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~$\Cal A(\Bbb T)$ endowed with the $\ell^1(\Bbb Z)$~norm of their Fourier coefficients, and by identifying its Gelfand transform as the injection of~$\Cal A(\Bbb T)$ into the algebra of continuous functions~$\Cal C(\Bbb T)$. The author begins by giving an elementary proof following {\it D.~J. Newman} [Proc. Am. Math. Soc.~48, 264‒265 (1975; Zbl 0296.42017)]. On a more abstract level, following {\it A. Hulanicki} [Invent. Math.~17, 135‒142 (1972; Zbl 0264.43007)], this proof yields in fact that the spectral radius of an element in~$\Cal A(\Bbb T)$ equals its spectral radius in~$\Cal C(\Bbb T)$, and this is enough to show that $\Cal A(\Bbb T)$~is inverse-closed in~$\Cal C(\Bbb T)$. Wiener’s lemma applies also in operator theory: the elements of~$\ell^1(\Bbb Z)$ operate by convolution on the sequence spaces~$\ell^p(\Bbb Z)$ and it turns out that, if the corresponding operator has an inverse, then the inverse operates again by convolution with an element of~$\ell^1(\Bbb Z)$. 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. { indent6.5mm \item{(1)} Multivariate periodic functions with a Fourier series that is absolutely convergent with respect to a submultiplicative weight~$ν$, if and only if the weight satisfies the Gel’fand-Raĭkov-Shilov condition: $ν(nk)^{1/n}@>>{n\to\infty}> 1$ for every index~$k$. \item{(2)} Operators on~$\ell^p(\Bbb Z^d)$ whose matrix is such that the supremum along diagonals forms an absolutely convergent series: this is {\it A. G. Baskakov}’s theorem [Funkts. Anal. Prilozh.~24, No.~3, 64‒65 (1990; Zbl 0728.47021)]. \item{(3)} The rotation algebra of time-frequency shifts $f\mapsto e^{2πiξ{\cdot}}f({\cdot}-x)$, where the~$(x,ξ)$ are elements of a lattice in~$\Bbb R^d\times\Bbb R^d$; they may be considered as operators by twisted convolution. This is due to the author and {\it M. Leinert} [J. Am. Math. Soc.~17, 1‒18 (2004; Zbl 1037.22012)]. \item{(4)} Convolution operators on a locally compact group, if and only if the group is amenable and symmetric. \item{(5)} The pseudodifferential operators with symbol in the Sjöstrand class~$M^{\infty,1}(\Bbb R^{2d})$. } 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 $ν$-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.

Stefan Neuwirth (Besançon)