arXiv subjects

Ole Christensen

Publications and source records attributed to Ole Christensen.

At least 19 recordsLinked to original sources

arXiv2025

Weaving Information Packets

The concept of weaving of frames for Hilbert spaces was introduced by Bemrose et al. in 2016. Two frames $\{f_k\}_{k\in I}, \{g_k\}_{k\in I}$ are woven if the ``mixed system" $\{f_k\}_{k\in \sigma} \cup \{g_k\}_{k\in I\setminus \sigma}$ is a frame for each index set $\sigma \subset I;$ that is, processing a signal using two woven frames yields a certain stability against loss of information. The concept easily extends to $N$ frames, for any integer $N>2.$ Unfortunately it is nontrivial to construct useful woven frames, and the literature is sparse concerning explicit constructions. In this paper we introduce so-called information packets, which contain as well frames as fusion frames as special case. The concept of woven frames immediately generalizes to information packets, and we demonstrate how to construct practically relevant woven information packets based on particular wavelet systems in $\ltr.$ Interestingly, we show that certain wavelet systems can be split into $N$ woven information packets, for any integer $N\ge 2.$ We finally consider corresponding questions for Gabor system in $\ltr,$ and prove that for any fixed $N\in \mn$ we can find a Gabor frame that can be split into $N$ woven information packets; however, in contrast to the wavelet case, the density conditions for Gabor system excludes the possibility of finding a single Gabor frame that works simultaneously for all $N\in \mn.$

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arXiv2025

Cyclic frames in finite-dimensional Hilbert spaces

Generalizing a definition by Kalra \cite{Kalra}, the purpose of this paper is to analyze cyclic frames in finite-dimensional Hilbert spaces. Cyclic frames form a subclass of the dynamical frames introduced and analyzed in detail by Aldroubi et al. in \cite{ACM} and subsequent papers; they are particularly interesting due to their attractive properties in the context of erasure problems. By applying an alternative approach, we are able to shed new light on general dynamical frames as well as cyclic frames. In particular, we provide a characterization of dynamical frames, which in turn leads to a characterization of cyclic frames.

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arXiv2023

The mystery of Carleson frames

In 2016 Aldroubi et al. constructed the first class of frames having the form $\{T^kφ\}_{k=0}^\infty$ for a bounded linear operator on the underlying Hilbert space. In this paper we show that a subclass of these frames has a number of additional remarkable features that have not been identified for any other frames in the literature. Most importantly, the subfamily obtained by selecting each Nth element from the frame is itself a frame, regardless of the choice of $N\in\mathbb{N}$. Furthermore, the frame property is kept upon removal of an arbitrarily finite number of elements.

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arXiv2021

A survey on frame representations via dynamical sampling

Dynamical sampling deals with representations of a frame $\{ f_k \}_{k=1}^\infty$ as an orbit $\{ T^n φ\}_{n=0}^\infty$ of a linear and possibly bounded operator $T$ acting on the underlying Hilbert space. It is known that the desire of boundedness of the operator $T$ puts severe restrictions on the frame $\{ f_k \}_{k=1}^\infty$. The purpose of the paper is to present an overview of the results in the literature and also discuss various alternative ways of representing a frame; in particular the class of considered frames can be enlarged drastically by allowing representations using only a subset $\{ T^{α(k)} φ\}^\infty_{k=1}$ of the operator orbit $\{ T^n φ\}_{n=0}^\infty$. In general it is difficult to specify appropriate values for the scalars $α(k)$ and the vector $φ;$ however, by accepting an arbitrarily small and controllable deviation between the given frame $\{ f_k \}_{k=1}^\infty$ and $\{ T^{α(k)} φ\}_{k=1}^\infty$ we will be able to do so.

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arXiv2021

Approximately dual pairs of wavelet frames

This paper deals with structural issues concerning wavelet frames and their dual frames. It is known that there exist wavelet frames $\{a^{j/2}ψ( a^j\cdot -kb)\}_{j,k\in \mathbb Z}$ in $L^2(\mathbb R)$ for which no dual frame has wavelet structure. We first generalize this result by proving that there exist wavelet frames for which no approximately dual frame has wavelet structure. Motivated by this we show that by imposing a very mild decay condition on the Fourier transform of the generator $ψ\in L^2(\mathbb R),$ a certain oversampling $\{a^{j/2}ψ( a^j\cdot -kb/N)\}_{j,k\in \mathbb Z}$ indeed has an approximately dual wavelet frame; most importantly, by choosing the parameter $N\in \mathbb N$ sufficiently large we can get as close to perfect reconstruction as desired, which makes the approximate dual frame pairs perform equally well as the classical dual frame pairs in applications.

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arXiv2021

Completion versus removal of redundancy by perturbation

A sequence $\{g_k\}_{k=1}^\infty$ in a Hilbert space $\cal H$ has the expansion property if each $f\in \overline{\text{span}} \{g_k\}_{k=1}^\infty$ has a representation $f= \sum_{k=1}^\infty c_k g_k$ for some scalar coefficients $c_k.$ In this paper we analyze the question whether there exist small norm-perturbations of $\{g_k\}_{k=1}^\infty$ which allow to represent all $f\in \cal H;$ the answer turns out to be yes for frame sequences and Riesz sequences, but no for general basic sequences. The insight gained from the analysis is used to address a somewhat dual question, namely, whether it is possible to remove redundancy from a sequence with the expansion property via small norm-perturbations; we prove that the answer is yes for frames $\{g_k\}_{k=1}^\infty$ such that $g_k\to 0$ as $k\to \infty,$ as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.

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arXiv2021

On approximate operator representations of sequences in Banach spaces

Generalizing results by Halperin et al., Grivaux recently showed that any linearly independent sequence $\{f_k\}_{k=1}^\infty$ in a separable Banach space $X$ can be represented as a suborbit $\{T^{α(k)}φ\}_{k=1}^\infty$ of some bounded operator $T: X\to X.$ In general, the operator $T$ and the powers $α(k)$ are not known explicitly. In this paper we consider approximate representations $\{f_k\}_{k=1}^\infty \approx \{T^{α(k)}φ\}_{k=1}^\infty$ of certain types of sequences $\{f_k\}_{k=1}^\infty.$ In contrast to the results in the literature we are able to be very explicit about the operator $T$ and suitable powers $α(k),$ and we do not need to assume that the sequences are linearly independent. The exact meaning of approximation is defined in a way such that $\{T^{α(k)}φ\}_{k=1}^\infty$ keeps essential features of $\{f_k\}_{k=1}^\infty,$ e.g., in the setting of atomic decompositions and Banach frames. We will present two different approaches. The first approach is universal, in the sense that it applies in general Banach spaces; the technical conditions are typically easy to verify in sequence spaces, but are more complicated in function spaces. For this reason we present a second approach, directly tailored to the setting of Banach function spaces. A number of examples prove that the results apply in arbitrary weighted $\ell^p$-spaces and $L^p$-spaces.

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arXiv2019

Approximate frame representations via iterated operator systems

It is known that it is a very restrictive condition for a frame $\{f_k\}_{k=1}^\infty$ to have a representation $ \{T^n φ\}_{n=0}^\infty$ as the orbit of a bounded operator $T$ under a single generator $φ\in\mathcal{H}.$ In this paper we prove that, on the other hand, any frame can be approximated arbitrarily well by a suborbit $\{T^{α(k)} φ\}_{k=1}^\infty$ of a bounded operator $T$. An important new aspect is that for certain important classes of frames, e.g., frames consisting of finitely supported vectors in $\ell^2(\mathbb{N}),$ we can be completely explicit about possible choices of the operator $T$ and the powers $α(k),k\in \mathbb{N}.$ A similar approach carried out in $L^2(\mathbb{R})$ leads to an approximation of a frame using suborbits of two bounded operators. The results are illustrated with an application to Gabor frames generated by a compactly supported function. The paper is concluded with an appendix which collects general results about frame representations using multiple orbits of bounded operators.

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arXiv2018

Frame Properties of Operator Orbits

We consider sequences in a Hilbert space $\mathcal H$ of the form $(T^nf_0)_{n\in I},$ with a linear operator $T$, the index set being either $I = \mathbb N$ or $I = \mathbb Z$, a vector $f_0\in \mathcal H$, and answer the following two related questions: (a) {\it Which frames for $\mathcal H$ are of this form with an at least closable operator $T$?} and (b) {\it For which bounded operators $T$ and vectors $f_0$ is $(T^nf_0)_{n\in I}$ a frame for $\mathcal H$?} As a consequence of our results, it turns out that an overcomplete Gabor or wavelet frame can never be written in the form $(T^nf_0)_{n\in\mathbb N}$ with a bounded operator $T$. The corresponding problem for $I = \mathbb Z$ remains open. Despite the negative result for Gabor and wavelet frames, the results demonstrate that the class of frames that can be represented in the form $(T^nf_0)_{n\in\mathbb N}$ with a bounded operator $T$ is significantly larger than what could be expected from the examples known so far.

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arXiv2018

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.

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arXiv2018

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 $\{T^n φ\}_{n=0}^\infty$ for some $φ\in {\cal H}$ 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 $\{T^n φ\}_{n=0}^\infty$ has the very particular property that $\{T^n φ\}_{n=0}^{N-1} \cup \{T^n φ\}_{n=N+\ell}^\infty$ is a frame for ${\cal H} $ for all $\ell\in {\mathbf N}_0$. We also prove that frames of the form $\{T^n φ\}_{n=0}^\infty$ are sensitive to the ordering of the elements and to norm-perturbations of the generator $φ$ and the operator $T.$ On the other hand positive stability results are obtained by considering perturbations of the generator $φ$ belonging to an invariant subspace on which $T$ is a contraction.

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arXiv2017

Gabor frames in $\ell^2(\mathbf Z)$ and linear dependence

We prove that an overcomplete Gabor frame in $ \ell^2(\mathbf Z)$ by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in $\ell^2(\mathbf Z)$ with modulation parameter $1/M$ and translation parameter $N$ for some $M,N\in \mathbf N,$ and generated by a finite sequence $g$ in $\ell^2(\mathbf Z)$ with $K$ nonzero entries.

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arXiv2017

Construction of scaling partitions of unity

Partitions of unity in ${\mathbf R}^d$ formed by (matrix) scales of a fixed function appear in many parts of harmonic analysis, e.g., wavelet analysis and the analysis of Triebel-Lizorkin spaces. We give a simple characterization of the functions and matrices yielding such a partition of unity. For invertible expanding matrices, the characterization leads to easy ways of constructing appropriate functions with attractive properties like high regularity and small support. We also discuss a class of integral transforms that map functions having the partition of unity property to functions with the same property. The one-dimensional version of the transform allows a direct definition of a class of nonuniform splines with properties that are parallel to those of the classical B-splines. The results are illustrated with the construction of dual pairs of wavelet frames.

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arXiv2017

B-spline approximations of the Gaussian, their Gabor frame properties, and approximately dual frames

We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order $N$ of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters $a,b>0$ with $ab<1,$ the scaled version of $B_N$ generates Gabor frames for $N$ sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to "almost perfect reconstruction" within any desired error tolerance whenever the product $ab$ is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the $L^p$-spaces, as well in the time-domain as in the frequency domain.

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arXiv2017

The unitary extension principle on locally compact abelian groups

The unitary extension principle (UEP) by Ron and Shen yields conditions for the construction of a multi-generated tight wavelet frame for $L^2(\mr^s)$ based on a given refinable function. In this paper we show that the UEP can be generalized to locally compact abelian groups. In the general setting, the resulting frames are generated by modulates of a collection of functions, via the Fourier transform this corresponds to a generalized shift-invariant system. Both the stationary and the nonstationary case are covered. We provide general constructions, based on B-splines on the group itself as well as on characteristic functions on the dual group. Finally, we consider a number of concrete groups and derive explicit constructions of the resulting frames.

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arXiv2017

An open problem concerning operator representations of frames

Recent research has shown that the properties of overcomplete Gabor frames and frames arising from shift-invariant systems form a precise match with certain conditions that are necessary for a frame in $L^2(\mathbf R)$ to have a representation $\{T^k φ\}_{k=0}^\infty$ for some bounded linear operator $T$ on $L^2(\mathbf R)$ and some $φ\in L^2(\mathbf R).$ However, for frames of this type the existence of such a representation has only been confirmed in the case of Riesz bases. This leads to several open questions connecting dynamical sampling, coherent states, frame theory, and operator theory. The key questions can either be considered in the general functional analytic context of operators on a Hilbert space, or in the specific situation of Gabor frames in $L^2(\mathbf R).$

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arXiv2017

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.

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arXiv2016

Explicit constructions and properties of generalized shift-invariant systems in $L^2(\mathbb{R})$

Generalized shift-invariant (GSI) systems, originally introduced by Hernández, 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 Calderón sum is not necessarily bounded by the lower frame bound. We identify a technical condition implying that the Calderón 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 Hernández 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).

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