arXiv subjects

Marcin Bownik

Publications and source records attributed to Marcin Bownik.

At least 19 recordsLinked to original sources

arXiv2026

Riesz sequences near the critical density

We construct Riesz sequences whose Beurling density is arbitrary close to the critical density in the setting of reproducing kernel Hilbert spaces on metric measure spaces. In particular, our results apply to exponential systems on unbounded sets, nonlocalized Gabor systems and general coherent systems arising from nilpotent Lie groups. The methods used in this paper are based on our previous work on the redundancy of frames and a selector form of Weaver's conjecture.

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arXiv2026

Triebel-Lizorkin spaces and expansive matrices

We survey recent classification theorems for expansive matrices that generate the same anisotropic homogeneous Triebel-Lizorkin function space or sequence space. The function spaces are classified precisely by those matrices for which their associated homogeneous quasi-norms on Euclidean space are equivalent, whereas the sequence spaces are classified by a strictly stronger condition. We unravel this discrepancy between function spaces and sequence spaces by showing that two sequence spaces are retracts of each other whenever the corresponding function spaces are the same.

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arXiv2026

Uniform discretization of continuous frames

Let $H$ be an infinite-dimensional separable Hilbert space and let $(X,d,μ)$ be a metric measure space satisfying the doubling and upper Alhfors regularity conditions at small scale. We prove that every bounded continuous tight frame $Ψ\colon X\rightarrow H$ can be sampled to obtain a frame for $H$, which is uniformly discrete and nearly tight. That is, for every $0<ε<1$, there exist a sampling sequence $\{x_n\}_{n\in\mathbb{N}}$ in $X$ and $r>0$ such that $\inf_{n\neq m}d(x_n,x_m)\geq r$ and $\{Ψ(x_n)\}_{n\in\mathbb{N}}$ is a frame whose ratio of frame bounds is less than $1+ε$. We apply our main result to show that for every nonzero function $g$ in $L^2(\mathbb{R}^d)$ there exists a uniformly discrete set $Λ$ such that the corresponding Gabor system $\{e^{2πibx}g(x-a)\}_{(a,b)\in Λ}$ is a nearly tight frame. We also prove that if $ψ\in L^2(\mathbb{R})$ satisfies the Calderón admissibility condition, then there exists a uniformly discrete set $Γ$ such that wavelet system $\{a^{1/2}ψ(ax-b)\}_{(a,b)\in Γ}$ is a nearly tight frame. Analogous discretization results for exponential frames and spectral subspaces of elliptic differential operators are presented as well.

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arXiv2025

Frame redundancy and Beurling density

We show that the frame measure function of a frame in certain reproducing kernel Hilbert spaces on metric measure spaces is given by the reciprocal of the Beurling density of its index set. In addition, we show that each such frame with Beurling density greater than one contains a subframe with Beurling density arbitrary close to one. This confirms that the concept of frame measure function as introduced by Balan and Landau is a meaningful quantitative definition for the redundancy of a large class of infinite frames. In addition, it shows that the necessary density conditions for sampling in reproducing kernel Hilbert spaces obtained by Führ, Gröchenig, Haimi, Klotz and Romero are optimal. As an application, we also settle the open questions of the existence of frames near the critical density for exponential frames on unbounded sets and for nonlocalized Gabor frames. The techniques used in this paper combine a selector form of Weaver's conjecture and various methods for quantifying the overcompleteness of frames.

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arXiv2025

Are MSF wavelets minimally supported?

Larson's problem asks ``Must the support of the Fourier transform of a wavelet contain a wavelet set?". We give an affirmative answer to a non-measurable variant of this question by proving that the Fourier transform of a wavelet must contain a possibly non-measurable wavelet set. We also provide background results on Larson's problem and propose two new related problems.

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arXiv2025

Extrapolation and Factorization of matrix weights

In this paper we prove the Jones factorization theorem and the Rubio de Francia extrapolation theorem for matrix $\mathcal A_p$ weights. These results answer longstanding open questions in the study of matrix weights. The proof requires the development of the theory of convex-set valued functions and measurable seminorm functions. In particular, we define a convex-set valued version of the Hardy Littlewood maximal operator and construct an appropriate generalization of the Rubio de Francia iteration algorithm, which is central to the proof of both results in the scalar case.

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arXiv2025

On exponential frames near the critical density

Given a relatively compact set $Ω\subseteq \mathbb{R}$ of Lebesgue measure $|Ω|$ and $\varepsilon > 0$, we show the existence of a set $Λ\subseteq \mathbb{R}$ of uniform density $D (Λ) \leq (1+\varepsilon) |Ω|$ such that the exponential system $\{ \exp(2πi λ\cdot) \mathbf{1}_Ω: λ\in Λ\}$ is a frame for $L^2 (Ω)$ with frame bounds $A |Ω|, B |Ω|$ for constants $A,B$ only depending on $\varepsilon$. This solves a problem on the frame bounds of an exponential frame near the critical density posed by Nitzan, Olevskii and Ulanovskii. We also prove an extension to locally compact abelian groups, which improves a result by Agora, Antezana and Cabrelli by providing frame bounds involving the spectrum.

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arXiv2025

Meyer wavelets for rational dilations

We show the existence of smooth band-limited multiresolution analysis (MRA) for any expansive dilation with real entries in any spatial dimension. We then prove the existence of orthonormal Meyer wavelets, which have smooth and compactly supported Fourier transform, for any expansive dilation with rational entries and any spatial dimension. This extends one dimensional results of Auscher. In a converse direction, we show that well-localized orthogonal MRA wavelets, such as Meyer wavelets, can only exist for expansive dilations with rational entries. This shows the optimality of our existence result and extends one dimensional result of Lemarié-Rieusset.

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arXiv2024

Selector form of Weaver's conjecture, Feichtinger's conjecture, and frame sparsification

We show an extension of a probabilistic result of Marcus, Spielman, and Srivastava, which resolved the Kadison-Singer problem, for block diagonal positive semidefinite random matrices. We use this result to show several selector results, which generalize their partition counterparts. This includes a selector form of Weaver's KS$_r$ conjecture for block diagonal trace class operators, which extends a selector result for Bessel sequences, or equivalently rank one matrices, due to Londner and the author. We also show a selector variant of Feichtinger's conjecture for a (possibly infinite) collection of Bessel sequences, extending earlier results for a single Bessel sequence. We prove a generalization of the $R_ε$ conjecture of Casazza, Tremain, and Vershynin for infinite collection of equal norm Bessel sequences. In particular, our selector result yields a conjectured asymptotically optimal bound for a single Bessel sequence in terms of Riesz sequence tightness parameter. We establish an iterated selector form of Weaver's KS$_2$ conjecture and show its applications. This includes a solution of an open problem on nearly unit norm Parseval frames of exponentials, which was posed by Londner and the author. We generalize a discretization result for continuous frames by Freeman and Speegle in two ways. First, we extend their result from the setting of rank one operators to positive trace operator valued measures. Second, we establish a nearly tight discretization of bounded continuous Parseval frames. In particular, our selector result yields an improvement of the result of Nitzan, Olevskii, and Ulanovskii and implies the existence of nearly tight exponential frames for unbounded sets with an explicit control on their frame redundancy.

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arXiv2023

Diagonals of self-adjoint operators II: Non-compact operators

Given a self-adjoint operator $T$ on a separable infinite-dimensional Hilbert space we study the problem of characterizing the set $\mathcal D(T)$ of all possible diagonals of $T$. For operators $T$ with at least two points in their essential spectrum $σ_{ess}(T)$, we give a complete characterization of $\mathcal D(T)$ for the class of self-adjoint operators sharing the same spectral measure as $T$ with a possible exception of multiplicities of eigenvalues at the extreme points of $σ_{ess}(T)$. We also give a more precise description of $\mathcal D(T)$ for a fixed self-adjoint operator $T$, albeit modulo the kernel problem for special classes of operators. These classes consist of operators $T$ for which an extreme point of the essential spectrum $σ_{ess}(T)$ is also an extreme point of the spectrum $σ(T)$. Our results generalize a characterization of diagonals of orthogonal projections by Kadison, Blaschke-type results of Müller and Tomilov, and Loreaux and Weiss, and a characterization of diagonals of operators with finite spectrum by the authors.

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arXiv2023

Diagonals of self-adjoint operators I: compact operators

Given a self-adjoint operator $T$ on a separable infinite-dimensional Hilbert space we study the problem of characterizing the set $\mathcal D(T)$ of all possible diagonals of $T$. For compact operators $T$, we give a complete characterization of diagonals modulo the kernel of $T$. That is, we characterize $\mathcal D(T)$ for the class of operators sharing the same nonzero eigenvalues (with multiplicities) as $T$. Moreover, we determine $\mathcal D(T)$ for a fixed compact operator $T$, modulo the kernel problem for positive compact operators with finite-dimensional kernel. Our results generalize a characterization of diagonals of trace class positive operators by Arveson and Kadison and diagonals of compact positive operators by Kaftal, Loreaux, and Weiss. The proof uses the technique of diagonal-to-diagonal results, which was pioneered in the earlier joint work of the authors with Siudeja.

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arXiv2023

On Akemann-Weaver Conjecture

Akemann and Weaver showed Lyapunov-type theorem for rank one positive semidefinite matrices which is an extension of Weaver's KS$_2$ conjecture that was proven by Marcus, Spielman, and Srivastava in their breakthrough solution of the Kadison-Singer problem. They conjectured that a similar result holds for higher rank matrices. We prove the conjecture of Akemann and Weaver by establishing Lyapunov-type theorem for trace class operators. In the process we prove a matrix discrepancy result for sums of hermitian matrices. This extends rank one result of Kyng, Luh, and Song who established an improved bound in Lyapunov-type theorem of Akemann and Weaver.

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arXiv2022

Stability of iterated dyadic filter banks

This paper examines the frame properties of finitely and infinitely iterated dyadic filter banks. It is shown that the stability of an infinitely iterated dyadic filter bank guarantees that of any associated finitely iterated dyadic filter bank with uniform bounds. Conditions under which the stability of finitely iterated dyadic filter banks with uniform bounds implies that of the infinitely iterated dyadic filter bank are also given. The main result describes a sufficient condition under which the infinitely iterated dyadic filter bank associated with a specific class of finitely supported filters is stable.

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arXiv2021

Marcinkiewicz averages of smooth orthogonal projections on sphere

We construct a single smooth orthogonal projection with desired localization whose average under a group action yields the decomposition of the identity operator. For any full rank lattice $Γ\subset\mathbb R^d$, a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain $\mathbb R^d/Γ$. We also show the existence of a highly localized smooth orthogonal projection, whose Marcinkiewicz average under the action of $SO(d)$, is a multiple of the identity on $L^2(\mathbb S^{d-1})$. As an application we construct highly localized continuous Parseval frames on the sphere.

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arXiv2021

Simultaneous dilation and translation tilings of $\mathbb R^n$

We solve the wavelet set existence problem. That is, we characterize the full-rank lattices $Γ\subset \mathbb R^n$ and invertible $n \times n$ matrices $A$ for which there exists a measurable set $W$ such that $\{W + γ: γ\in Γ\}$ and $\{A^j(W): j\in \mathbb Z\}$ are tilings of $\mathbb R^n$. The characterization is a non-obvious generalization of the one found by Ionascu and Wang, which solved the problem in the case $n = 2$. As an application of our condition and a theorem of Margulis, we also strengthen a result of Dai, Larson, and the second author on the existence of wavelet sets by showing that wavelet sets exist for matrix dilations, all of whose eigenvalues $λ$ satisfy $|λ| \ge 1$. As another application, we show that the Ionascu-Wang characterization characterizes those dilations whose product of two smallest eigenvalues in absolute value is $\ge 1$.

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arXiv2021

A characterization of spaces of homogeneous type induced by continuous ellipsoid covers of $\mathbb R^n$

We study the relationship between the concept of a continuous ellipsoid $Θ$ cover of $\mathbb{R}^n$, which was introduced by Dahmen, Dekel, and Petrushev, and the space of homogeneous type induced by $Θ$. We characterize the class of quasi-distances on $\mathbb{R}^n$ (up to equivalence) which correspond to continuous ellipsoid covers. This places firmly continuous ellipsoid covers as a subclass of spaces of homogeneous type on $\mathbb{R}^n$ satisfying quasi-convexity and $1$-Ahlfors-regularity.

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arXiv2020

Parseval wavelet frames on Riemannian manifold

We construct Parseval wavelet frames in $L^2(M)$ for a general Riemannian manifold $M$ and we show the existence of wavelet unconditional frames in $L^p(M)$ for $1 < p <\infty$. This is made possible thanks to smooth orthogonal projection decomposition of the identity operator on $L^2(M)$, which was recently proven by the authors in arXiv:1803.03634. We also show a characterization of Triebel-Lizorkin $\mathbf F_{p,q}^s(M)$ and Besov $\mathbf B_{p,q}^s(M)$ spaces on compact manifolds in terms of magnitudes of coefficients of Parseval wavelet frames. We achieve this by showing that Hestenes operators are bounded on manifolds $M$ with bounded geometry.

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arXiv2020

A PDE Characterization of Anisotropic Hardy Spaces

We obtain a differential characterization for the anisotropic Hardy space $H_A^p$ by identifying it with a parabolic Hardy space associated with a general continuous group. This allows $H_A^p$ to be defined using a parabolic differential equation of Calderon and Torchinsky. We also provide a classification of dilations corresponding to equivalent anisotropic Hardy spaces with respect to linear transformations.

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