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Wiener’s lemma and memory localization. (English) Zbl 1248.47040

Wiener’s Tauberian lemma 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, some of which have recently proved to be very useful in the study of localized frames; cf. A. Aldroubi, A. Baskakov and I. Krishtal [J. Funct. Anal. 255, No. 7, 1667–1691 (2008; Zbl 1155.42007)], R. Balan et al. [J. Fourier Anal. Appl. 12, No. 2, 105–143 (2006; Zbl 1096.42014)], M. Fornasier and K. Gröchenig [Constructive Approximation 22, No. 3, 395–415 (2005; Zbl 1130.41304)], F. Futamura [Proc. Am. Math. Soc. 137, No. 12, 4187–4197 (2009; Zbl 1205.42029)], K. Gröchenig [J. Fourier Anal. Appl. 10, No. 2, 105–132 (2004; Zbl 1055.42018)], I. A. Krishtal and K. A. Okoudjou [J. Approx. Theory 153, No. 2, 212–224 (2008; Zbl 1283.42045)], Q.-Y. Sun [Adv. Comput. Math. 28, No. 4, 301–329 (2008; Zbl 1218.42015)]). In this paper, the author reverses the emphasis and uses frames, fusion frames, and the like to obtain an extension of Wiener’s lemma itself for new classes of operators in Hilbert spaces.
The article is organized as follows. Section 1 is introductory. Section 2 exhibits a few known versions of the noncommutative Wiener lemma and introduces the notation. In Section 3, the result is proved for frames, and in Section 4 for fusion frames. Finally, Section 5 is devoted to the case of the newly introduced fuzzy fusion frames and the most general case of g-frames.

MSC:

47B99 Special classes of linear operators
42C15 General harmonic expansions, frames
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