 
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, vol. 25, no. 4, pp. 20372063, August 2019.
A convolution algebra is a topological vector space 𝓧 that is closed under the convolution operation. It is said to be inverseclosed 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 ℓ_{1} algebras that are inverseclosed—these are henceforth referred to as the GelfandRaikovShilov (GRS) spaces. Our starting point here is the observation that the space 𝓢(ℤ^{d}) of rapidly decreasing sequences, which is not Banach but nuclear, is an inverseclosed convolution algebra. This property propagates to the more constrained space of exponentially decreasing sequences 𝓔(ℤ^{d}) that we prove to be nuclear as well. Using a recent extended version of the GRS condition, we then show that 𝓔(ℤ^{d}) is actually the smallest inverseclosed convolution algebra. This allows us to describe the hierarchy of the inverseclosed convolution algebras from the smallest, 𝓔(ℤ^{d}), to the largest, ℓ_{1}(ℤ^{d}). In addition, we prove that, in contrast to 𝓢(ℤ^{d}), all members of 𝓔(ℤ^{d}) admit welldefined convolution inverses in 𝓢'(ℤ^{d}) with the "unstable" scenario (when some frequencies are vanishing) giving rise to inverse filters with slowlyincreasing impulse responses. Finally, we use those results to reveal the decay and reproduction properties of an extended family of cardinal spline interpolants.

@ARTICLE(http://bigwww.epfl.ch/publications/fageot1902.html,
AUTHOR="Fageot, J. and Unser, M. and Ward, J.P.",
TITLE="Beyond {W}iener's Lemma: {N}uclear Convolution Algebras and the
Inversion of Digital Filters",
JOURNAL="The Journal of {F}ourier Analysis and Applications",
YEAR="2019",
volume="25",
number="4",
pages="20372063",
month="August",
note="")
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2019
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