Weighted Coorbit Spaces and Banach Frames on Homogeneous Spaces

This article is concerned with frame constructions on domains and manifolds. The starting point is a unitary group representation which is square integrable modulo a suitable subgroup and therefore gives rise to a generalized continuous wavelet transform. Then generalized coorbit spaces can be defined by collecting all functions for which this wavelet transform is contained in a weighted L_p-space. Moreover, we show that a judicious discretization of the representation leads to an atomic decomposition and to Banach frames for these coorbit spaces.     

Adaptive application of operators in standard representation

Recently adaptive wavelet methods have been developed which can be shown to exhibit an asymptotically optimal accuracy/work balance for a wide class of variational problems including classical elliptic boundary value problems, boundary integral equations as well as certain classes of noncoercive problems such as saddle point problems. A core ingredient of these schemes is the approximate application of the involved operators in standard wavelet representation. Optimal computational complexity could be shown under the assumption that the entries in properly compressed standard representations are known or computable in average at unit cost. In this paper we propose concrete computational strategies and show under which circumstances this assumption is justified in the context of elliptic boundary value problems. Dedicated to Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 41A25, 41A46, 65F99, 65N12, 65N55. This work has been supported in part by the Deutsche Forschungsgemeinschaft SFB 401, the first and third author are supported in part by the European Community's Human Potential Programme under contract HPRN-CT-202-00286 (BREAKING COMPLEXITY). The second author acknowledges the financial support provided through the European Union's Human Potential Programme, under contract HPRN-CT-2002-00285 (HASSIP) and through DFG grant DA 360/4–1.     

Struktur‐ und magnetochemische Untersuchungen an KCuGaF6

Structural and Magnetochemical Studies on KCuGaF₆ The crystal structure of KCuGaF₆ was determined on the base of X‐ray single crystal data (wR₂ = 0.084 for 2476 independent reflections). The compound crystallizes with a = 728.56(4), b = 989.51(6), c = 676.27(3) pm, β = 93.120(5)°, Z = 4 in space group P2₁/c of the pyrochlore related KCuCrF₆ type. The octahedral coordinations [GaF₆] and [CuF₆] are slightly resp. strongly distorted (mean values Ga‐F: 188.2 pm resp. Cu‐F: 188.2/200.1/227.6 pm). The longest distances Ga‐F and the shortest ones Cu‐F are found within octahedral chains of these two kinds of atoms, running along [100] and [001], resp., and being mutually bridged as well (M‐F‐M in between 114 and 145°). The magnetic mole susceptibilities measured at powders and at a single crystal follow the isotropic Heisenberg model for S = ¹/₂, if effects of chain disrupture are considered in the form of some paramagnetic portion. No indication of threedimensional magnetic order is observed down to T = 2 K and low magnetic fields H < 100 G. KCuGaF₆ (J/k = —71 K for the powder) is distinguished this way from the chain structure compounds KCuAlF₆ und Na₂CuScF₇ (J/k = —76 resp. —59 K) which were also magnetically studied and yield similar antiferromagnetic exchange constants J/k.     

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.     

Frames and Coorbit Theory on Homogeneous Spaces with a Special Guidance on the Sphere

The topic of this article is a generalization of the theory of coorbit spaces and related frame constructions to Banach spaces of functions or distributions over domains and manifolds. As a special case one obtains modulation spaces and Gabor frames on spheres. Group theoretical considerations allow first to introduce generalized wavelet transforms. These are then used to define coorbit spaces on homogeneous spaces, which consist of functions having their generalized wavelet transform in some weighted L_p space. We also describe natural ways of discretizing those wavelet transforms, or equivalently to obtain atomic decompositions and Banach frames for the corresponding coorbit spaces. Based on these facts we treat aspects of nonlinear approximation and show how the new theory can be applied to the Gabor transform on spheres. For the S¹ we exhibit concrete examples of admissible Gabor atoms which are very closely related to uncertainty minimizing states.     

Besov regularity for second order elliptic boundary value problems with variable coefficients

This paper is concerned with some theoretical foundations for adaptive numerical methods for elliptic boundary value problems. The approximation order that can be achieved by such an adaptive method is determined by certain Besov regularity of the weak solution. We study Besov regularity for second order elliptic problems in bounded domains in ℝ ^d . The investigations are based on intermediate Schauder estimates and on some potential theoretic framework. Moreover, we use characterizations of Besov spaces by wavelet expansions. This work has been supported by the Deutsche Forschungsgemeinschaft (Da 360/1-1)     

Adaptive wavelet methods and sparsity reconstruction for inverse heat conduction problems

This paper is concerned with the numerical treatment of inverse heat conduction problems. In particular, we combine recent results on the regularization of ill-posed problems by iterated soft shrinkage with adaptive wavelet algorithms for the forward problem. The analysis is applied to an inverse parabolic problem that stems from the industrial process of melting iron ore in a steel furnace. Some numerical experiments that confirm the applicability of our approach are presented.