arXiv subjects

Franz Luef

Publications and source records attributed to Franz Luef.

At least 37 records · Page 2Linked to original sources

arXiv2023

Local Structure and effective Dimensionality of Time Series Data Sets

The goal of this paper is to develop novel tools for understanding the local structure of systems of functions, e.g. time-series data points, such as the total correlation function, the Cohen class of the data set, the data operator and the average lack of concentration. The Cohen class of the data operator gives a time-frequency representation of the data set. Furthermore, we show that the von Neumann entropy of the data operator captures local features of the data set and that it is related to the notion of the effective dimensionality. The accumulated Cohen class of the data operator gives us a low-dimensional representation of the data set and we quantify this in terms of the average lack of concentration and the von Neumann entropy of the data operator by an application of a Berezin-Lieb inequality. The framework for our approach is provided by quantum harmonic analysis.

math.FA↗

arXiv2023

$τ$-quantization and $τ$-Cohen classes distributions of Feichtinger operators

We investigate the $τ$-quantizations and Cohen's class distributions of a suitable class of trace-class operators, called Feichtinger's operators, and show that it is a convenient substitute for the class of Schwartz operators. Many well-known concepts and results for functions in time-frequency analysis have an operator-analog in our setting, e.g. that Cohen's classes are convolutions of Wigner functions with distributions or characterization of the class of Schwartz operators as an intersection of weighted variants of the class of Feichtinger operators.

math.FA↗

arXiv2022

Deformations and Balian-Low theorems for Gabor frames on the adeles

We generalize Feichtinger and Kaiblinger's theorem on linear deformations of uniform Gabor frames to the setting of a locally compact abelian group $G$. More precisely, we show that Gabor frames over lattices in the time-frequency plane of $G$ with windows in the Feichtinger algebra are stable under small deformations of the lattice by an automorphism of $G \times \widehat{G}$. The topology we use on the automorphisms is the Braconnier topology. We characterize the groups in which the Balian--Low theorem for the Feichtinger algebra holds as exactly the groups with noncompact identity component. This generalizes a theorem of Kaniuth and Kutyniok on the zeros of the Zak transform on locally compact abelian groups. We apply our results to a class of number-theoretic groups, including the adele group associated to a global field.

math.FA↗

arXiv2022

Uncertainty principle via variational calculus on modulation spaces

We approach uncertainty principles of Cowling-Price-Heis-\\enberg-type as a variational principle on modulation spaces. In our discussion we are naturally led to compact localization operators with symbols in modulation spaces. The optimal constant in these uncertainty principles is the smallest eigenvalue of the inverse of a compact localization operator. The Euler-Lagrange equations for the associated functional provide equations for the eigenfunctions of the smallest eigenvalue of these compact localization operators. As a by-product of our proofs we derive a generalization to mixed-norm spaces of an inequality for Wigner and Ambiguity functions due do Lieb.

math.FA↗

arXiv2022

Yang-Mills connections and sigma-models on quantum Heisenberg manifolds

We construct a spectral triple on a quantum Heisenberg manifold, which generalizes the results of Chakraborty and Shinha, and associate to it an energy functional on the set of projections, following the approach of Mathai-Rosenberg to non-linear sigma models. The spectral triples that we construct extend the We derive a lower bound for this energy functional that is linked on the topological charge of the projection which depends on the curvature of a compatible connection. A detailed study of this lower bound is given for the Kang projection in quantum Heisenberg manifolds. These results display an intriguing interplay between non-linear sigma models and Yang-Mills theory on quantum Heisenberg manifolds, unlike in the well-studied case of noncommutative tori.

math.OA↗

arXiv2021

Affine Quantum Harmonic Analysis

We develop a quantum harmonic analysis framework for the affine group. This encapsulates several examples in the literature such as affine localization operators, covariant integral quantizations, and affine quadratic time-frequency representations. In the process, we develop a notion of admissibility for operators and extend well known results to the operator setting. A major theme of the paper is the interaction between operator convolutions, affine Weyl quantization, and admissibility.

math.FA↗

arXiv2020

A Wiener Tauberian theorem for operators and functions

We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian theorems are related to localization operators, Toeplitz operators, isomorphism theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization operators and Tauberian theorems we note that the analogue of Pitt's Tauberian theorem in our setting implies compactness results for Toeplitz operators in terms of the Berezin transform. In addition, we extend the results on Toeplitz operators to other reproducing kernel Hilbert spaces induced by the short-time Fourier transform, known as Gabor spaces. Finally, we establish the equivalence of Wiener's Tauberian theorem and the condition in the characterization of compactness of localization operators due to Fernández and Galbis.

math.FA↗

arXiv2020

Duality of Gabor frames and Heisenberg modules

Given a locally compact abelian group $G$ and a closed subgroup $Λ$ in $G\times\widehat{G}$, Rieffel associated to $Λ$ a Hilbert $C^*$-module $\mathcal{E}$, known as a Heisenberg module. He proved that $\mathcal{E}$ is an equivalence bimodule between the twisted group $C^*$-algebra $C^*(Λ,\textsf{c})$ and $C^*(Λ^\circ,\bar{\textsf{c}})$, where $Λ^{\circ}$ denotes the adjoint subgroup of $Λ$. Our main goal is to study Heisenberg modules using tools from time-frequency analysis and pointing out that Heisenberg modules provide the natural setting of the duality theory of Gabor systems. More concretely, we show that the Feichtinger algebra ${\textbf{S}}_{0}(G)$ is an equivalence bimodule between the Banach subalgebras ${\textbf{S}}_{0}(Λ,\textsf{c})$ and ${\textbf{S}}_{0}(Λ^{\circ},\bar{\textsf{c}})$ of $C^*(Λ,\textsf{c})$ and $C^*(Λ^\circ,\bar{\textsf{c}})$, respectively. Further, we prove that ${\textbf{S}}_{0}(G)$ is finitely generated and projective exactly for co-compact closed subgroups $Λ$. In this case the generators $g_1,\ldots,g_n$ of the left ${\textbf{S}}_{0}(Λ)$-module ${\textbf{S}}_{0}(G)$ are the Gabor atoms of a multi-window Gabor frame for $L^2(G)$. We prove that this is equivalent to $g_1,\ldots,g_n$ being a Gabor super frame for the closed subspace generated by the Gabor system for $Λ^{\circ}$. This duality principle is of independent interest and is also studied for infinitely many Gabor atoms. We also show that for any non-rational lattice $Λ$ in $\mathbb{R}^{2m}$ with volume ${s}(Λ)<1$ there exists a Gabor frame generated by a single atom in ${\textbf{S}}_{0}(\mathbb{R}^m)$.

math.FA↗

arXiv2019

Polyanalytic Toeplitz operators: isomorphisms, symbolic calculus and approximation of Weyl operators

We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.

math-ph↗

arXiv2019

The Affine Wigner Distribution

We examine the affine Wigner distribution from a quantization perspective with an emphasis on the underlying group structure. One of our main results expresses the scalogram as (affine) convolution of affine Wigner distributions. We strive to unite the literature on affine Wigner distributions and we provide the connection to the Mellin transform in a rigorous manner. Moreover, we present an affine ambiguity function and show how this can be used to illuminate properties of the affine Wigner distribution. In contrast with the usual Wigner distribution, we demonstrate that the affine Wigner distribution is never an analytic function. Our approach naturally leads to several applications, one of which is an approximation problem for the affine Wigner distribution. We show that the deviation for a symbol to be an affine Wigner distribution can be expressed purely in terms of intrinsic operator-related properties of the symbol. Finally, we present an affine positivity conjecture regarding the non-negativity of the affine Wigner distribution.

math-ph↗

arXiv2019

Modulation Spaces as a Smooth Structure in Noncommutative Geometry

We demonstrate that a class of modulation spaces are examples of a smooth structure on the noncommutative 2-torus in the sense of recent developments in KK-theory. In addition, we prove that this class of modulation spaces can be represented as corners in operator linking algebras.

math.OA↗

arXiv2019

Gabor Duality Theory for Morita Equivalent $C^*$-algebras

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent $C^*$-algebras where the equivalence bimodule is a finitely generated projective Hilbert $C^*$-module. These Hilbert $C^*$-modules are equipped with some extra structure and are called Gabor bimodules. We formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group $C^*$-algebras of a lattice in phase space. We lift all these results to the matrix algebra level and in the description of the module frames associated to a matrix Gabor bimodule we introduce $(n,d)$-matrix frames, which generalize superframes and multi-window frames. Density theorems for $(n,d)$-matrix frames are established, which extend the ones for multi-window and super Gabor frames. Our approach is based on the localization of a Hilbert $C^*$-module with respect to a trace.

math.OA↗

arXiv2019

A twisted local index formula for curved noncommutative two tori

We consider the Dirac operator of a general metric in the canonical conformal class on the noncommutative two torus, twisted by an idempotent (representing the $K$-theory class of a general noncommutative vector bundle), and derive a local formula for the Fredholm index of the twisted Dirac operator. Our approach is based on the McKean-Singer index formula, and explicit heat expansion calculations by making use of Connes' pseudodifferential calculus. As a technical tool, a new rearrangement lemma is proved to handle challenges posed by the noncommutativity of the algebra and the presence of an idempotent in the calculations in addition to a conformal factor.

math.QA↗

arXiv2019

A Large Scale Approach to Decomposition Spaces

Decomposition spaces are a class of function spaces constructed out of well-behaved coverings and partitions of unity of a set. The structure of the covering of the set determines the properties of the decomposition space. Besov spaces, shearlet spaces and modulation spaces are well-known decomposition spaces. In this paper we focus on the geometric aspects of decomposition spaces and utilize that these are naturally captured by the large scale properties of a metric space, the covered space, associated to a covering of a set. We demonstrate that decomposition spaces constructed out of quasi-isometric covered spaces have many geometric features in common. The notion of geometric embedding is introduced to formalize the way one decomposition space can be embedded into another decomposition space while respecting the geometric features of the coverings. Some consequences of the large scale approach to decomposition spaces are (i) comparison of coverings of different sets, (ii) study of embeddings of decomposition spaces based on the geometric features and the symmetries of the coverings and (iii) the use of notions from large scale geometry, such as asymptotic dimension or hyperbolicity, to study the properties of decomposition spaces.

math.FA↗

arXiv2019

Yang-Mills connections on quantum Heisenberg manifolds

We investigate critical points and minimizers of the Yang-Mills functional YM on quantum Heisenberg manifolds $D^c_{μν}$, where the Yang-Mills functional is defined on the set of all compatible linear connections on finitely generated projective modules over the QHMs. A compatible linear connection which is both a critical point and minimizer of YM is called a Yang-Mills connection. In this paper, we investigate Yang-Mills connections with constant curvature. We are interested in Yang-Mills connections on the following classes of modules over the QHMs: (i) Abadie's module $Ξ$ of trace $2μ$ and its submodules; (ii) modules $Ξ^\prime$ of trace $2ν$; (iii) tensor product modules of the form $P E^c_{μν}\otimes Ξ$, where $E^c_{μν}$ is Morita equivalent to $D^c_{μν}$ and $P$ is a projection in $E^c_{μν}$. We present a characterization of critical points and minimizers of YM, and provide a class of new Yang-Mills connections with constant curvature via concrete examples. In particular, we show that every Yang-Mills connection $\nabla$ on $Ξ$ over $D^c_{μν}$ with constant curvature should have a certain form of the curvature such that $Θ_\nabla(X,Y)=Θ_\nabla(X,Z)=0$ and $Θ(Y,Z)=\frac{πi}μ Id_E$. Also we show that these Yang-Mills connections with constant curvature do not provide global minima but only local minima, and give two other examples which show that the critical points and minimizers of YM depend crucially on the geometric structure of the QHMs and of the projective modules over them. Furthermore, we construct the Grassmannian connection on the projective modules $Ξ^\prime$ with trace $2ν$ over the QHMs and compute its corresponding curvature, and we construct a tensor product connections on $P E^c_{μν}\otimes Ξ$ whose coupling constant is $2ν$ and characterize the critical points of YM for this projective module.

math.OA↗

arXiv2019

Metaplectic transformations and finite group actions on noncommutative tori

In this article we describe extensions of some K-theory classes of Heisenberg modules over higher-dimensional noncommutative tori to projective modules over crossed products of noncommutative tori by finite cyclic groups, aka noncommutative orbifolds. The two dimensional case was treated by Echterhoff, Lück, Phillips and Walters. Our approach is based on the theory of metaplectic transformations of the representation theory of the Heisenberg group. We also describe the generators of the K-groups of the crossed products of flip actions by $\mathbb{Z}_2$ on 3-dimensional noncommutative tori.

math.OA↗

arXiv2018

A duality principle for groups II: Multi-frames meet super-frames

The duality principle for group representations developed in \cite{DHL-JFA, HL_BLM} exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: The Wexler-Raz biorthogonality and the Fundamental Identity of Gabor analysis. In this paper we will show that these fundamental properties remain to be true for general projective unitary group representations. The main purpose of this paper is present a more general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pairs. In particular, for the Gabor representations $π_Λ$ and $π_{Λ^{o}}$ with respect to a pair of dual time-frequency lattices $Λ$ and $Λ^{o}$ in $\R^{d}\times \R^{d}$ we have that $\{π_Λ(m, n)g_{1} \oplus ... \oplus π_Λ(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a frame for $L^{2}(\R^{d})\oplus ... \oplus L^{2}(\R^{d})$ if and only if $\cup_{i=1}^{k}\{π_{Λ^{o}}(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a Riesz sequence, and $\cup_{i=1}^{k}\{π_Λ(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a frame for $L^{2}(\R^{d})$ if and only if $\{π_{Λ^{o}}(m, n)g_{1} \oplus ... \oplus π_{Λ^{o}}(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a Riesz sequence. This appears to be new even in the context of Gabor analysis.

math.FA↗

arXiv2018

Sampling and periodization of generators of Heisenberg modules

This paper considers generators of Heisenberg modules in the case of twisted group $C^*$-algebras of closed subgroups of locally compact abelian groups and how the restrction and/or periodization of these generators yield generators for other Heisenberg modules. Since generators of Heisenberg modules are exactly the generators of (multi-window) Gabor frames, our methods are going to be from Gabor analy\-sis. In the latter setting the procedure of restriction and periodization of generators is well known. Our results extend this established part of Gabor analy\-sis to the general setting of locally compact abelian groups. We give several concrete examples where we demonstrate some of the consequences of our results. Finally, we show that vector bundles over an irrational noncommutative torus may be approximated by vector bundles for finite-dimensional matrix algebras that converge to the irrational noncommutative torus with respect to the module norm of the generators, where the matrix algebras converge in the quantum Gromov-Hausdorff distance to the irrational noncommutative torus.

math.OA↗