New Trends in Applied Harmonic Analysis, Volume 2
Harmonic Analysis, Geometric Measure Theory, and Applications
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Chapter
Frames of Iterations and Vector-Valued Model Spaces
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Article
Correction to: The structure of group preserving operators
A correction to this paper has been published: https://doi.org/10.1007/s43670-021-00010-6
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Article
Multi-orbital Frames Through Model Spaces
We characterize the normal operators A on \(\ell ^2\) ℓ 2 ...
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Chapter
Local-to-Global Frames and Applications to the Dynamical Sampling Problem
In this paper, we consider systems of vectors in a Hilbert space ℋ \(\mathcal {H}\) ...
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Dynamical sampling on finite index sets
We consider bounded operators A acting iteratively on a finite set of vectors {fi: i ∈ I} in a Hilbert space ℌ and address the problem of providing necessary and sufficient conditions for the collection of iterat...
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The potential added value of Regional Climate Models in South America using a multiresolution approach
This paper aims to identify those regions within the South American continent where the Regional Climate Models (RCMs) have the potential to add value (PAV) compared to their coarser-resolution global forcing....
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Chapter
Data Approximation with Time-Frequency Invariant Systems
In this paper we prove the existence of a time-frequency space that best approximates a given finite set of data. Here best approximation is in the least square sense, among all time-frequency spaces with no m...
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Book
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Book
New Trends in Applied Harmonic Analysis
Sparse Representations, Compressed Sensing, and Multifractal Analysis
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Chapter
Visible and Invisible Cantor Sets
In this chapter we study for which Cantor sets there exists a gauge-function h, such that the h−Hausdorff measure—is positive and finite. We show that the collection of sets for which this is true is dense in the...
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Invariance of a Shift-Invariant Space in Several Variables
In this article we study invariance properties of shift-invariant spaces in higher dimensions. We state and prove several necessary and sufficient conditions for a shift-invariant space to be invariant under a...
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A Dimension Reduction Scheme for the Computation of Optimal Unions of Subspaces
Given a set of points F in a high dimensional space, the problem of finding a union of subspaces ∪i Vi ⊆ RN that best explains the data F increases dramatically with the dimension of RN. In this article, we study...
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Invariance of a Shift-Invariant Space
A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications....
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Book
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Sampling in a Union of Frame Generated Subspaces
A new paradigm in sampling theory has been developed recently by Lu and Do. In this new approach the classical linear model is replaced by a non-linear, but structured, model consisting of a union of subspaces...
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Optimal Non-Linear Models for Sparsity and Sampling
Given a set of vectors (the data) in a Hilbert space ℋ, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspa...
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Density of the Set of Generators of Wavelet Systems
Given a function ψ in \({\cal L}^2({\Bbb R}^d),\) the affine (wavelet) system generated by ψ, associated to an inverti...
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Functions in Sampling Spaces
Sampling theory in spaces other than the space of band-limited functions has recently received considerable attention. This is in part because the band-limitedness assumption is not very realistic in many appl...
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Chapter
Learning the Right Model from the Data
In this chapter we discuss the problem of finding the shift-invariant space model that best fits a given class of observed data F. If the data is known to belong to a fixed—but unknown—shift-invariant space V(Φ) ...
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Accuracy of several multidimensional refinable distributions
Compactly supported distributions f1,..., fr on ℝd are fefinable if each fi is a finite linear combination of the rescaled and translated distributions fj(Ax−k), where the translates k are taken along a lattice Γ...
