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Karlheinz Gröchenig

Publications and source records attributed to Karlheinz Gröchenig.

At least 55 records · Page 3Linked to original sources
arXiv2017

Sampling Theorems for Shift-invariant Spaces, Gabor Frames, and Totally Positive Functions

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as $ \hat g(ξ)= \prod_{j=1}^n (1+2πiδ_jξ)^{-1} \, e^{-c ξ^2}$ for $δ_1,\ldots,δ_n\in \mathbb{R}, c >0$ (in which case $g$ is called totally positive of Gaussian type). In analogy to Beurling's sampling theorem for the Paley-Wiener space of entire functions, we prove that every separated set with lower Beurling density $>1$ is a sampling set for the shift-invariant space generated by such a $g$. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a subtle connection between sampling in shift-invariant spaces and the theory of Gabor frames, we show that the set of phase-space shifts of $g$ with respect to a rectangular lattice $α\mathbb{Z} \times β\mathbb{Z}$ forms a frame, if and only if $αβ<1$. This solves an open problem going back to Daubechies in 1990 for the class of totally positive functions of Gaussian type. The proof strategy involves the connection between sampling in shift-invariant spaces and Gabor frames, a new characterization of sampling sets "without inequalities" in the style of Beurling, new properties of totally positive functions, and the interplay between zero sets of functions in a shift-invariant space and functions in the Bargmann-Fock space.

math.FA↗
arXiv2017

Density of Sampling and Interpolation in Reproducing Kernel Hilbert Spaces

We derive necessary density conditions for sampling and for interpolation in general reproducing kernel Hilbert spaces satisfying some natural conditions on the geometry of the space and the reproducing kernel. If the volume of shells is small compared to the volume of balls (weak annular decay property) and if the kernel possesses some off-diagonal decay or even some weaker form of localization, then there exists a critical density $D$ with the following property: a set of sampling has density $\geq D$, whereas a set of interpolation has density $\leq D$. The main theorem unifies many known density theorems in signal processing, complex analysis, and harmonic analysis. For the special case of bandlimited function we recover Landau's fundamental density result. In complex analysis we rederive a critical density for generalized Fock spaces. In harmonic analysis we obtain the first general result about the density of coherent frames.

math.FA↗
arXiv2017

Sampling and Reconstruction in Distinct Subspaces Using Oblique Projections

We study reconstruction operators on a Hilbert space that are exact on a given reconstruction subspace. Among those the reconstruction operator obtained by the least squares fit has the smallest operator norm, and therefore is most stable with respect to noisy measurements. We then construct the operator with the smallest possible quasi-optimality constant, which is the most stable with respect to a systematic error appearing before the sampling process (model uncertainty). We describe how to vary continuously between the two reconstruction methods, so that we can trade stability for quasi-optimality. As an application we study the reconstruction of a compactly supported function from nonuniform samples of its Fourier transform.

math.NA↗
arXiv2017

Orthonormal Bases in the Orbit of Square-Integrable Representations of Nilpotent Lie Groups

Let $G$ be a connected, simply connected nilpotent group and $π$ be a square-integrable irreducible unitary representation modulo its center $Z(G)$ on $L^2(\mathbf{R}^d)$. We prove that under reasonably weak conditions on $G$ and $π$ there exist a discrete subset $Γ$ of $G/Z(G)$ and some (relatively) compact set $F \subseteq \mathbf{R}^d$ such that $$\bigl \{ |F|^{-1/2} \hspace{2pt} π(γ) 1_F \mid γ\in Γ\bigr\}$$ forms an orthonormal basis of $L^2(\mathbf{R}^d)$. This construction generalizes the well-known example of Gabor orthonormal bases in time-frequency analysis. The main theorem covers graded Lie groups with one-dimensional center. In the presence of a rational structure, the set $Γ$ can be chosen to be a uniform subgroup of $G/Z$.

math.RT↗
arXiv2016

A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators

This chapter offers a detailed survey on intrinsically localized frames and the corresponding matrix representation of operators. We re-investigate the properties of localized frames and the associated Banach spaces in full detail. We investigate the representation of operators using localized frames in a Galerkin-type scheme. We show how the boundedness and the invertibility of matrices and operators are linked and give some sufficient and necessary conditions for the boundedness of operators between the associated Banach spaces.

math.FA↗
arXiv2016

The Cramer-Wold theorem on quadratic surfaces and Heisenberg uniqueness pairs

Two measurable sets $S, Λ\subseteq \mathcal{R}^d$ form a Heisenberg uniqueness pair, if every bounded measure $μ$ with support in S whose Fourier transform vanishes on Λ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathcal{R}^d$. As a corollary we obtain a new, surprising version of the classical Cramér-Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients .

math.CA↗
arXiv2016

Stability of Gabor frames under small time Hamiltonian evolutions

We consider Hamiltonian deformations of Gabor systems, where the window evolves according to the action of a Schrödinger propagator and the phase-space nodes evolve according to the corresponding Hamiltonian flow. We prove the stability of the frame property for small times and Hamiltonians consisting of a quadratic polynomial plus a potential in the Sjöstrand class with bounded second order derivatives. This answers a question raised in [de Gosson, M. Symplectic and Hamiltonian Deformations of Gabor Frames. Appl. Comput. Harmon. Anal. Vol. 38 No.2, (2015) p.196--221.]

math-ph↗
arXiv2016

Completeness of Gabor systems

We investigate the completeness of Gabor systems with respect to several classes of window functions on rational lattices. Our main results show that the time-frequency shifts of every finite linear combination of Hermite functions with respect to a rational lattice are complete in L2(R), thus generalizing a remark of von Neumann (and proved by Bargmann, Perelomov et al.). An analogous result is proven for functions that factor into certain rational functions and the Gaussian. The results are also interesting from a conceptual point of view since they show a vast difference between the completeness and the frame property of a Gabor system. In the terminology of physics we prove new results about the completeness of coherent state subsystems.

math-ph↗
arXiv2015

What is Variable Bandwidth?

We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator $A_pf = - (pf')'$ where $p>0$ is a strictly positive function. Denote by $c_Λ (A_p)$ the orthogonal projection of $A_p$ corresponding to the spectrum of $A_p$ in $Λ$, the range of this projection is the space of functions of variable bandwidth with spectral set in $Λ$. We will develop the basic theory of these function spaces. First, we derive (nonuniform) sampling theorems, second, we prove necessary density conditions in the style of Landau. Roughly, for a spectrum $Λ= [0,Ω] $ the main results say that, in a neighborhood of $x\in R$, a function of variable bandwidth behaves like a bandlimited function with local bandwidth $(Ω/ p(x))^{1/2}$. Although the formulation of the results is deceptively similar to the corresponding results for classical bandlimited functions, the methods of proof are much more involved. On the one hand, we use the oscillation method from sampling theory and frame theoretic methods, on the other hand, we need the precise spectral theory of Sturm-Liouville operators and the scattering theory of one-dimensional Schrödinger operators.

math.FA↗
arXiv2015

Implementation of discretized Gabor frames and their duals

The usefulness of Gabor frames depends on the easy computability of a suitable dual window. This question is addressed under several aspects: several versions of Schulz's iterative algorithm for the approximation of the canonical dual window are analyzed for their numerical stability. For Gabor frames with totally positive windows or with exponential B-splines a direct algorithm yields a family of exact dual windows with compact support. It is shown that these dual windows converge exponentially fast to the canonical dual window.

math.NA↗
arXiv2015

New atomic decompositons for Bergman spaces on the unit ball

We derive atomic decompositions and frames for weighted Bergman spaces of several complex variables on the unit ball in the spirit of Coifman, Rochberg, and Luecking. In contrast to our predecessors, we use group theoretic methods, in particular the representation theory of the discrete series of $SU(n,1)$ and its covering groups. One of the benefits is a much larger class of admissible

math.CV↗
arXiv2015

Deformation of Gabor systems

We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurling's characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames "without inequalities" from lattices to non-uniform sets.

math.FA↗
arXiv2015

Weaving Frames

We study an intriguing question in frame theory we call "Weaving Frames" that is partially motivated by preprocessing of Gabor frames. Two frames $\{φ_i\}_{i\in I}$ and $\{ψ_i \}_{i\in I}$ for a Hilbert space ${\mathbb H}$ are woven if there are constants $0<A \le B $ so that for every subset $σ\subset I$, the family $\{φ_i\}_{i\in σ} \cup \{ψ_i\}_{i\in σ^c}$ is a frame for $\mathbb H$ with frame bounds $A,B$. Fundamental properties of woven frames are developed and key differences between weaving Riesz bases and weaving frames are considered. In particular, it is shown that a Riesz basis cannot be woven with a redundant frame. We also introduce an apparently weaker form of weaving but show that it is equivalent to weaving. Weaving frames has potential applications in wireless sensor networks that require distributed processing under different frames, as well as preprocessing of signals using Gabor frames.

math.FA↗
arXiv2015

Generalized Metaplectic Operators and the Schrödinger Equation with a Potential in the Sjöstrand Class

It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential opertators in a Sjöstrand class enjoys the same decay properties. We study the behavior of these generalized metaplectic operators and represent them by Fourier integral operators. Our main result shows that the one-parameter group generated by a Hamiltonian operator with a potential in the Sjöstrand class consists of generalized metaplectic operators. As a consequence, the Schrödinger equation preserves the phase-space concentration, as measured by modulation space norms.

math.AP↗
arXiv2014

On Minimal Trajectories for Mobile Sampling of Bandlimited Fields

We study the design of sampling trajectories for stable sampling and the reconstruction of bandlimited spatial fields using mobile sensors. The spectrum is assumed to be a symmetric convex set. As a performance metric we use the path density of the set of sampling trajectories that is defined as the total distance traveled by the moving sensors per unit spatial volume of the spatial region being monitored. Focussing first on parallel lines, we identify the set of parallel lines with minimal path density that contains a set of stable sampling for fields bandlimited to a known set. We then show that the problem becomes ill-posed when the optimization is performed over all trajectories by demonstrating a feasible trajectory set with arbitrarily low path density. However, the problem becomes well-posed if we explicitly specify the stability margins. We demonstrate this by obtaining a non-trivial lower bound on the path density of an arbitrary set of trajectories that contain a sampling set with explicitly specified stability bounds.

cs.IT↗
arXiv2014

Time-Frequency Localization Operators and a Berezin Transform

Time-frequency localization operators are a quantization procedure that maps symbols on $R^{2d}$ to operators and depends on two window functions. We study the range of this quantization and its dependence on the window functions. If the short-time Fourier transform of the windows does not have any zero, then the range is dense in the Schatten $p$-classes. The main tool is new version of the Berezin transform associated to operators on $L^2(R^d)$. Although some results are analogous to results about Toeplitz operators on spaces of holomorphic functions, the absence of a complex structure requires the development of new methods that are based on time-frequency analysis.

math.FA↗
arXiv2014

On accumulated spectrograms

We study the eigenvalues and eigenfunctions of the time-frequency localization operator $H_Ω$ on a domain $Ω$ of the time-frequency plane. The eigenfunctions are the appropriate prolate spheroidal functions for an arbitrary domain $Ω$. Indeed, in analogy to the classical theory of Landau-Slepian-Pollak, the number of eigenvalues of $H_Ω$ in $[1-δ, 1]$ is equal to the measure of $Ω$ up to an error term depending on the perimeter of the boundary of $Ω$. Our main results show that the spectrograms of the eigenfunctions corresponding to the large eigenvalues (which we call the accumulated spectrogram) form an approximate partition ofunity of the given domain $Ω$. We derive both asymptotic, non-asymptotic, and weak $L^2$ error estimates for the accumulated spectrogram. As a consequence the domain $Ω$ can be approximated solely from the spectrograms of eigenfunctions without information about their phase.

math.CA↗