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Karlheinz Gröchenig

Publications and source records attributed to Karlheinz Gröchenig.

At least 19 recordsLinked to original sources
arXiv2026

Gabor Frames of Totally Positive Functions: A Complete Characterization

We prove that the set of time-frequency shifts $\{e^{2πi βl t} g(t-αk) : k,l \in \mathbb{Z}\}$ with a continuous, integrable totally positive function $g$ and lattice parameters $α,β>0$ generates a frame for $L^2(\mathbb{R})$ if and only if $αβ<1$. This fully settles the so-called frame set problem for the class of totally positive functions. As a closely related result we prove a sharp Kadets-type theorem for every shift-invariant space generated by a continuous totally positive function. The proofs are based on Fredholm theory and limit-operator theory. A formalization of our main result in Lean 4 is also provided.

math.FA↗
arXiv2026

Quantitative Constraints for Stable Sampling on the Sphere

We derive quantitative volume constraints for sampling measures $μ_t$ on the unit sphere $\mathbb{S}^d$ that satisfy Marcinkiewicz-Zygmund inequalities of order $t$. Using precise localization estimates for Jacobi polynomials, we obtain explicit upper and lower bounds on the $μ_t$-mass of geodesic balls at the natural scale $t^{-1}$. Whereas constants are typically left implicit in the literature, we place special emphasis on fully explicit constants, and the results are genuinely quantitative. Moreover, these bounds yield quantitative constraints for the $s$-dimensional Hausdorff volume of Marcinkiewicz-Zygmund sampling sets and, in particular, optimal lower bounds for the length of Marcinkiewicz-Zygmund curves.

math.NA↗
arXiv2025

Quantitative estimates: How well does the discrete Fourier transform approximate the Fourier transform on $\mathbb{R}$

In order to compute the Fourier transform of a function $f$ on the real line numerically, one samples $f$ on a grid and then takes the discrete Fourier transform. We derive exact error estimates for this procedure in terms of the decay and smoothness of $f$. The analysis provides a new recipe of how to relate the number of samples, the sampling interval, and the grid size.

math.NA↗
arXiv2025

The Lifting Property for Frame Multipliers and Toeplitz Operators

Frame multipliers are an abstract version of Toeplitz operators in frame theory and consist of a composition of a multiplication operator with the analysis and synthesis operators. Whereas the boundedness properties of frame multipliers on Banach spaces associated to a frame, so-called coorbit spaces, are well understood, their invertibility is much more difficult. We show that frame multipliers with a positive symbol are Banach space isomorphisms between the corresponding coorbit spaces. The results resemble the lifting theorems in the theory of Besov spaces and modulation spaces. Indeed, the application of the abstract lifting theorem to Gabor frames yields a new lifting theorem between modulation spaces. A second application to Fock spaces yields isomorphisms between weighted Fock spaces. The main techniques are the theory of localized frames and existence of inverse-closed matrix algebras.

math.FA↗
arXiv2025

A Characterization of Metaplectic Time-Frequency Representations

We characterize all time-frequency representations that satisfy a general covariance property: any weak*-continuous bilinear mapping that intertwines time-frequency shifts on the configuration space with time-frequency shifts on phase space is a multiple of a metaplectic time-frequency representation.

math.FA↗
arXiv2025

More Uncertainty Principles for Metaplectic Time-Frequency Representations

We develop a method for the transfer of an uncertainty principle for the short-time Fourier transform or a Fourier pair to an uncertainty principle for a sesquilinear or quadratic metaplectic time-frequency representation. In particular, we derive Beurling-type and Hardy-type uncertainty principles for metaplectic time-frequency representations.

math.FA↗
arXiv2024

Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation

We build on our recent results on the Lipschitz dependence of the extreme spectral values of one-parameter families of pseudodifferential operators with symbols in a weighted Sjöstrand class. We prove that larger symbol classes lead to Hölder continuity with respect to the parameter. This result is then used to investigate the behavior of frame bounds of families of Gabor systems $\mathcal{G}(g,αΛ)$ with respect to the parameter $α>0$, where $Λ$ is a set of non-uniform, relatively separated time-frequency shifts, and $g\in M^1_s(\mathbb{R}^d)$, $0\leq s\leq 2$. In particular, we show that the frame bounds depend continuously on $α$ if $g\in M^1(\mathbb{R}^d)$, and are Hölder continuous if $g\in M^1_s(\mathbb{R}^d)$, $0<s\leq 2$, with the Hölder exponent explicitly given.

math.FA↗
arXiv2024

Benedicks-type uncertainty principle for metaplectic time-frequency representations

Metaplectic Wigner distributions are joint time-frequency representations that are parametrized by a symplectic matrix and generalize the short-time Fourier transform and the Wigner distribution. We investigate the question which metaplectic Wigner distributions satisfy an uncertainty principle in the style of Benedicks and Amrein-Berthier. That is, if the metaplectic Wigner distribution is supported on a set of finite measure, must the functions then be zero? While this statement holds for the short-time Fourier transform, it is false for some other natural time-frequency representations. We provide a full characterization of the class of metaplectic Wigner distributions which exhibit an uncertainty principle of this type, both for sesquilinear and quadratic versions.

math.FA↗
arXiv2023

Sampling theorems with derivatives in shift-invariant spaces generated by periodic exponential B-splines

We derive sufficient conditions for sampling with derivatives in shift-invariant spaces generated by a periodic exponential B-spline. The sufficient conditions are expressed with a new notion of measuring the gap between consecutive samples. These conditions are near optimal, and, in particular, they imply the existence of sampling sets with lower Beurling density arbitrarily close to the necessary density.

math.FA↗
arXiv2023

Variable Bandwidth via Wilson bases

We introduce a new concept of variable bandwidth that is based on the truncation of Wilson expansions. For this model we derive both (nonuniform) sampling theorems, the complete reconstruction of $f$ from its samples, and necessary density conditions for sampling.

math.FA↗
arXiv2023

t-design curves and mobile sampling on the sphere

In analogy to classical spherical t-design points, we introduce the concept of t-design curves on the sphere. This means that the line integral along a t-design curve integrates polynomials of degree t exactly. For low degrees we construct explicit examples. We also derive lower asymptotic bounds on the lengths of t-design curves. Our main results prove the existence of asymptotically optimal t-design curves in the Euclidean 2-sphere and the existence of t-design curves in the d-sphere.

math.MG↗
arXiv2023

Spectral Subspaces of Sturm-Liouville Operators and Variable Bandwidth

We study spectral subspaces of the Sturm-Liouville operator $f \mapsto -(pf')'$ on $\mathbb{R}$, where $p$ is a positive, piecewise constant function. Functions in these subspaces can be thought of as having a local bandwidth determined by $1/\sqrt{p}$. Using the spectral theory of Sturm-Liouville operators, we make the reproducing kernel of these spectral subspaces more explicit and compute it completely in certain cases. As a contribution to sampling theory, we then prove necessary density conditions for sampling and interpolation in these subspaces and determine the critical density that separates sets of stable sampling from sets of interpolation.

math.CA↗
arXiv2023

Lipschitz Continuity of Spectra of Pseudodifferential Operators in a Weighted Sjöstrand Class and Gabor Frame Bounds

We study one-parameter families of pseudodifferential operators whose Weyl symbols are obtained by dilation and a smooth deformation of a symbol in a weighted Sjöstrand class. We show that their spectral edges are Lipschitz continuous functions of the dilation or deformation parameter. Suitably local estimates hold also for the edges of every spectral gap. These statements extend Bellissard's seminal results on the Lipschitz continuity of spectral edges for families of operators with periodic symbols to a large class of symbols with only mild regularity assumptions. The abstract results are used to prove that the frame bounds of a family of Gabor systems $\mathcal{G}(g,αΛ)$, where $Λ$ is a set of non-uniform time-frequency shifts, $α>0$, and $g\in M^1_2(\mathbb{R}^d)$, are Lipschitz continuous functions in $α$. This settles a question about the precise blow-up rate of the condition number of Gabor frames near the critical density.

math.FA↗
arXiv2022

Gauss Quadrature for Freud Weights, Modulation Spaces, and Marcinkiewicz-Zygmund Inequalities

We study Gauss quadrature for Freud weights and derive worst case error estimates for functions in a family of associated Sobolev spaces. For the Gaussian weight $e^{-πx^2}$ these spaces coincide with a class of modulation spaces which are well-known in (time-frequency) analysis and also appear under the name of Hermite spaces. Extensions are given to more general sets of nodes that are derived from Marcinkiewicz-Zygmund inequalities. This generalization can be interpreted as a stability result for Gauss quadrature.

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