decompositions, their connection to reproducing kernel Hilbert spaces (RKHS) has not been systematically explored within the algebraic signal processing framework. In this paper, we develop a comprehensive theory showing that the range of integral operators naturally induces RKHS convolutional signal models whose reproducing kernels are determined by a box product of the operator symbols. We characterize the algebraic and spectral properties of these induced RKHS and show that polynomial filtering with integral operators corresponds to iterated box products, giving rise to a unital kernel algebra. This perspective yields pointwise RKHS representations of filters via the reproducing property, providing an alternative to operator-based implementations. Our results establish precise connections between eigendecompositions and RKHS representations in graphon signal processing, extend naturally to directed graphons, and enable novel spatial--spectral localization results. Furthermore, we show that when the spectral domain is a subset of the original domain of the signals, optimal filters for regularized learning problems admit finite-dimensional RKHS representations, providing a principled foundation for learnable filters in integral-operator-based neural architectures.