Wiener’s lemma: Theme and variations. An introduction to spectral invariance and its applications. (English) Zbl 1215.46034
Forster, Brigitte (ed.) et al., Four short courses on harmonic analysis. Wavelets, frames, time-frequency methods, and applications to signal and image analysis. Basel: Birkhäuser (ISBN 978-0-8176-4890-9/hbk; 978-0-8176-4891-6/ebook). Applied and Numerical Harmonic Analysis, 175-244 (2010).
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
For the entire collection see [Zbl 1182.42002].
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 \(M^{\infty,1}(\mathbb R^{2d})\).
For the entire collection see [Zbl 1182.42002].
Reviewer: Stefan Neuwirth (Besançon)
MSC:
46J10 | Banach algebras of continuous functions, function algebras |
47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |
42A20 | Convergence and absolute convergence of Fourier and trigonometric series |
43A20 | \(L^1\)-algebras on groups, semigroups, etc. |
47B34 | Kernel operators |
47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
47G10 | Integral operators |
47G30 | Pseudodifferential operators |
94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |