Applied harmonic analysis, frame theory, and operator theoryCopyright (c) 2022 Washington University in St. Louis All rights reserved.
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis
Recent Events in Applied harmonic analysis, frame theory, and operator theoryen-usWed, 02 Feb 2022 13:06:08 PST3600The Discretization Problem for continuous frames.
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/16
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/16Mon, 18 Jul 2016 16:00:00 PDT
There is a long history of creating frames by sampling continuous frames. For instance, Gabor frames are formed by sampling the short time Fourier transform at a lattice. Continuous frames often arise naturally in mathematics and physics, but the sampled frames are usually more useful in application. Using the results of Marcus-Spielman-Srivastava in their solution of the Kadison-Singer problem, we prove that every bounded continuous frame may be sampled to obtain a frame. This is joint work with Darrin Speegle.
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Daniel FreemanBessel sequences from iterated operator actions
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/15
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/15Mon, 18 Jul 2016 18:00:00 PDT
The subject of the talk is motivated by questions related to the new field of Dynamical Sampling which was recently initiated by A. Aldroubi et al. In short, the general Dynamical Sampling problem deals with sequences $(A^kx_i)_{i\in I,\,k=0,\ldots,K_i}$, where $A$ is a linear operator and the $x_i$ are vectors. The question then is for which $A$, $x_i$, and $K_i$ the sequence constitutes a frame for the underlying Hilbert space. Clearly, for this it is necessary that each of the subsequences $(A^kx_i)_{k=0,\ldots,K_i}$ is a Bessel sequence. As this is only interesting when $K_i = \infty$, we consider sequences of the form $(A^kx)_{k\in\mathbb N}$. In order to exploit a comprehensive spectral theory, we restrict ourselves to normal operators $A$ and completely characterize the normal operators $A$ and vectors $x$ for which $(A^kx)_{k\in\mathbb N}$ is a Bessel sequence. The characterization is formulated in terms of the measure $\mu_x := \|E(\cdot)x\|^2$, where $E$ is the spectral measure of the operator $A$: {\it The sequence $(A^kx)_{k\in\mathbb N}$ is a Bessel sequence if and only if {\rm (i)} $\mu_x$ is concentrated on the closed unit disc $\overline{\mathbb D}$, {\rm (ii)} The restriction of $\mu_x$ to the unit circle is absolutely continuous with respect to arc length measure with $L^\infty$-density function, {\rm (iii)} $\mu_x|\mathbb D$ is a Carleson measure.} If time permits, we will apply the results to a certain sampling framework with the heat equation in the background.
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Friedrich PhilippOff the grid low-rank matrix recovery
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/14
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/14Tue, 19 Jul 2016 16:00:00 PDT
Matrix sensing problems capitalize on the assumption that a data matrix of interest is low-rank or it can be well-approximated by a low-rank matrix. This low dimensional structure sometimes arises because the data matrix is obtained by sampling a â€œsmooth" function on a regular (or structured) grid. However, in many practical situations the measurements are taken on an irregular grid (that is accurately known). This results in what we call an "unstructured data matrix" that is a worse fit for the low rank model compared to its regular counterpart and results in degraded reconstruction via rank penalization techniques. In this talk, we propose a modified low-rank matrix recovery work-flow that admits unstructured observations. Specifically, we incorporate into the nuclear-norm minimization problem a regularization operator that (sufficiently accurately) maps structured data to unstructured data. As a result, we are able to compensate for data irregularity. We establish recovery error bounds for our method. Furthermore, we will present numerical experiments, both in the matrix sensing and matrix completion settings, including applications to seismic trace interpolation to demonstrate the potential of the approach. This is joint work with Oscar Lopez, Rajiv Kumar, and Felix Herrmann.
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Ozgur YilmazDuals of Gabor systems and weighted exponentials at the critical density
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/13
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/13Mon, 18 Jul 2016 17:00:00 PDT
A Gabor system can only be a Riesz basis when the Beurling density of its index set is exactly 1. There exist Gabor systems that are Schauder bases but not Riesz basis, but it is not known whether every Gabor Schauder basis must have density 1. For lattice Gabor systems there is a complete characterization in terms of the Zak transform of the generating atom. We investigate the properties of subsets of lattice Gabor systems and systems of weighted exponentials at the critical density, which can still be complete and even minimal. Can these systems be bases? We will show that many properties of the dual system are tied to the behavior of the zeros of the atom and the number of lattice elements that are missing from the lattice.
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Christopher HeilCounterexamples to the B-spline conjecture for Gabor frames
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/12
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/12Mon, 18 Jul 2016 17:30:00 PDT
Frame set problems in Gabor analysis are classical problems that ask the question for which sampling and modulation rates the corresponding time-frequency shifts of a generating window allow for stable reproducing formulas of $L^2$-functions. In this talk we show that the frame set conjecture for B-splines of order two and greater is false. The arguments are based on properties of the Zak transform (also known as the Bloch-Floquet transform and Weil-Brezin transform). Our proof shows that, somewhat surprisingly, even nice Gabor windows in the Feichtinger algebra can have frame sets with a very complicated structure.
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Jakob LemvigOn invariant Graph subspaces
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/11
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/11Tue, 19 Jul 2016 17:30:00 PDT
For unbounded \(2\times 2\) block operator matrices \(B=\begin{pmatrix}A_0&W1\\W_0&A_1\end{pmatrix}\) we investigate the relation between pairs of reducing graph subspaces, solutions to the Riccati equation \[A_1X-XA_0-XW_1X+W_0=0\] and block diagonalization of the operator \(B\). Under mild additional assumptions, we show that such a pair of subspaces decomposes the operator matrix if and only if its domain is invariant for the angular operators associated with the graphs. As a byproduct of our considerations, we suggest a new block diagonalization procedure that resolves related domain issues. In the case when only a single invariant graph subspace is available, we obtain block triangular representations for the operator matrix \(B\). As an application, we provide a way for block diagonalization of a massless two dimensional graphene Hamiltonian. \vspace{6pt} This talk is based on joint work with K.~A.~Makarov and A.~Seelmann.
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Stephan SchmitzKernel-based function approximation on Grassmannians
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/10
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/10Tue, 19 Jul 2016 17:00:00 PDT
Function approximation on compact Riemannian manifolds based on eigenfunctions of the Laplace-Beltrami operator has been widely studied in the recent literature. In this talk, we present numerical experiments associated to the approximation schemes for the Grassmann manifold. Our numerical computations illustrate and match the theoretical findings in the literature.
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Martin EhlerDynamical Sampling and Systems of Iterative Action of Operators
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/9
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/9Tue, 19 Jul 2016 15:00:00 PDT
We consider frames and Bessel systems generated by iterations of the form $\{A^ng: g\in \G,\, 0\le n,
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Akram AldroubiFourier Series on Fractals and the Kaczmarz Algorithm
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/8
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/8Tue, 19 Jul 2016 18:00:00 PDT
The Kaczmarz algorithm is a well-known technique for solving systems of linear equations. We show that the algorithm can be used to construct Fourier series expansions for functions which are square-integrable with respect to a fractal measure (or any singular measure) on the unit interval. The algorithm also gives rise to reproducing kernel subspaces of the Hardy space of the disc with specified boundary representations, which in turn is related to the spectral theory of the backward shift on the Hardy space. This is joint work with John Herr and Palle Jorgensen.
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Eric WeberOptimality beyond the Welch bound: orthoplectic Grassmannian frames as weighted complex projective 2-designs
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/7
https://openscholarship.wustl.edu/iwota2016/special/AppliedHarmonicAnalysis/7Mon, 18 Jul 2016 15:00:00 PDT
Despite being pursued with a lot of dedication in frame theory and quantum information theory, maximal equiangular tight frames have only been confirmed to exist for lowest dimensions. This is especially puzzling since maximal sets of mutually unbiased bases, which are also known to be optimal packings but have a slightly larger number of vectors, are known to exist in complex Hilbert spaces of any prime power dimension. This talk presents a method by which we can construct a set of $M^2+1$ vectors in $M$ complex dimensions that form a tight complex orthoplectic Grassmannian frame that is in addition related to a weighted complex projective 2-design with optimality properties for quantum state tomography.
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Bernhard Bodmann