Wiener’s Lemma asserts that the pointwise inverse of a periodic function with absolutely convergent Fourier series has again an absolutely convergent Fourier series. Today, Wiener’s lemma is usually proved by gathering such functions up into a commutative Banach algebra \(\mathcal A(\mathbb T)\) endowed with the \(\ell^1(\mathbb Z)\) norm of their Fourier coefficients, and by identifying its Gelfand transform as the injection of \(\mathcal A(\mathbb T)\) into the algebra of continuous functions \(\mathcal C(\mathbb T)\).
The author begins by giving an elementary proof following D. J.Newman [Proc.Am.Math.Soc. 48, 264–265 (1975; Zbl 0296.42017)]. On a more abstract level, following A. Hulanicki [Invent.Math. 17, 135–142 (1972; Zbl 0264.43007)], this proof yields in fact that the spectral radius of an element in \(\mathcal A(\mathbb T)\) equals its spectral radius in \(\mathcal C(\mathbb T)\), and this is enough to show that \(\mathcal A(\mathbb T)\) is inverse-closed in \(\mathcal C(\mathbb T)\). Wiener’s lemma applies also in operator theory: the elements of \(\ell^1(\mathbb Z)\) operate by convolution on the sequence spaces \(\ell^p(\mathbb Z)\) and it turns out that, if the corresponding operator has an inverse, then the inverse operates again by convolution with an element of \(\ell^1(\mathbb Z)\). Note that the matrix of convolution operators is Toeplitz: it is constant along diagonals.
The author proceeds by describing many generalisations of Wiener’s lemma: it still holds in the following five frameworks. 6.5mm

(1)

Multivariate periodic functions with a Fourier series that is absolutely convergent with respect to a submultiplicative weight \(\nu\), if and only if the weight satisfies the Gel’fand-Raĭkov-Shilov condition: \(\nu(nk)^{1/n}@>>{n\to\infty}> 1\) for every index \(k\).

(2)

Operators on \(\ell^p(\mathbb Z^d)\) whose matrix is such that the supremum along diagonals forms an absolutely convergent series: this is A. G. Baskakov’s theorem [Funkts.Anal.Prilozh. 24, No. 3, 64–65 (1990; Zbl 0728.47021)].

(3)

The rotation algebra of time-frequency shifts \(f\mapsto e^{2\pi i\xi{\cdot}}f({\cdot}-x)\), where the \((x,\xi)\) are elements of a lattice in \(\mathbb R^d\times\mathbb R^d\); they may be considered as operators by twisted convolution. This is due to the author and M. Leinert [J. Am.Math.Soc. 17, 1–18 (2004; Zbl 1037.22012)].

(4)

Convolution operators on a locally compact group, if and only if the group is amenable and symmetric.

(5)

The pseudodifferential operators with symbol in the Sjöstrand class \(M^{\infty,1}(\mathbb R^{2d})\).

The results in (2–5) admit weighted counterparts as in (1); the space on which the operators act may also be weighted if the weight is \(\nu\)-moderate. The author succeeds in motivating all these developments by concrete questions in signal analysis of discrete and continuous time-invariant and time-varying systems, such as mobile communication and transmission of information, and even in providing a mathematical justification of certain engineering intuitions.
For the entire collection see [Zbl 1182.42002].