10 research outputs found
Nonlinear tensor product approximation of functions
We are interested in approximation of a multivariate function
by linear combinations of products
of univariate functions , . In the case it is a
classical problem of bilinear approximation. In the case of approximation in
the space the bilinear approximation problem is closely related to the
problem of singular value decomposition (also called Schmidt expansion) of the
corresponding integral operator with the kernel . There are known
results on the rate of decay of errors of best bilinear approximation in
under different smoothness assumptions on . The problem of multilinear
approximation (nonlinear tensor product approximation) in the case is
more difficult and much less studied than the bilinear approximation problem.
We will present results on best multilinear approximation in under mixed
smoothness assumption on
Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type
Coorbit space theory is an abstract approach to function spaces and their
atomic decompositions. The original theory developed by Feichtinger and
Gr{\"o}chenig 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 the first named author 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 precise 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.Comment: to appear in Journal of Functional Analysi