material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We give characterizations for homogeneous and inhomogeneous Besov-Lizorkin-Triebel spaces [28,30,31] in terms of continuous local means for the full range of parameters. In particular, we prove characterizations using tent spaces (Lusin functions) and spaces involving the Peetre maximal function in order to apply the classical coorbit space theory due to Feichtinger and Gröchenig [7,8,9,13,14]. This results in atomic decompositions and wavelet bases for homogeneous spaces. In particular we give sufficient conditions for suitable wavelets in terms of moment, decay and smoothness conditions. The latter are indeed new and received their name from the fact that quantities related to the classical Peetre maximal function are involved. This leads to the main intention of the paper. We use the established characterizations for the homogeneous spaces in order to embed them in the abstract framework of coorbit space theory originally due to Feichtinger and Gröchenig [7,8,9,13,14] in the 80s. This connection was already observed by them in [7,13,14]. They worked with Triebel's equivalent continuous normings from [29] and the results on tent spaces which were introduced more or less at the same time by Coifman, Meyer, Stein [5] to interpret Lizorkin-Triebel spaces as coorbits. On the one hand the present paper gives a late justification and on the other hand we observe that Peetre type spaces on G are a much better choice for this issue. Their two-sided translation invariance is immediate and much more transparent as we will show in Section 4.1. Furthermore, generalizations in different directions are now possible. In a forthcoming paper we will show how to apply a generalized coorbit space theory due to Fornasier and Rauhut [11] in order to recover inhomogeneous spaces based on the characterizations given here. Moreover, the extension of the results to quasi-Banach spaces using a theory developed by Rauhut in [21,22] is possible.Once we have interpreted classical homogeneous Besov-Lizorkin-Triebel spaces as certain coorbits, we are able to benefit from the achievements of the abstract theory in [7,8,9,13,14]. The main feature is a powerful discretization machinery which leads in an abstract universal way to atomic decompositions. We are now able to apply this method which results in atomic decompositions and wavelet bases for homogeneous spaces. More precisely, sufficient conditions in terms of vanishing moments, decay, and smoothness properties of the respective wavelet function are given. Compact support of the used atoms does not play any role here. In particular, we specify the order of a suitable orthonormal spline wavelet system depending on the parameters of the respective space.The paper is organized as follows. After giving some preliminaries we start in Section 2 with the definition of classical Besov-Lizorkin-Triebel spaces and their characterization via continuous local means. In Section 3 we give a brief introduction to abstract coorbit space theory which is applied in Section 4 on the ax + b-group G. We recov...

Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gröchenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces and the recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and Rauhut (2005) [24] that one may replace coherent states related to the group representation by more general abstract continuous frames. In the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces. A unified approach towards atomic decompositions and Banach frames with new results for general coorbit spaces is presented. In the second part we apply the abstract setting to a specific framework and study coorbits of what we call Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel spaces of various types of interest as coorbits. We obtain several old and new wavelet characterizations based on explicit smoothness, decay, and vanishing moment assumptions of the respective wavelet. As main examples we obtain results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed smoothness and even mixtures of the mentioned ones. Due to the generality of our approach, there are many more examples of interest where the abstract coorbit space theory is applicable.

No abstract

We investigate the approximation of d-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness s > 0 on the d-dimensional torus, where the approximation error is measured in the L 2 −norm. In other words, we study the approximation numbers of the Sobolev embeddings H, with particular emphasis on the dependence on the dimension d. For any fixed smoothness s > 0, we find the exact asymptotic behavior of the constants as d → ∞. We observe super-exponential decay of the constants in d, if n, the number of linear samples of f , is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by low rank approximations. We present some surprising results for the socalled "preasymptotic" decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.