Duality Principles in Frame Theory

Duality principles in Gabor theory such as the Ron–Shen duality principle and the Wexler–Raz biorthogonality relations play a fundamental role for analyzing Gabor systems. In this article we present a general approach to derive duality principles in abstract frame theory. For each sequence in a separable Hilbert space we define a corresponding sequence dependent only on two orthonormal bases. Then we characterize exactly properties of the first sequence in terms of the associated one, which yields duality relations for the abstract frame setting. In the last part we apply our results to Gabor systems.     

Glycosylation site analysis of human alpha-1-acid glycoprotein (AGP) by capillary liquid chromatography-electrospray mass spectrometry

A new anionic surfactant (RapiGest SF) was successfully used for site-specific analysis of glycosylation in human alpha-1-acid glycoprotein (AGP). By means of this analytical approach combined with capillary HPLC-mass spectrometry (and tandem mass spectrometry), the N-linked glycosylation pattern of AGP was explored. On the basis of mass matching and MS/MS experiments ca 80 different AGP-derived glycopeptides were identified. Glycosylation shows a markedly different pattern for the various glycosylation sites. At sites I and II, triantennary complex-type oligosaccharides predominate and at sites III, IV and V, tetra-antennary complex-type oligosaccharides predominate. Sites IV and V show the presence of additional N-acetyl lactosamine (Gal-GlcNAc) units (even higher degree of branching and/or longer antennae are also present).     

Construction of Compactly Supported Shearlet Frames

Shearlet tight frames have been extensively studied in recent years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital settings. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial.     

Formation of solvated ions in the atmospheric interface of an electrospray ionization triple-quadrupole mass spectrometer

A simple method capable of generating and investigating various solvent clusters and solvated ions was developed. The technique opens a door to studying these complexes on commercially available instruments. Formation of the desired solvated ion in the gas phase was achieved by introducing the appropriate volatile solvent vapour into the curtain gas stream. Capabilities of the technique are illustrated by generating alkali, alkaline earth and transition metal cations solvated by various volatile compounds such as water, methanol and acetonitrile. Depending on the ligands and on the experimental conditions, clusters of 2-100 molecules may be observed. Isotope labelling suggests that these are formed by a re-solvation process in the curtain gas region.     

Uptake of magnetic nanoparticles into cells for cell tracking

A challenge for future applications in nanotechnology is the functional integration of nano-sized materials into cellular structures. Here we investigated superparamagnetic Fe₃O₄ iron oxide nanoparticles coated with a lipid bilayer for uptake into cells and for targeting subcellular compartments. It was found that magnetic nanoparticles (MNPs) are effectively taken up into cells and make cells acquire magnetic activity. Biotin-conjugated MNPs were further functionalized by binding of the fluorescent tag streptavidin–fluorescein isothiocyanate (FITC) and, following uptake into cells, shown to confer magnetic activity and fluorescence labeling. Such FITC-MNPs were localized in the lysosomal compartment of cells which suggests a receptor-mediated uptake mechanism.     

Shearlet coorbit spaces and associated Banach frames

In this paper, we study the relationships of the newly developed continuous shearlet transform with the coorbit space theory. It turns out that all the conditions that are needed to apply the coorbit space theory can indeed be satisfied for the shearlet group. Consequently, we establish new families of smoothness spaces, the shearlet coorbit spaces. Moreover, our approach yields Banach frames for these spaces in a quite natural way. We also study the approximation power of best n-term approximation schemes and present some first numerical experiments.     

Parabolic Molecules

Anisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significant attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities. In this paper, with the introduction of the notion of parabolic molecules, we aim to provide a comprehensive framework which includes customarily employed representation systems based on parabolic scaling such as curvelets and shearlets. It is shown that pairs of parabolic molecules have the fundamental property to be almost orthogonal in a particular sense. This result is then applied to analyze parabolic molecules with respect to their ability to sparsely approximate data governed by anisotropic features. For this, the concept of sparsity equivalence is introduced which is shown to allow the identification of a large class of parabolic molecules providing the same sparse approximation results as curvelets and shearlets. Finally, as another application, smoothness spaces associated with parabolic molecules are introduced providing a general theoretical approach which even leads to novel results for, for instance, compactly supported shearlets.     

A generalization of Gram–Schmidt orthogonalization generating all Parseval frames

Given an arbitrary finite sequence of vectors in a finite-dimensional Hilbert space, we describe an algorithm, which computes a Parseval frame for the subspace generated by the input vectors while preserving redundancy exactly. We further investigate several of its properties. Finally, we apply the algorithm to several numerical examples.      

Classification of edges using compactly supported shearlets

We analyze the detection and classification of singularities of functions $f={\chi }_S$, where $S?{\mathbb{R}}^d$ and $d=2,3$. It will be shown how the set ?S can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if ?S is a $d?1$ dimensional piecewise smooth manifold with $d=2$ or 3, we will classify smooth and non-smooth components of ?S. This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show that our bounds are optimal. Along the way, we also obtain novel results on the characterization of wavefront sets in 3 dimensions by compactly supported shearlets. Finally, geometric properties of ?S such as curvature are described in terms of the continuous shearlet transform of f.     

Compactly supported shearlets are optimally sparse

Cartoon-like images, i.e., C² functions which are smooth apart from a C² discontinuity curve, have by now become a standard model for measuring sparse (nonlinear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit sparse approximations within this model, which are optimal up to a log-factor. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications.In this paper, we present the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of (non-tight) shearlet frames with shearlet generators having compact support and satisfying some weak directional vanishing moment conditions.