Beyond Wiener’s Lemma: Nuclear Convolution Algebras and the Inversion of Digital Filters

Abstract

A convolution algebra is a topological vector space X that is closed under the convolution operation. It is said to be inverse-closed if each element of X whose spectrum is bounded away from zero has a convolution inverse that is also part of the algebra. The theory of discrete Banach convolution algebras is well established with a complete characterization of the weighted ℓ1 algebras that are inverse-closed—these are henceforth referred to as the Gelfand–Raikov–Shilov (GRS) spaces. Our starting point here is the observation that the space S(Zd) of rapidly decreasing sequences, which is not Banach but nuclear, is an inverse-closed convolution algebra. This property propagates to the more constrained space of exponentially decreasing sequences E(Zd) that we prove to be nuclear as well. Using a recent extended version of the GRS condition, we then show that E(Zd) is actually the smallest inverse-closed convolution algebra. This allows us to describe the hierarchy of the inverse-closed convolution algebras from the smallest, E(Zd) , to the largest, ℓ1(Zd). In addition, we prove that, in contrast to S(Zd) , all members of E(Zd) admit well-defined convolution inverses in S′(Zd) with the “unstable” scenario (when some frequencies are vanishing) giving rise to inverse filters with slowly-increasing impulse responses. Finally, we use those results to reveal the decay and reproduction properties of an extended family of cardinal spline interpolants.