871 research outputs found
On biunimodular vectors for unitary matrices
A biunimodular vector of a unitary matrix is a vector v \in
\mathbb{T}^n\subset\bc^n such that as well. Over the
last 30 years, the sets of biunimodular vectors for Fourier matrices have been
the object of extensive research in various areas of mathematics and applied
sciences. Here, we broaden this basic harmonic analysis perspective and extend
the search for biunimodular vectors to arbitrary unitary matrices. This search
can be motivated in various ways. The main motivation is provided by the fact,
that the existence of biunimodular vectors for an arbitrary unitary matrix
allows for a natural understanding of the structure of all unitary matrices
Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization
We establish wavelet characterizations of homogeneous Besov spaces on
stratified Lie groups, both in terms of continuous and discrete wavelet
systems.
We first introduce a notion of homogeneous Besov space in
terms of a Littlewood-Paley-type decomposition, in analogy to the well-known
characterization of the Euclidean case. Such decompositions can be defined via
the spectral measure of a suitably chosen sub-Laplacian. We prove that the
scale of Besov spaces is independent of the precise choice of Littlewood-Paley
decomposition. In particular, different sub-Laplacians yield the same Besov
spaces.
We then turn to wavelet characterizations, first via continuous wavelet
transforms (which can be viewed as continuous-scale Littlewood-Paley
decompositions), then via discretely indexed systems. We prove the existence of
wavelet frames and associated atomic decomposition formulas for all homogeneous
Besov spaces , with and .Comment: 39 pages. This paper is to appear in Journal of Function Spaces and
Applications. arXiv admin note: substantial text overlap with arXiv:1008.451
Wavelet Coorbit Spaces viewed as Decomposition Spaces
In this paper we show that the Fourier transform induces an isomorphism
between the coorbit spaces defined by Feichtinger and Gr\"ochenig of the mixed,
weighted Lebesgue spaces with respect to the quasi-regular
representation of a semi-direct product with suitably
chosen dilation group , and certain decomposition spaces
(essentially as
introduced by Feichtinger and Gr\"obner), where the localized ,,parts`` of a
function are measured in the -norm.
This equivalence is useful in several ways: It provides access to a
Fourier-analytic understanding of wavelet coorbit spaces, and it allows to
discuss coorbit spaces associated to different dilation groups in a common
framework. As an illustration of these points, we include a short discussion of
dilation invariance properties of coorbit spaces associated to different types
of dilation groups
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