arXiv subjects

Franz Luef

Publications and source records attributed to Franz Luef.

At least 19 recordsLinked to original sources

arXiv2026

A Time-Frequency Framework for GKP Codes

We develop a time--frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $M^\infty$ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus. Multi-window Gabor analysis then represents the logical vector by a finite block of adjoint-lattice coefficients. We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction. We further construct normalizable GKP approximants as lattice-envelope Gabor multipliers and establish weak-$*$ convergence and asymptotically isometric encoding. Finally, we recover displacement syndromes from phase relations between translated coefficient blocks and quantify their stability under additive perturbations.

math-ph↗

arXiv2026

Harmoniq: Efficient Data Augmentation on a Quantum Computer Inspired by Harmonic Analysis

Quantum machine learning has attracted significant interest in recent years. Most existing approaches, however, are variational in nature and require extensive parameter optimization subroutines. Here, we propose a conceptually distinct quantum machine learning approach that goes beyond the variational paradigm. Harmoniq takes a recently developed data augmentation technique from quantum harmonic analysis and implements it as a stochastic mixture of n-qubit circuits with at most quadratic depth each. A key strength of Harmoniq is its modularity: viewed as a quantum process acting on density matrices, it can readily be combined with other quantum data processing and learning subroutines. A subsequent case study demonstrates this modularity by combining Harmoniq with stochastic amplitude encoding for the input density matrix and quantum PCA on the output density matrix. This results in a promising signal denoising pipeline that works particularly well in the small sample size regime.

quant-ph↗

arXiv2025

Bessel duality of Gabor systems: A von Neumann algebraic perspective

Bessel duality of regular Gabor systems states that a Gabor system over a lattice is a Bessel sequence if and only if the corresponding Gabor system over the adjoint lattice is a Bessel sequence. We show that this fundamental result of time-frequency analysis can be deduced from a theorem in the theory of bimodules over von Neumann algebras, namely that under certain conditions, their left and right bounded vectors coincide.

math.FA↗

arXiv2025

Quantum Harmonic Analysis and the Structure in Data: Augmentation

In this short note, we study the impact of data augmentation on the smoothness of principal components of high-dimensional datasets. Using tools from quantum harmonic analysis, we show that eigenfunctions of operators corresponding to augmented data sets lie in the modulation space $M^1(\mathbb{R}^d)$, guaranteeing smoothness and continuity. Numerical examples on synthetic and audio data confirm the theoretical findings. While interesting in itself, the results suggest that manifold learning and feature extraction algorithms can benefit from systematic and informed augmentation principles.

math.FA↗

arXiv2025

Approximation properties of operator coorbit spaces and sparsity classes

Extensions of coorbit spaces for functions to operators have been introduced by two different groups in \cite{doelumcskr24} and \cite{k\"obaLOC25}, where one is based on the coorbit theory of Feichtinger-Gr\"ochening while the other is based on the theory of localized frames. We show that for certain Gabor g-frames the co-orbit spaces in \cite{k\"obaLOC25} conincide with the ones in \cite{doelumcskr24} and we refer to this class of operators as operator coorbit spaces. Based on the description of operator coorbit spaces in terms of Gabor g-frames we provide operator dictionaries for these spaces that allow us to define sparsity classes in this setting. We establish that these sparsity classes also coincide with the operator coorbit spaces, which holds, in particular for all Feichtinger operators, a nice class of mixed states. Numerical examples confirm the expected approximation quality by few terms for appropriately chosen operators.

math.FA↗

arXiv2024

Beurling density theorems for sampling and interpolation on the flat cylinder

We consider the Fock space weighted by $e^{-\alpha |z|^{2}}$, of entire and quasi-periodic (modulo a weight dependent on $\nu $) functions on ${C}$. The quotient space $\mathbb{C}/\mathbb{Z}$, called `The flat cylinder', is represented by the vertical strip $[0,1)\times \mathbb{R}$, which tiles ${C}$ by ${Z}$-translations and is therefore a fundamental domain for $\mathbb{C}/\mathbb{Z}$. Our main result gives a complete characterization of the sets $Z\subset \Lambda \left( \mathbb{Z}\right) $ that are sets of sampling or interpolation, in terms of concepts of upper and lower Beurling densities, $ D^{+}(Z)$ and $D^{-}(Z)$, adapted to the geometry of $\mathbb{C}/\mathbb{Z}$. The critical `Nyquist density' is the real number $\frac{\alpha }{\pi }$, meaning that the condition $D^{-}(Z)>\frac{\alpha }{\pi }$ characterizes sets of sampling, while the condition $D^{+}(Z)<\frac{\alpha }{\pi }$ characterizes sets of interpolation. The results can be reframed as a complete characterization of Gabor frames and Riesz basic sequences (given by arbitrary discrete sets in $Z\subset \Lambda \left( \mathbb{Z}\right) $), with time-periodized Gaussian windows (theta-Gaussian), for spaces of functions $f$, measurable in $\mathbb{R}$, square-integrable in $(0,1)$, and quasi-periodic with respect to integer translations.

math.FA↗

arXiv2024

Fourier restriction for Schatten class operators and functions on phase space

We formulate a variant of Fourier restriction for operators in Schatten classes, where the Fourier-Wigner transform of a bounded operator replaces the Fourier transform of a function. The Fourier-Wigner transform is closely related to the group Fourier transform of the Heisenberg group. The first result shows that Fourier-Wigner restriction for Schatten class operators is equivalent to the restriction of the symplectic Fourier transform of functions on phase space. We deduce various Schatten class results for the quantum Fourier extension operator and answer a conjecture by Mishra and Vemuri concerning the Weyl transform of measures in the affirmative.

math.FA↗

arXiv2024

Linear Deformations of Heisenberg Modules and Gabor Frames

Heisenberg modules over noncommutative tori may also be viewed as Gabor frames. Building on this fact, we relate to deformations of noncommutative tori a bundle of Banach spaces induced by Heisenberg modules. The construction of this bundle of Banach spaces rests on deformation results of Gabor frames with windows in Feichtinger's algebra due to Feichtinger and Kaiblinger. We extend some of these results to Heisenberg modules, \eg we establish an analog of the results by Feichtinger-Kaiblinger and a Balian-Low theorem. Finally, we extend our results to several generators on the bundle of Heisenberg modules and show that they provide a generalized Fell's condition on the bundle of noncommutative tori.

math.OA↗

arXiv2024

On the structure of multivariate Gabor systems and a result on Gaussian Gabor frames

We introduce an equivalence relation on the set of lattices in $\mathbb{R}^{2d}$ such that equivalent lattices support identical structures of Gabor systems, up to unitary equivalence, a notion we define. These equivalence classes are parameterized by symplectic forms on $\mathbb{R}^{2d}$ and they consist of lattices related by symplectic transformations. This implies that $2d^2 - d$ parameters suffice to describe the possible structures of Gabor systems over lattices in $\mathbb{R}^{2d}$, as opposed to the $4d^2$ degrees of freedom in the choice of lattice. We also prove that (modulo a minor complication related to complex conjugation) symplectic transformations are the only linear transformations of the time-frequency plane which implement equivalences of this kind, thereby characterizing symplectic transformations as the structure-preserving transformations of the time-frequency plane in the context of Gabor analysis. In addition, we investigate the equivalence classes that have separable lattices as representatives and find that the parameter space in this case is $d^2$-dimensional. We provide an explicit example showing that non-separable lattices with irrational lattice points can behave exactly like separable and rational ones. This approach also allows us to formulate and prove a higher-dimensional variant of the Lyubarskii-Seip-Wallst\'en Theorem for Gaussian Gabor frames. This gives us, for a large class of lattices in $\mathbb{R}^{2d}$ (including all symplectic ones), necessary and sufficient conditions for $d$-parameter families of Gaussians to generate Gabor frames.

math.FA↗

arXiv2024

On the transcendentality condition for Gaussian Gabor frames and Hermite super/multiwindow frames

We give a criterion for higher-dimensional Gaussian Gabor frames, which is a reformulation of one of the main results in in a previous article by the first and last authors in more explicit terms. We use this formulation in order to extend a result of Romero, Ulanovskii, and Zlotnikov to lattices given by irrational rotations. We also show that this density criterion for Gaussian Gabor frames is generic in a certain sense. In addition, we also extend the methods of the first and last named authors to the pseudoeffective threshold which gives a condition for uniqueness in the Bargmann-Fock space. We also use this viewpoint to study super and multi-window Gabor frames with Hermitian windows. In particular, we find a density criterion for transcendental lattices.

math.FA↗

arXiv2024

Wiener's Tauberian theorem in classical and quantum harmonic analysis

We investigate Wiener's Tauberian theorem from the perspective of limit functions, which results in several new versions of the Tauberian theorem. Based on this, we formulate and prove analogous Tauberian theorems for operators in the sense of quantum harmonic analysis. Using these results, we characterize the class of slowly oscillating operators and show that this class is strictly larger than the class of compact operators. Finally, we discuss uniform versions of Wiener's Tauberian theorem and its operator analogue and provide an application of this in operator theory.

math.FA↗

arXiv2024

Quantum Time-Frequency Analysis and Pseudodifferential Operators

We introduce Quantum Time-Frequency Analysis, which expands the approach of Quantum Harmonic Analysis to include modulations of operators in addition to translations. This is done by a projective representation of double-phase space, and we consider the associated matrix coefficients and integrated representation. This leads to the polarised Cohen's class, which is an isomorphism from Hilbert-Schmidt operators to a reproducing kernel Hilbert space, and has orthogonality relations similar to many objects in classical time-frequency analysis. By considering a class of windows for the polarised Cohen's class that is smaller than the class of Hilbert-Schmidt operators, then we find spaces of modulation spaces of operators, and we consider the properties of these spaces, including discretisation results and mapping properties between function modulation spaces. We also compare modulation spaces of operators to known symbol classes for pseudodifferential operators. In many cases, using rank-one examples of operators, we recover familiar objects and results from classical time-frequency analysis and the theory of pseudodifferential operators.

math.FA↗

arXiv2023

On Wigdersons' approach to the uncertainty principle

We revisit the uncertainty principle from the point of view suggested by A. Wigderson and Y. Wigderson. This approach is based on a primary uncertainty principle from which one can derive several inequalities expressing the impossibility of a simultaneous sharp localization in time and frequency. Moreover, it requires no specific properties of the Fourier transform and can therefore be easily applied to all operators satisfying the primary uncertainty principle. A. Wigderson and Y. Wigderson also suggested many generalizations to higher dimensions and stated several conjectures which we address in the present paper. We argue that we have to consider a more general primary uncertainty principle to prove the results suggested by the authors. As a by-product we obtain some new inequalities akin to the Cowling-Price uncertainty principle, a generalization of the Heisenberg uncertainty principle, and derive the entropic uncertainty principle from the primary uncertainty principles.

math.FA↗

arXiv2023

Measure-operator convolutions and applications to mixed-state Gabor multipliers

For the Weyl-Heisenberg group, convolutions between functions and operators were defined by Werner as a part of a framework called quantum harmonic analysis. We show how recent results by Feichtinger can be used to extend this definition to include convolutions between measures and operators. Many properties of function-operator convolutions carry over to this setting and allow us to prove novel results on the distribution of eigenvalues of mixed-state Gabor multipliers and derive a version of the Berezin-Lieb inequality for lattices. New results on the continuity of Gabor multipliers with respect to lattice parameters, masks and windows as well as their ability to approximate localization operators are also derived using this framework.

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

Time-Frequency Analysis and Coorbit Spaces of Operators

We introduce an operator valued Short-Time Fourier Transform for certain classes of operators with operator windows, and show that the transform acts in an analogous way to the Short-Time Fourier Transform for functions, in particular giving rise to a family of vector-valued reproducing kernel Banach spaces, the so called coorbit spaces, as spaces of operators. As a result of this structure the operators generating equivalent norms on the function modulation spaces are fully classified. We show that these operator spaces have the same atomic decomposition properties as the function spaces, and use this to give a characterisation of the spaces using localisation operators.

math.FA↗

arXiv2022

Time-frequency analysis on flat tori and Gabor frames in finite dimensions

We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori \begin{equation*} \mathbb{T}_{N}^2=\mathbb{R}^2/(\mathbb{Z}\times N\mathbb{Z})=[0,1]\times \lbrack 0,N] \end{equation*} act as phase spaces. We work on an $N$-dimensional subspace $S_{N}$ of distributions periodic in time and frequency in the dual $S_0'(\mathbb{R})$ of the Feichtinger algebra $S_0(\mathbb{R})$ and equip it with an inner product. To construct the Hilbert space $S_{N}$ we apply a suitable double periodization operator to $S_0(\mathbb{R})$. On $S_{N}$, the STFT is applied as the usual STFT defined on $S_0'(\mathbb{R})$. This STFT is a continuous extension of the finite discrete Gabor transform from the lattice onto the entire flat torus. As such, sampling theorems on flat tori lead to Gabor frames in finite dimensions. For Gaussian windows, one is lead to spaces of analytic functions and the construction allows to prove a necessary and sufficient Nyquist rate type result, which is the analogue, for Gabor frames in finite dimensions, of a well known result of Lyubarskii and Seip-Wallst{\'e}n for Gabor frames with Gaussian windows and which, for $N$ odd, produces an explicit \emph{full spark Gabor frame}. The compactness of the phase space, the finite dimension of the signal spaces and our sampling theorem offer practical advantages in some applications. We illustrate this by discussing a problem of current research interest: recovering signals from the zeros of their noisy spectrograms.

math.FA↗