Gabor Analysis, Noncommutative Tori and Feichtinger's Algebra
نویسنده
چکیده
We point out a connection between Gabor analysis and noncommutative analysis. Especially, the strong Morita equivalence of noncommutative tori appears as underlying setting for Gabor analysis, since the construction of equivalence bimodules for noncommutative tori has a natural formulation in the notions of Gabor analysis. As an application we show that Feichtinger’s algebra is such an equivalence bimodule. Furthermore, we present Connes’s construction of projective modules for noncommutative tori and the relevance of a generalization of Wiener’s lemma for twisted convolution by Gröchenig and Leinert. Finally we indicate an approach to the biorthogonality relation of Wexler-Raz on the existence of dual atoms of a Gabor frame operator based on results about Morita equivalence.
منابع مشابه
Projective Modules over Noncommutative Tori Are Multi-window Gabor Frames for Modulation Spaces
In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to exploit the deeper properties of Gabor frames. Furthermore, we a...
متن کاملOn Spectral Invariance of Non–Commutative Tori
Around 1980 Connes extended the notions of geometry to the noncommutative setting. Since then non-commutative geometry has turned into a very active area of mathematical research. As a first non-trivial example of a noncommutative manifold Connes discussed subalgebras of rotation algebras, the socalled non-commutative tori. In the last two decades researchers have unrevealed the relevance of no...
متن کاملTitle of dissertation : GABOR FRAMES FOR QUASICRYSTALS AND K - THEORY
Title of dissertation: GABOR FRAMES FOR QUASICRYSTALS AND K-THEORY Michael Kreisel, Doctor of Philosophy, 2015 Dissertation directed by: Professor Jonathan Rosenberg Department of Mathematics We study the connection between Gabor frames for quasicrystals, the topology of the hull ΩΛ of a quasicrystal Λ and the K-theory of an associated twisted groupoid algebra. In particular, we construct a fin...
متن کاملOn complex and noncommutative tori
The “noncommutative geometry” of complex algebraic curves is studied. As first step, we clarify a morphism between elliptic curves, or complex tori, and C-algebras Tθ = {u, v | vu = e2πiθuv}, or noncommutative tori. The main result says that under the morphism isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra...
متن کاملStrong Morita Equivalence of Higher-dimensional Noncommutative Tori
Let n ≥ 2 and Tn be the space of n × n real skew-symmetric matrices. For each θ ∈ Tn the corresponding n-dimensional noncommutative torus Aθ is defined as the universal C-algebra generated by unitaries U1, · · · , Un satisfying the relation UkUj = e(θkj)UjUk, where e(t) = e. Noncommutative tori are one of the canonical examples in noncommutative differential geometry [12, 2]. One may also consi...
متن کامل