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Marcin Bownik

Publications and source records attributed to Marcin Bownik.

At least 37 records · Page 2Linked to original sources
arXiv2020

Variable Anisotropic Singular Integral Operators

We introduce the class of variable anisotropic singular integral operators associated to a continuous multi-level ellipsoid cover $Θ$ of $\mathbb{R}^n$ introduced by Dahmen, Dekel, and Petrushev \cite{ddp}. This is an extension of the classical isotropic singular integral operators on $\mathbb{R}^n$ of arbitrary smoothness and their anisotropic analogues for general expansive matrices introduced by the first author \cite{b}. We establish the boundedness of variable anisotropic singular integral operators $T$ on the Hardy spaces with pointwise variable anisotropy $H^p(Θ)$, which were developed by Dekel, Petrushev, and Weissblat \cite{dpw}. In contrast with the general theory of Hardy spaces on spaces of homogenous type, our results work in the full range $0<p\leq 1$.

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arXiv2020

Wavelets on compact abelian groups

Multiresolution analysis (MRA) on compact abelian group $G$ has been constructed with epimorphism as a dilation operator. We show a characterization of scaling sequences of an MRA on $L^p(G)$, $1\le p<\infty$. With the help of this scaling sequence we construct a wavelet orthonormal basis of $L^2(G)$.

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arXiv2019

Multiplication-invariant operators and the classification of LCA group frames

In this paper we study the properties of multiplication invariant (MI) operators acting on subspaces of the vector-valued space $L^2(X;\mathcal H)$. We characterize such operators in terms of range functions by showing that there is an isomorphism between the category of MI spaces (with MI operators as morphisms) and the category of measurable range functions whose morphisms are measurable range operators. We investigate how global properties of an MI operator are reflected by local pointwise properties of its corresponding range operator. We also establish several results about frames generated by multiplications in $L^2(X;\mathcal H)$. This includes the classification of frames of multiplications with respect to unitary equivalence by measurable fields of Gramians. Finally, we show applications of our results in the study of abelian group frames and translation-invariant (TI) operators acting on subspaces of $L^2(\mathcal G)$, where $\mathcal G$ is a locally compact group.

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arXiv2019

On syndetic Riesz sequences

Applying the solution to the Kadison-Singer problem, we show that every subset $\mathcal{S}$ of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials $\left\{ e^{iλx}\right\} _{λ\in Λ}$ such that $Λ\subset\mathbb{Z}$ is a set with gaps between consecutive elements bounded by ${\displaystyle \frac{C}{\left|\mathcal{S}\right|}}$. In the case when $\mathcal{S}$ is an open set we demonstrate, using quasicrystals, how such $Λ$ can be deterministically constructed.

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arXiv2019

Open problems in wavelet theory

We present a collection of easily stated open problems in wavelet theory and we survey the current status of answering them. This includes a problem of Larson on minimally supported frequency wavelets. We show that it has an affirmative answer for MRA wavelets.

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arXiv2018

Measurable selector in Kadison's carpenter's theorem

We show the existence of a measurable selector in Carpenter's Theorem due to Kadison. This solves a problem posed by Jasper and the first author. As an application we obtain a characterization of all possible spectral functions of shift-invariant subspaces of $L^2(\mathbb R^d)$ and Carpenter's Theorem for type I$_\infty$ von Neumann algebras.

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arXiv2018

Smooth orthogonal projections on Riemannian manifold

We construct a decomposition of the identity operator on a Riemannian manifold $M$ as a sum of smooth orthogonal projections subordinate to an open cover of $M$. This extends a decomposition of the real line by smooth orthogonal projection due to Coifman, Meyer and Auscher, Weiss, Wickerhauser, and a similar decomposition when $M$ is the sphere by the first two authors.

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arXiv2018

The Kadison-Singer Problem

We give self-contained presentation of results related to the Kadison-Singer problem, which was recently solved by Marcus, Spielman, and Srivastava. This problem connects with unusually large number of areas including: operator algebras (pure states), set theory (ultrafilters), operator theory (paving), random matrix theory, linear and multilinear algebra, algebraic combinatorics (real stable polynomials), algebraic curves, frame theory, harmonic analysis (Fourier frames), and functional analysis.

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arXiv2018

Lyapunov's Theorem for continuous frames

Akemann and Weaver (2014) have shown a remarkable extension of Weaver's $KS_r$ Conjecture (2004) in the form of approximate Lyapunov's theorem. This was made possible thanks to the breakthrough solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava (2015). In this paper we show a similar type of Lyapunov theorem for continuous frames on non-atomic measure spaces. In contrast with discrete frames, the proof of this result does not rely on the recent solution of the Kadison-Singer problem.

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arXiv2018

Continuous frames and the Kadison-Singer problem

In this paper we survey a recent progress on continuous frames inspired by the solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava. We present an extension of Lyapunov's theorem for discrete frames due to Akemann and Weaver and a similar extension for continuous frames by the author. We also outline a solution of the discretization problem, which was originally posed by Ali, Antoine, and Gazeau, and recently solved by Freeman and Speegle.

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arXiv2017

On Wilson bases in L2(R^d)

A Wilson system is a collection of finite linear combinations of time frequency shifts of a square integrable function. In this paper we use the fact that a Wilson system is a shift-invariant system to explore its relationship with Gabor systems. It is well known that, starting from a tight Gabor frame for $L^{2}(\mathbb{R})$ with redundancy $2$, one can construct an orthonormal Wilson basis for $L^{2}(\mathbb{R})$ whose generator is well localized in the time-frequency plane. In this paper, we show that one can construct multi-dimensional orthonormal Wilson bases starting from tight Gabor frames of redundancy $2^k$ where $k=1, 2, ..., d$. These results generalize most of the known results about the existence of orthonormal Wilson bases.

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arXiv2016

Improved bounds in Weaver and Feichtinger Conjectures

We sharpen the constant in the $KS_2$ conjecture of Weaver \cite{We}, which was validated by Marcus, Spielman, and Srivastava \cite{MSS} in their solution of the Kadison--Singer problem. We then apply this result to prove optimal asymptotic bounds on the size of partitions in the Feichtinger conjecture.

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arXiv2016

The Schur-Horn theorem for unbounded operators with discrete spectrum

We characterize diagonals of unbounded self-adjoint operators on a Hilbert space H that have only discrete spectrum, i.e., with empty essential spectrum. Our result extends the Schur-Horn theorem from a finite dimensional setting to an infinite dimensional Hilbert space, analogous to Kadison's theorem for orthogonal projections, Kaftal and Weiss' results for positive compact operators, and Bownik and Jasper's characterization for operators with finite spectrum. Furthermore, we show that if a symmetric unbounded operator E on H has a nondecreasing unbounded diagonal, then any sequence that weakly majorizes this diagonal is also a diagonal of E.

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arXiv2016

Wavelets for non-expanding dilations and the lattice counting estimate

We show that problems of existence and characterization of wavelets for non-expanding dilations are intimately connected with the geometry of numbers; more specifically, with a bound on the number of lattice points in balls dilated by the powers of a dilation matrix $A \in \mathrm{GL}(n,\mathbb{R})$. This connection is not visible for the well-studied class of expanding dilations since the desired lattice counting estimate holds automatically. We show that the lattice counting estimate holds for all dilations $A$ with $\left|\det{A}\right|\ne 1$ and for almost every lattice $Γ$ with respect to the invariant probability measure on the set of lattices. As a consequence, we deduce the existence of minimally supported frequency (MSF) wavelets associated with such dilations for almost every choice of a lattice. Likewise, we show that MSF wavelets exist for all lattices and and almost every choice of a dilation $A$ with respect to the Haar measure on $\mathrm{GL}(n,\mathbb{R})$.

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arXiv2014

Diagonals of self-adjoint operators with finite spectrum

Given a finite set $X\subseteq\R$ we characterize the diagonals of self-adjoint operators with spectrum $X$. Our result extends the Schur-Horn theorem from a finite dimensional setting to an infinite dimensional Hilbert space analogous to Kadison's theorem for orthogonal projections and the second author's result for operators with three point spectrum.

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arXiv2013

The Schur-Horn theorem for operators with finite spectrum

We characterize the set of diagonals of the unitary orbit of a self-adjoint operator with a finite spectrum. Our result extends the Schur-Horn theorem from a finite dimensional setting to an infinite dimensional Hilbert space analogous to Kadison's theorem for orthogonal projections and the second author's result for operators with three point spectrum.

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