Wiener's lemma: pictures at an exhibition (Q2888979)

This is an expository article that covers different versions and generalizations of the celebrated Wiener's lemma which states that if \(a\in C(\mathbb{T})\) has a Fourier series given by NEWLINE\[NEWLINEa(t)\simeq \sum_{n\in \mathbb Z}a_n e^{2\pi i n t }, NEWLINE\]NEWLINE and the sequence \(\left( a_n \right)\) belongs to \(\ell^1(\mathbb Z)\) and \(a(t)\neq 0\) for all \(t\in\mathbb{T},\) where \(\mathbb{T}=(0,1],\) then \(b=1/a\in C(\mathbb{T})\) has an absolutely convergent Fourier series \(b(t)= \sum_{n\in \mathbb Z}b_n e^{2\pi i n t }, \) the sequence \(\left( b_n \right)\) also belongs to \(\ell^1(\mathbb Z),\)NEWLINENEWLINEThe papers presents different generalizations and analogues of Wiener's lemma of which we shall mention only two just for illustration. The first example is in terms of matrices and the second deals with \({\mathcal B}\)-valued functions.NEWLINENEWLINELet \(A_a\) be a bi-infinite Laurent matrix comprised of the Fourier coefficients of the function \(a(t),\) i.e., NEWLINE\[NEWLINE\begin{aligned} A_a= \begin{pmatrix} \ddots & \ddots & \ddots &\ddots & \ddots \\ \dots & a_{0} & a_{1} & a_{2} & \dots \\ \ddots & a_{-1} & a_{0} & a_{1} & \ddots \\ \ddots & a_{-2} & a_{-1} & a_{0} & \ddots \\ \ddots & \ddots & \ddots &\ddots & \ddots \end{pmatrix}. \end{aligned}NEWLINE\]NEWLINE A bi-infinite Laurent matrix is said to have summable diagonals if its rows belong to \(\ell^1 (\mathbb Z).\) An analogue of Wiener's lemma is the following: Assume that a bi-infinite Laurnet matrix has summable diagonals and defines a bounded invertible operator on \(\ell^1 (\mathbb Z).\) Then its inverse \(A_a^{-1}\) is a Laurent matrix that also has summable diagonals.NEWLINENEWLINEThe second example deals with \({\mathcal B}\)-valued functions, where \({\mathcal B}\) denotes a Banach algebra consisting of bounded operators acting on a Banach space \({\mathcal X}\). Let \(a\in C( \mathbb{T}^d, {\mathcal B})\) has a Fourier series given by NEWLINE\[NEWLINEa(t)\simeq \sum_{n\in \mathbb Z^d}a_n e^{2\pi i \langle n, t\rangle }, NEWLINE\]NEWLINE where the sequence \(\left( a_n \right)_{n\in \mathbb Z^d}\) belongs to \(\ell^1(\mathbb Z^d, {\mathcal B}),\) and let \(a(t)\) be invertible in \({\mathcal B}\) for all \(t\in \mathbb{T}^d.\) Then \(b=a^{-1}\in C( \mathbb{T}^d, {\mathcal B})\) has an absolutely convergent Fourier series \(b(t)= \sum_{n\in \mathbb Z^d}b_n e^{2\pi i \langle n, t\rangle }, \) and the sequence \(\left( b_n \right)_{n\in \mathbb Z^d}\) also belongs to \(\ell^1(\mathbb Z^d, {\mathcal B}).\)NEWLINENEWLINEOther generalizations aiming at loosening the periodicity requirement for functions in \(C( \mathbb R^d, {\mathcal B})\) are considered. In the last section of the paper the author discusses applications of Wiener's lemma to frame theory in Hilbert spaces, in particular to frame localization and g-frames. First, it is shown that Wiener's lemma is used to prove that certain types of frame localization are preserved by the dual frames. Second, it is illustrated how frame theory leads to a few Wiener-type results.