Operator representations of frames: boundedness, duality, and stability

The purpose of the paper is to analyze frames $\{f_k\}_{k\in \mathbf Z}$ having the form $\{T^kf_0\}_{k\in\mathbf Z}$ 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 $T$ in terms of shift-invariance of a certain sequence space. One of the consequences is a characterization of the case where the representation $\{f_k\}_{k\in \mathbf Z}=\{T^kf_0\}_{k\in\mathbf Z}$ can be achieved for an operator $T$ that has an extension to a bounded bijective operator $\widetilde{T}: \cal H \to \cal H.$ In this case we also characterize all the dual frames that are representable in terms of iterations of an operator $V;$ in particular we prove that the only possible operator is $V=(\widetilde{T}^*)^{-1}.$ Finally, we consider stability of the representation $\{T^kf_0\}_{k\in\mathbf Z};$ 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 L2(R)L^2(\mathbb{R})

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-tt Distribution

In this paper, we consider maximum likelihood estimations of the degree of freedom parameter $\nu$, the location parameter $\mu$ and the scatter matrix $\Sigma$ of the multivariate Student-$t$ distribution. In particular, we are interested in estimating the degree of freedom parameter $\nu$ 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 $\nu \rightarrow \infty$, for which the Student-$t$ 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 $\nu$, the first algorithm is an accelerated EM algorithm known from the literature. However, since we do not fix $\nu$, we cannot apply standard convergence results for the EM algorithm. The other two algorithms differ from this algorithm in the iteration step for $\nu$. We show how the objective function behaves for the different updates of $\nu$ 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-$t$ noise

Dynamical sampling and frame representations with bounded operators

The purpose of this paper is to study frames for a Hilbert space ${\cal H},$ having the form $\{T^n \varphi\}_{n=0}^\infty$ for some $\varphi \in {\cal H}$ and an operator $T: {\cal H} \to {\cal H}.$ We characterize the frames that have such a representation for a bounded operator $T,$ and discuss the properties of this operator. In particular, we prove that the image chain of $T$ has finite length $N$ in the overcomplete case; furthermore $\{T^n \varphi\}_{n=0}^\infty$ has the very particular property that $\{T^n \varphi\}_{n=0}^{N-1} \cup \{T^n \varphi\}_{n=N+\ell}^\infty$ is a frame for ${\cal H}$ for all $\ell\in {\mathbf N}_0$. We also prove that frames of the form $\{T^n \varphi\}_{n=0}^\infty$ are sensitive to the ordering of the elements and to norm-perturbations of the generator $\varphi$ and the operator $T.$ On the other hand positive stability results are obtained by considering perturbations of the generator $\varphi$ belonging to an invariant subspace on which $T$ is a contraction.Comment: Accepted for publication in J. Math. Anal. App