33 research outputs found
Classification of General Sequences by Frame-Related Operators
This note is a survey and collection of results, as well as presenting some
original research. For Bessel sequences and frames, the analysis, synthesis and
frame operators as well as the Gram matrix are well-known, bounded operators.
We investigate these operators for arbitrary sequences, which in general lead
to possibly unbounded operators. We characterize various classes of sequences
in terms of these operators and vice-versa. Finally, we classify these
sequences by operators applied on orthonormal bases
On Various R-duals and the Duality Principle
The duality principle states that a Gabor system is a frame if and only if
the corresponding adjoint Gabor system is a Riesz sequence. In general Hilbert
spaces and without the assumption of any particular structure, Casazza,
Kutyniok and Lammers have introduced the so-called R-duals that also lead to a
characterization of frames in terms of associated Riesz sequences; however, it
is still an open question whether this abstract theory is a generalization of
the duality principle. In this paper we prove that a modified version of the
R-duals leads to a generalization of the duality principle that keeps all the
attractive properties of the R-duals. In order to provide extra insight into
the relations between a given sequence and its R-duals, we characterize all the
types of R-duals that are available in the literature for the special case
where the underlying sequence is a Riesz basis
Weighted frames, weighted lower semi frames and unconditionally convergent multipliers
In this paper we ask when it is possible to transform a given sequence into a
frame or a lower semi frame by multiplying the elements by numbers. In other
words, we ask when a given sequence is a weighted frame or a weighted lower
semi frame and for each case we formulate a conjecture. We determine several
conditions under which these conjectures are true. Finally, we prove an
equivalence between two older conjectures, the first one being that any
unconditionally convergent multiplier can be written as a multiplier of Bessel
sequences by shifting of weights, and the second one that every unconditionally
convergent multiplier which is invertible can be written as a multiplier of
frames by shifting of weights. We also show that these conjectures are also
related to one of the newly posed conjectures.Comment: 14 page
Operator representations of sequences and dynamical sampling
This paper is a contribution to the theory of dynamical sampling. Our purpose
is twofold. We first consider representations of sequences in a Hilbert space
in terms of iterated actions of a bounded linear operator. This generalizes
recent results about operator representations of frames, and is motivated by
the fact that only very special frames have such a representation. As our
second contribution we give a new proof of a construction of a special class of
frames that are proved by Aldroubi et al. to be representable via a bounded
operator. Our proof is based on a single result by Shapiro \& Shields and
standard frame theory, and our hope is that it eventually can help to provide
more general classes of frames with such a representation.Comment: Accepted for publication in Sampl. Theory Signal Image Proces