Fading memory and the convolution theorem

TL;DR

This paper generalizes the convolution theorem for causal, time-invariant filters, linking fading memory properties with convolution representations and exploring their topological and functional analysis aspects.

Contribution

It extends the convolution theorem to a broader class of weighted norms and characterizes convolution representations via topological notions like minimal continuity.

Findings

Fading memory is equivalent to convolution representations under generalized weighted norms.

Minimal continuity and minimal fading memory characterize convolution representations in finite-dimensional codomains.

Embeddings of reproducing kernel Hilbert spaces are guaranteed when input and output spaces are Hilbert spaces.

Abstract

Several topological and analytical notions of continuity and fading memory for causal and time-invariant filters are introduced, and the relations between them are analyzed. A significant generalization of the convolution theorem that establishes the equivalence between the fading memory property and the availability of convolution representations of linear filters is proved. This result extends a previous similar characterization to a complete array of weighted norms in the definition of the fading memory property. Additionally, the main theorem shows that the availability of convolution representations can be characterized, at least when the codomain is finite-dimensional, not only by the fading memory property but also by the reunion of two purely topological notions that are called minimal continuity and minimal fading memory property. Finally, when the input space and the codomain of a linear functional are Hilbert spaces, it is shown that minimal continuity and the minimal fading memory property guarantee the existence of interesting embeddings of the associated reproducing kernel Hilbert spaces.