J Fourier Anal Appl (2011) 17:674–690 DOI 10.1007/s00041-010-9152-3 Wiener’s Lemma and Memory Localization Ilya A. Krishtal Received: 8 February 2010 / Revised: 7 August 2010 / Published online: 27 October 2010 © Springer Science+Business Media, LLC 2010 Abstract We obtain several versions of the non-commutative Wiener’s lemma for different matrix-like representations of localized operators in Hilbert spaces. Keywords Wiener’s lemma · Resolutions of the identity · Frames · Fusion frames · g-frames Mathematics Subject Classification 47B99 · 42C15 1 Introduction Wiener’s Tauberian lemma [35] is a classical result in harmonic analysis which states that if a periodic function f has an absolutely convergent Fourier series and never vanishes then the function 1/f also has an absolutely convergent Fourier series. This result has many extensions (see [1, 3, 4, 6–8, 15, 17, 20, 21, 23, 24, 26–30, 32, 34] and references therein), some of which have recently proved to be very useful in the study of localized frames [1, 5, 14, 15, 18, 22, 31]. In this paper we reverse the emphasis and use frames, fusion frames, and the like to obtain an extension of Wiener’s lemma itself for new classes of operators in Hilbert spaces. A standard reformulation of Wiener’s lemma states that if an invertible operator is defined by a (bi-infinite) Laurent matrix with summable diagonals then the inverse has the same property. In [6, 17, 23] it was shown (independently) that the matrix does not have to be Laurent. Hence, if one interprets a matrix entry a ij as a “mem- ory cell”, that is information on what impact an event at “time” j has on the state Communicated by Karlheinz Gröchenig. The author was supported in part by the NSF grant DMS-0908239. I.A. Krishtal ( ) Dept. of Mathematical Sciences, Northern Illinois University, DeKalb, IL 610115, USA e-mail: krishtal@math.niu.edu