Beyond Wiener's Lemma: Nuclear Convolution Algebras and the Inversion of Digital Filters
J. Fageot, M. Unser, J.P. Ward
Journal of Fourier Analysis and Applications, in press.
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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.