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

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.