36 research outputs found
Operator representations of frames: boundedness, duality, and stability
The purpose of the paper is to analyze frames
having the form for some linear operator T:
\mbox{span} \{f_k\}_{k\in \mathbf Z} \to \mbox{span}\{f_k\}_{k\in \mathbf Z}.
A key result characterizes boundedness of the operator in terms of
shift-invariance of a certain sequence space. One of the consequences is a
characterization of the case where the representation can be achieved for an operator that has an
extension to a bounded bijective operator
In this case we also characterize all the dual frames that are representable in
terms of iterations of an operator in particular we prove that the only
possible operator is Finally, we consider stability
of the representation rather surprisingly, it
turns out that the possibility to represent a frame on this form is sensitive
towards some of the classical perturbation conditions in frame theory. Various
ways of avoiding this problem will be discussed. Throughout the paper the
results will be connected with the operators and function systems appearing in
applied harmonic analysis, as well as with general group representations
Explicit constructions and properties of generalized shift-invariant systems in
Generalized shift-invariant (GSI) systems, originally introduced by
Hern\'andez, Labate & Weiss and Ron & Shen, provide a common frame work for
analysis of Gabor systems, wavelet systems, wave packet systems, and other
types of structured function systems. In this paper we analyze three important
aspects of such systems. First, in contrast to the known cases of Gabor frames
and wavelet frames, we show that for a GSI system forming a frame, the
Calder\'on sum is not necessarily bounded by the lower frame bound. We identify
a technical condition implying that the Calder\'on sum is bounded by the lower
frame bound and show that under a weak assumption the condition is equivalent
with the local integrability condition introduced by Hern\'andez et al. Second,
we provide explicit and general constructions of frames and dual pairs of
frames having the GSI-structure. In particular, the setup applies to wave
packet systems and in contrast to the constructions in the literature, these
constructions are not based on characteristic functions in the Fourier domain.
Third, our results provide insight into the local integrability condition
(LIC).Comment: Adv. Comput. Math., to appea
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
Alternatives to the EM Algorithm for ML-Estimation of Location, Scatter Matrix and Degree of Freedom of the Student- Distribution
In this paper, we consider maximum likelihood estimations of the degree of
freedom parameter , the location parameter and the scatter matrix
of the multivariate Student- distribution. In particular, we are
interested in estimating the degree of freedom parameter that determines
the tails of the corresponding probability density function and was rarely
considered in detail in the literature so far. We prove that under certain
assumptions a minimizer of the negative log-likelihood function exists, where
we have to take special care of the case , for which
the Student- distribution approaches the Gaussian distribution. As
alternatives to the classical EM algorithm we propose three other algorithms
which cannot be interpreted as EM algorithm. For fixed , the first
algorithm is an accelerated EM algorithm known from the literature. However,
since we do not fix , we cannot apply standard convergence results for the
EM algorithm. The other two algorithms differ from this algorithm in the
iteration step for . We show how the objective function behaves for the
different updates of and prove for all three algorithms that it decreases
in each iteration step. We compare the algorithms as well as some accelerated
versions by numerical simulation and apply one of them for estimating the
degree of freedom parameter in images corrupted by Student- noise
A multiobjective optimization approach to compute the efficient frontier in data envelopment analysis
Data envelopment analysis is a linear programming-based operations research technique for performance measurement of decision-making units. In this paper, we investigate data envelopment analysis from a multiobjective point of view to compute both the efficient extreme points and the efficient facets of the technology set simultaneously. We introduce a dual multiobjective linear programming formulation of data envelopment analysis in terms of input and output prices and propose a procedure based on objective space algorithms for multiobjective linear programmes to compute the efficient frontier. We show that using our algorithm, the efficient extreme points and facets of the technology set can be computed without solving any optimization problems. We conduct computational experiments to demonstrate that the algorithm can compute the efficient frontier within seconds to a few minutes of computation time for real-world data envelopment analysis instances. For large-scale artificial data sets, our algorithm is faster than computing the efficiency scores of all decision-making units via linear programming
Dynamical sampling and frame representations with bounded operators
The purpose of this paper is to study frames for a Hilbert space
having the form for some
and an operator We characterize the frames that
have such a representation for a bounded operator and discuss the
properties of this operator. In particular, we prove that the image chain of
has finite length in the overcomplete case; furthermore has the very particular property that is a frame for
for all . We also prove that frames of the
form are sensitive to the ordering of the
elements and to norm-perturbations of the generator and the operator
On the other hand positive stability results are obtained by considering
perturbations of the generator belonging to an invariant subspace on
which is a contraction.Comment: Accepted for publication in J. Math. Anal. App