Stephan Dahlke’s research while affiliated with Philipps University of Marburg and other places

This paper is concerned with near-optimal approximation of a given univariate function with elements of a polynomially enriched wavelet frame, a so-called quarklet frame. Inspired by hp-approximation techniques of Binev, we use the underlying tree structure of the frame elements to derive an adaptive algorithm that, under standard assumptions concerning the local errors, can be used to create approximations with an error close to the best tree approximation error for a given cardinality. We support our findings by numerical experiments demonstrating that this approach can be used to achieve inverse-exponential convergence rates.

In recent years, cone-adapted shearlets have been established as a powerful tool for the detection of directional information in images. It has also turned out that an interpretation of cone-adapted shearlets in the realm of coorbit space theory exists. Then, cone-adapted shearlets are defined by requiring a certain sparsity pattern of the frame expansion coefficients. In this note, we discuss some structural properties of these spaces.

Ecosystem functions and services are severely threatened by unprecedented global loss in biodiversity. To counteract these trends, it is essential to develop systems to monitor changes in biodiversity for planning, evaluating, and implementing conservation and mitigation actions. However, the implementation of monitoring systems suffers from a trade‐off between grain (i.e., the level of detail), extent (i.e., the number of study sites), and temporal repetition. Here, we present an applied and realized networked sensor system for integrated biodiversity monitoring in the Nature 4.0 project as a solution to these challenges, which considers plants and animals not only as targets of investigation, but also as parts of the modular sensor network by carrying sensors. Our networked sensor system consists of three main closely interlinked components with a modular structure: sensors, data transmission, and data storage, which are integrated into pipelines for automated biodiversity monitoring. We present our own real‐world examples of applications, share our experiences in operating them, and provide our collected open data. Our flexible, low‐cost, and open‐source solutions can be applied for monitoring individual and multiple terrestrial plants and animals as well as their interactions. Ultimately, our system can also be applied to area‐wide ecosystem mapping tasks, thereby providing an exemplary cost‐efficient and powerful solution for biodiversity monitoring. Building upon our experiences in the Nature 4.0 project, we identified ten key challenges that need to be addressed to better understand and counteract the ongoing loss of biodiversity using networked sensor systems. To tackle these challenges, interdisciplinary collaboration, additional research, and practical solutions are necessary to enhance the capability and applicability of networked sensor systems for researchers and practitioners, ultimately further helping to ensure the sustainable management of ecosystems and the provision of ecosystem services.

This paper is concerned with near-optimal approximation of a given function $f \in L_2([0,1])$ with elements of a polynomially enriched wavelet frame, a so-called quarklet frame. Inspired by hp-approximation techniques of Binev, we use the underlying tree structure of the frame elements to derive an adaptive algorithm that, under standard assumptions concerning the local errors, can be used to create approximations with an error close to the best tree approximation error for a given cardinality. We support our findings by numerical experiments demonstrating that this approach can be used to achieve inverse-exponential convergence rates.

Time-frequency analysis deals with signals for which the underlying spectral characteristics change over time. The essential tool is the short-time Fourier transform, which localizes the Fourier transform in time by means of a window function. In a white noise model, we derive rate-optimal and adaptive estimators of signals in modulation spaces, which measure smoothness in terms of decay properties of the short-time Fourier transform. The estimators are based on series expansions by means of Gabor frames and on thresholding the coefficients. The minimax rates have interesting new features, and the derivation of the lower bounds requires the use of test functions which approximately localize both in time and in frequency. Simulations and applications to audio recordings illustrate the practical relevance of our methods. We also discuss the best N -term approximation and the approximation of variational problems in modulation spaces by Gabor frame expansions.

This paper is concerned with the regularity of solutions to parabolic evolution equations. Special attention is paid to the smoothness in the specific anisotropic scale $\ B^{r\mathbf{a}}_{\tau,\tau}, \ \frac{1}{\tau}=\frac{r}{d}+\frac{1}{p}\$ of Besov spaces where $\mathbf{a}$ measures the anisotropy. The regularity in these spaces determines the approximation order that can be achieved by fully space-time adaptive approximation schemes. In particular, we show that for the heat equation our results significantly improve previous results by Aimar and Gomez [3].

In this paper we prove that under some conditions on the parameters the one-dimensional Triebel-Lizorkin spaces $F^{s}_{r,q}(\mathbb{R})$ can be described in terms of quarklets. So for functions from Triebel-Lizorkin spaces we obtain a quarkonial decomposition as well as a new equivalent quasi-norm. For that purpose we use quarklets that are constructed out of biorthogonal compactly supported Cohen-Daubechies-Feauveau spline wavelets, where the primal generator is a cardinal B-spline. Moreover we introduce some sequence spaces apposite to our quarklet system and study their properties. Finally we also obtain a quarklet characterization for the Triebel-Lizorkin-Morrey spaces $\mathcal{E}^{s}_{u,r,q}(\mathbb{R})$.

In [3], Antoine and Vandergheynst propose a group-theoretic approach to continuous wavelet frames on the sphere. The frame is constructed from a single so-called admissible function by applying the unitary operators associated to a representation of the Lorentz group, which is square-integrable modulo the nilpotent factor of the Iwasawa decomposition. We prove necessary and sufficient conditions for functions on the sphere, which ensure that the corresponding system is a frame. We strengthen a similar result in [3] by providing a complete and detailed proof.