arXiv subjects

Eric S. Weber

Publications and source records attributed to Eric S. Weber.

17 recordsLinked to original sources

arXiv2025

The Kaczmarz Algorithm in Hilbert $C^{*}$-modules

The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert $C^*$-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert $C(X)$-modules can be generated by the Kaczmarz algorithm and realized as orbits of bounded operators.

math.FA↗

arXiv2024

Operator orbit frames and frame-like Fourier expansions

Frames in a Hilbert space that are generated by operator orbits are vastly studied because of the applications in dynamic sampling and signal recovery. We demonstrate in this paper a representation theory for frames generated by operator orbits that provides explicit constructions of the frame and the operator when the operators are not surjective. It is known that the Kaczmarz algorithm for stationary sequences in Hilbert spaces generates a frame that arises from an operator orbit where the operator is not surjective. In this paper, we show that every frame generated by a not surjective operator in any Hilbert space arises from the Kaczmarz algorithm. Furthermore, we show that the operators generating these frames are similar to rank one perturbations of unitary operators. After this, we describe a large class of operator orbit frames that arise from Fourier expansions for singular measures. Moreover, we classify all measures that possess frame-like Fourier expansions arising from two-sided operator orbit frames. Finally, we show that measures that possess frame-like Fourier expansions arising from two-sided operator orbits are weighted Lebesgue measure with weight satisfying a weak $A_{2}$ condition, even in the non-frame case. We also use these results to classify measures with other types of frame-like Fourier expansions.

math.FA↗

arXiv2024

Fourier series for singular measures in higher dimensions

For multi-variable finite measure spaces, we present in this paper a new framework for non-orthogonal $L^2$ Fourier expansions. Our results hold for probability measures $μ$ with finite support in $\mathbb{R}^d$ that satisfy a certain disintegration condition that we refer to as ``slice-singular''. In this general framework, we present explicit $L^{2}(μ)$-Fourier expansions, with Fourier exponentials having positive Fourier frequencies in each of the d coordinates. Our Fourier representations apply to every $f \in L^2(μ)$, are based on an extended Kaczmarz algorithm, and use a new recursive $μ$ Rokhlin disintegration representation. In detail, our Fourier series expansion for $f$ is in terms of the multivariate Fourier exponentials $\{e_n\}$, but the associated Fourier coefficients for $f$ are now computed from a Kaczmarz system $\{g_n\}$ in $L^{2}(μ)$ which is dual to the Fourier exponentials. The $\{g_n\}$ system is shown to be a Parseval frame for $L^{2}(μ)$. Explicit computations for our new Fourier expansions entail a detailed analysis of subspaces of the Hardy space on the polydisk, dual to $L^{2}(μ)$, and an associated d-variable Normalized Cauchy Transform. Our results extend earlier work for measures $μ$ in one and two dimensions, i.e., $d=1 (μ$ singular), and $d=2 (μ$ assumed slice-singular). Here our focus is the extension to the cases of measures $μ$ in dimensions $d >2$. Our results are illustrated with the use of explicit iterated function systems (IFSs), including the IFS generated Menger sponge for $d=3$.

math.FA↗

arXiv2023

Fiduciary Responsibility: Facilitating Public Trust in Automated Decision Making

Automated decision-making systems are being increasingly deployed and affect the public in a multitude of positive and negative ways. Governmental and private institutions use these systems to process information according to certain human-devised rules in order to address social problems or organizational challenges. Both research and real-world experience indicate that the public lacks trust in automated decision-making systems and the institutions that deploy them. The recreancy theorem argues that the public is more likely to trust and support decisions made or influenced by automated decision-making systems if the institutions that administer them meet their fiduciary responsibility. However, often the public is never informed of how these systems operate and resultant institutional decisions are made. A ``black box'' effect of automated decision-making systems reduces the public's perceptions of integrity and trustworthiness. The result is that the public loses the capacity to identify, challenge, and rectify unfairness or the costs associated with the loss of public goods or benefits. The current position paper defines and explains the role of fiduciary responsibility within an automated decision-making system. We formulate an automated decision-making system as a data science lifecycle (DSL) and examine the implications of fiduciary responsibility within the context of the DSL. Fiduciary responsibility within DSLs provides a methodology for addressing the public's lack of trust in automated decision-making systems and the institutions that employ them to make decisions affecting the public. We posit that fiduciary responsibility manifests in several contexts of a DSL, each of which requires its own mitigation of sources of mistrust. To instantiate fiduciary responsibility, a Los Angeles Police Department (LAPD) predictive policing case study is examined.

cs.CY↗

arXiv2022

The Persistence Landscapes of Affine Fractals

We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations. Given an iterated function system of affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes which intertwines the action of the iterated function system. This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. We present several examples of the theory as well as confirm the main results through simulations.

math.AT↗

arXiv2019

Almost-nowhere intersection of Cantor sets, and sufficient sampling of their cumulative distribution functions

Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere (with respect to their respective invariant measures) intersection.

math.CA↗

arXiv2019

Stability of the Kaczmarz Reconstruction for Stationary Sequences

The Kaczmarz algorithm is an iterative method to reconstruct an unknown vector $f$ from inner products $\langle f , φ_{n} \rangle $. We consider the problem of how additive noise affects the reconstruction under the assumption that $\{ φ_{n} \}$ form a stationary sequence. Unlike other reconstruction methods, such as frame reconstructions, the Kaczmarz reconstruction is unstable in the presence of noise. We show, however, that the reconstruction can be stabilized by relaxing the Kaczmarz algorithm; this relaxation corresponds to Abel summation when viewed as a reconstruction on the unit disc. We show, moreover, that for certain noise profiles, such as those that lie in $H^{\infty}(\mathbb{D})$ or certain subspaces of $H^{2}(\mathbb{D})$, the relaxed version of the Kaczmarz algorithm can fully remove the corruption by noise in the inner products. Using the spectral representation of stationary sequences, we show that our relaxed version of the Kaczmarz algorithm also stabilizes the reconstruction of Fourier series expansions in $L^2(μ)$ when $μ$ is singular.

math.FA↗

arXiv2019

A Kaczmarz Algorithm for Solving Tree Based Distributed Systems of Equations

The Kaczmarz algorithm is an iterative method for solving systems of linear equations. We introduce a modified Kaczmarz algorithm for solving systems of linear equations in a distributed environment, i.e. the equations within the system are distributed over multiple nodes within a network. The modification we introduce is designed for a network with a tree structure that allows for passage of solution estimates between the nodes in the network. We prove that the modified algorithm converges under no additional assumptions on the equations. We demonstrate that the algorithm converges to the solution, or the solution of minimal norm, when the system is consistent. We also demonstrate that in the case of an inconsistent system of equations, the modified relaxed Kaczmarz algorithm converges to a weighted least squares solution as the relaxation parameter approaches $0$.

math.NA↗

arXiv2018

The Dual Kaczmarz Algorithm

The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector $x$ in a (separable) Hilbert space from the inner-products $\{\langle x, ϕ_{n} \rangle\}$. The Kaczmarz algorithms defines a sequence of approximations from the sequence $\{\langle x, ϕ_{n} \rangle\}$; these approximations only converge to $x$ when $\{ϕ_{n}\}$ is ${effective}$. We dualize the Kaczmarz algorithm so that $x$ can be obtained from $\{\langle x, ϕ_{n} \rangle\}$ by using a second sequence $\{ψ_{n}\}$ in the reconstruction. This allows for the recovery of $x$ even when the sequence $\{ϕ_{n}\}$ is not effective; in particular, our dualization yields a reconstruction when the sequence $\{ϕ_{n}\}$ is $almost$ $effective$. We also obtain some partial results characterizing when the sequence of approximations from $\{\langle x, ϕ_{n} \rangle\}$ using $\{ψ_{n}\}$ converges to $x$, in which case $\{(ϕ_n, ψ_n)\}$ is called an $effective$ $pair$.

math.FA↗

arXiv2018

A Characterization of Boundary Representations of Positive Matrices in the Hardy Space via the Abel Product

Spectral measures give rise to a natural harmonic analysis on the unit disc via a boundary representation of a positive matrix arising from a spectrum of the measure. We consider in this paper the reverse: for a positive matrix in the Hardy space of the unit disc we consider which measures, if any, yield a boundary representation of the positive matrix. We prove a characterization of those representing measures via a matrix identity by introducing a new operator product called the Abel Product.

math.FA↗

arXiv2017

A Paley-Wiener Type Theorem for Singular Measures on $\mathbb{T}$

For a fixed singular Borel probability measure $μ$ on $\mathbb{T}$, we give several characterizations of when an entire function is the Fourier transform of some $f \in L^2(μ)$. The first characterization is given in terms of criteria for sampling functions of the form $\hat{f}$ when $f \in L^2(μ)$. The second characterization is given in terms of criteria for interpolation of bounded sequences on $\mathbb{N}_{0}$ by $\hat{f}$. Both characterizations use the construction of Fourier series for $f \in L^2(μ)$ demonstrated in Herr and Weber via the Kaczmarz algorithm and classical results concerning the Cauchy transform of $μ$.

math.CV↗

arXiv2017

A matrix characterization of boundary representations of positive matrices in the Hardy space

Spectral measures give rise to a natural harmonic analysis on the unit disc via a boundary representation of a positive matrix arising from a spectrum of the measure. We consider in this paper the reverse: for a positive matrix in the Hardy space of the unit disc we consider which measures, if any, yield a boundary representation of the positive matrix. We introduce a potential characterization of those measures via a matrix identity and show that the characterization holds in several important special cases.

math.FA↗

arXiv2016

A Fast Fourier Transform for Fractal Approximations

We consider finite approximations of a fractal generated by an iterated function system of affine transformations on $\mathbb{R}^d$ as a discrete set of data points. Considering a signal supported on this finite approximation, we propose a Fast (Fractal) Fourier Transform by choosing appropriately a second iterated function system to generate a set of frequencies for a collection of exponential functions supported on this finite approximation. Since both the data points of the fractal approximation and the frequencies of the exponential functions are generated by iterated function systems, the matrix representing the Discrete Fourier Transform (DFT) satisfies certain recursion relations, which we describe in terms of Diţǎ's construction for large Hadamard matrices. These recursion relations allow for the DFT matrix calculation to be reduced in complexity to O(N log N ), as in the case of the classical FFT.

math.FA↗

arXiv2016

Fourier Series for Singular Measures

Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $μ$ on $[0,1)$, every $f\in L^2(μ)$ possesses a Fourier series of the form $f(x)=\sum_{n=0}^{\infty}c_ne^{2πinx}$. We show that the coefficients $c_{n}$ can be computed in terms of the quantities $\hat{f}(n) = \int_{0}^{1} f(x) e^{-2πi n x} d μ(x)$. We also demonstrate a Shannon-type sampling theorem for functions that are in a sense $μ$-bandlimited.

math.FA↗

arXiv2016

Positive Matrices in the Hardy Space with Prescribed Boundary Representations via the Kaczmarz Algorithm

For a singular probability measure $μ$ on the circle, we show the existence of positive matrices on the unit disc which admit a boundary representation on the unit circle with respect to $μ$. These positive matrices are constructed in several different ways using the Kaczmarz algorithm. Some of these positive matrices correspond to the projection of the Szegő kernel on the disc to certain subspaces of the Hardy space corresponding to the normalized Cauchy transform of $μ$. Other positive matrices are obtained which correspond to subspaces of the Hardy space after a renormalization, and so are not projections of the Szegő kernel. We show that these positive matrices are a generalization of a spectrum or Fourier frame for $μ$, and the existence of such a positive matrix does not require $μ$ to be spectral.

math.FA↗

arXiv2016

Sampling in de Branges Spaces and Naimark Dilation

We consider the problem of sampling in de Branges spaces and develop some necessary conditions and some sufficient conditions for sampling sequences, which generalize some well-known sampling results in the Paley-Wiener space. These conditions are obtained by identifying the main construction with Naimark dilation of frames--embedding the de Branges space into a larger de Branges space while embedding the kernel functions associated with a sampling sequence into a Riesz basis for the larger space.

math.CV↗

arXiv2015

Fourier Frames for the Cantor-4 Set

The measure supported on the Cantor-4 set constructed by Jorgensen-Pedersen is known to have a Fourier basis, i.e. that it possess a sequence of exponentials which form an orthonormal basis. We construct Fourier frames for this measure via a dilation theory type construction. We expand the Cantor-4 set to a 2 dimensional fractal which admits a representation of a Cuntz algebra. Using the action of this algebra, an orthonormal set is generated on the larger fractal, which is then projected onto the Cantor-4 set to produce a Fourier frame.

math.FA↗