1,524 research outputs found
Frames, semi-frames, and Hilbert scales
Given a total sequence in a Hilbert space, we speak of an upper (resp. lower)
semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently,
for an upper semi-frame, the frame operator is bounded, but has an unbounded
inverse, whereas a lower semi-frame has an unbounded frame operator, with
bounded inverse. For upper semi-frames, in the discrete and the continuous
case, we build two natural Hilbert scales which may yield a novel
characterization of certain function spaces of interest in signal processing.
We present some examples and, in addition, some results concerning the duality
between lower and upper semi-frames, as well as some generalizations, including
fusion semi-frames and Banach semi-frames.Comment: 27 pages; Numerical Functional Analysis and Optimization, 33 (2012)
in press. arXiv admin note: substantial text overlap with arXiv:1101.285
Riesz-like bases in rigged Hilbert spaces
The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are
generalized to a rigged Hilbert space \D[t] \subset \H \subset
\D^\times[t^\times]. A Riesz-like basis, in particular, is obtained by
considering a sequence \{\xi_n\}\subset \D which is mapped by a one-to-one
continuous operator T:\D[t]\to\H[\|\cdot\|] into an orthonormal basis of the
central Hilbert space \H of the triplet. The operator is, in general, an
unbounded operator in \H. If has a bounded inverse then the rigged
Hilbert space is shown to be equivalent to a triplet of Hilbert spaces
Heisenberg modules as function spaces
Let be a closed, cocompact subgroup of , where
is a second countable, locally compact abelian group. Using localization of
Hilbert -modules, we show that the Heisenberg module
over the twisted group -algebra
due to Rieffel can be continuously and densely embedded into the Hilbert space
. This allows us to characterize a finite set of generators for
as exactly the generators of multi-window
(continuous) Gabor frames over , a result which was previously known
only for a dense subspace of . We show that
as a function space satisfies two properties that
make it eligible for time-frequency analysis: Its elements satisfy the
fundamental identity of Gabor analysis if is a lattice, and their
associated frame operators corresponding to are bounded.Comment: 24 pages; several changes have been made to the presentation, while
the content remains essentially unchanged; to appear in Journal of Fourier
Analysis and Application
Basic definition and properties of Bessel multipliers
This paper introduces the concept of Bessel multipliers. These operators are
defined by a fixed multiplication pattern, which is inserted between the
Analysis and synthesis operators. The proposed concept unifies the approach
used for Gabor multipliers for arbitrary analysis/synthesis systems, which form
Bessel sequences, like wavelet or irregular Gabor frames. The basic properties
of this class of operators are investigated. In particular the implications of
summability properties of the symbol for the membership of the corresponding
operators in certain operator classes are specified. As a special case the
multipliers for Riesz bases are examined and it is shown that multipliers in
this case can be easily composed and inverted. Finally the continuous
dependence of a Bessel multiplier on the parameters (i.e. the involved
sequences and the symbol in use) is verified, using a special measure of
similarity of sequences.Comment: 15 pages; Paper was cut from 27 to 15 pages and got a new titl
Gabor Duality Theory for Morita Equivalent -algebras
The duality principle for Gabor frames is one of the pillars of Gabor
analysis. We establish a far-reaching generalization to Morita equivalent
-algebras where the equivalence bimodule is a finitely generated
projective Hilbert -module. These Hilbert -modules are equipped with
some extra structure and are called Gabor bimodules. We formulate a duality
principle for standard module frames for Gabor bimodules which reduces to the
well-known Gabor duality principle for twisted group -algebras of a
lattice in phase space. We lift all these results to the matrix algebra level
and in the description of the module frames associated to a matrix Gabor
bimodule we introduce -matrix frames, which generalize superframes and
multi-window frames. Density theorems for -matrix frames are
established, which extend the ones for multi-window and super Gabor frames. Our
approach is based on the localization of a Hilbert -module with respect to
a trace.Comment: 36 page
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