Wiener’s Lemma: Theme and Variations. An Introduction to Spectral Invariance and Its Applications

TLDR: It is proved Wiener’s Lemma is proved and equivalent formulations about convolution operators are discussed and the underlying abstract concepts from Banach algebras are extracted.

Abstract: Wiener’s Lemma is a classical statement about absolutely convergent Fourier series and remains one of the driving forces in the development of Banach algebra theory. In the first part of the chapter—the theme—we discuss Wiener’s Lemma in detail. We prove Wiener’s Lemma and discuss equivalent formulations about convolution operators.We then extract the underlying abstract concepts from Banach algebras. In the second part of the chapter—the variations—we discuss several, mostly noncommutative reincarnations of Wiener’s Lemma. We will develop some of the theoretical background and explain why Wiener’s Lemma is still useful and inspiring. The topics cover weighted versions of Wiener’s Lemma, infinite matrix algebras, noncommutative tori and time-frequency analysis, convolution operators on noncommutative groups, and time-varying systems and pseudodifferential operators.