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Joseph Rosenblatt

Publications and source records attributed to Joseph Rosenblatt.

14 recordsLinked to original sources
arXiv2026

The Zero Set of an Electric Field from a Finite Number of Point Charges

We consider the structure of the zero set in ${\mathbb R}^3$ of the electric vector field ${\mathbf F}=(X,Y,Z)$ from a finite set of point charges. We are most interested in the case where the point charges all lie in a plane, and we consider just the zero set in ${\mathbb R}^2$ of the electric vector field ${\mathbf F}=(X,Y)$ from the finite set of point charges. The conjecture is that the zero set of ${\mathbf F}=(X,Y)$ in ${\mathbb R}^2$ is finite. We show fairly easily that this conjecture is true in a Special Case: when the point charges for ${\mathbf F}=(X,Y)$ lie on a line, and we consider the possible zeros throughout ${\mathbb R}^2$. However, even in this Special Case, it is hard to get complete structural information about the zero sets of $X$ and $Y$ separately. We describe structural information about the asymptotic directions at infinity of these two zero sets, and relate this to the interlacing of the zero sets of sequences of polynomials. Then we consider the General Case where the point charges can be anywhere in the plane. As in the Special Case, we construct sequences of polynomials whose zero sets include the asymptotic directions at infinity of the zero sets of $X$ and $Y$ separately. But now the asymptotic directions are not necessarily interlacing, and the structure of the zero sets of these polynomials is less evident. Nonetheless, using these polynomials it might be possible to show that the zeros of ${\mathbf F}=(X,Y)$ are bounded, and then perhaps also, as a result, confirm the conjecture that the zero set of ${\mathbf F}=(X,Y)$ is finite.

math.CA↗
arXiv2024

Pointwise Convergence of Sequences of Singular Measures

We investigate the almost everywhere convergence of sequences of convolution operators given by probability measures $μ_n$ on $\mathbb R$. If this sequence of operators constitutes an approximate identity on a particular class of functions $\mathcal F$, under what additional conditions do we have $μ_n \ast f \to f$ a.e. for all $f \in \mathcal F$? We focus on the particular case of a sequence of contractions $C_{t_n}μ$ of a single probability measure $μ$, with $t_n \to 0$, so that that the sequence of operators is an approximate identity.

math.DS↗
arXiv2023

Moving Averages

We consider the convergence of moving averages in the general setting of ergodic theory or stationary ergodic processes. We characterize when there is universal convergence of moving averages based on complete convergence to zero of the standard ergodic averages. Using a theorem of Hsu-Robbins (1947) for independent, identically distributed processes, we prove for any bounded measurable function $f$ on a standard probability space $(X,\mathcal{B},μ)$, there exists a Bernoulli shift $T$, such that all moving averages $M(v_n, L_n)^T f = \frac{1}{L_n} \sum_{i=v_n+1}^{v_n+L_n} f \circ T^i$ with $L_n\geq n$ converge a.e. to $\int_X f dμ$. We refresh the reader about the cone condition established by Bellow, Jones, Rosenblatt (1990) which guarantees convergence of certain moving averages for all $f \in L^1(μ)$ and ergodic measure preserving maps $T$. We show given $f \in L^1(μ)$ and ergodic measure preserving $T$, there exists a moving average $M(v_n,L_n)^T f$ with $L_n$ strictly increasing such that $(v_n,L_n)$ does not satisfy the cone condition, but pointwise convergence holds a.e. We show for any non-zero $f\in L^1(μ)$, there is a generic class of ergodic maps $T$ such that each map has an associated moving average $M(v_n, L_n)^T f$ which does not converge pointwise. It is known if $f\in L^2(μ)$ is mean-zero, then there exist solutions $T$ and $g\in L^1(μ)$ to the coboundary equation: $f = g - g\circ T$. This implies $f$ and $T$ produce universal moving averages. We show this does not generalize to $L^p(μ)$ for $p<2$ by explicitly defining functions $f\in \cap_{p<2}L^p(μ)$ such that for each ergodic measure preserving $T$, there exists a moving average $M(v_n, L_n)^T f$ with $L_n\geq n$ such that these moving averages do not converge pointwise. Several of the results are generalized to the case of moving averages with polynomial growth.

math.DS↗
arXiv2022

Directional ergodicity and weak mixing for actions of $\mathbb R^d$ and $\mathbb Z^d$

We define notions of direction $L$ ergodicity, weak mixing, and mixing for a measure preserving $\mathbb Z^d$ action $T$ on a Lebesgue probability space $(X,μ)$, where $L\subseteq\mathbb R^d$ is a linear subspace. For $\mathbb R^d$ actions these notions clearly correspond to the same properties for the restriction of $T$ to $L$. For $\mathbb Z^d$ actions $T$ we define them by using the restriction of the unit suspension $\widetilde T$ to the direction $L$ and to the subspace of $L^2(\widetilde X,\widetilde μ)$ perpendicular to the suspension rotation factor. We show that for $\mathbb Z^d$ actions these properties are spectral invariants, as they clearly are for $\mathbb R^d$ actions. We show that for weak mixing actions $T$ in both cases, directional ergodicity implies directional weak mixing. For ergodic $\mathbb Z^d$ actions $T$ we explore the relationship between directional properties defined via unit suspensions and embeddings of $T$ in $\mathbb R^d$ actions. Genericity questions and the structure of non-ergodic and non-weakly mixing directions are also addressed.

math.DS↗
arXiv2022

Asymptotic Directions for the Zero Sets of the Components of an Electrical Field from a Finite Number of Point Charges on the Plane Part II

We study the structure of the zero set of a nontrivial finite point charge electrical field $F = (X,Y)$ in the plane $\mathbb R^2$. We establish equations satisfied by the possible directions for the zero sets \{X = 0\} and $\{Y = 0\}$ separately, and we show that there are only finitely many possible asymptotic directions for both of these zero sets. We suspect that the set of asymptotic directions for \{X = 0\} and the set of asymptotic directions for $\{Y = 0\}$ are (essentially) distinct.

math.CA↗
arXiv2022

Uniform distributions on curves and quantization

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous probability distribution by a discrete distribution. It has broad application in signal processing and data compression. In this paper, first we define the uniform distributions on different curves such as a line segment, a circle, and the boundary of an equilateral triangle. Then, we give the exact formulas to determine the optimal sets of $n$-means and the $n$th quantization errors for different values of $n$ with respect to the uniform distributions defined on the curves. In each case, we further calculate the quantization dimension and show that it is equal to the dimension of the object; and the quantization coefficient exists as a finite positive number. This supports the well-known result of Bucklew and Wise (1982), which says that for a Borel probability measure $P$ with non-vanishing absolutely continuous part the quantization coefficient exists as a finite positive number

math.PR↗
arXiv2020

Optimal Quantization via Dynamics

Quantization for probability distributions refers broadly to estimating a given probability measure by a discrete probability measure supported by a finite number of points. We consider general geometric approaches to quantization using stationary processes arising in dynamical systems, followed by a discussion of the special cases of stationary processes: random processes and Diophantine processes. We are interested in how close stationary process can be to giving optimal $n$-means and $n^{th}$ optimal mean distortion errors. We also consider different ways of measuring the degree of approximation by quantization, and their advantages and disadvantages in these different contexts.

math.DS↗
arXiv2019

Geometric and Measure-Theoretic Shrinking Targets in Dynamical Systems

We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described.

math.DS↗
arXiv2019

Existence and Non-existence of Solutions to the Coboundary Equation for Measure Preserving Systems

Let $(X,\mathcal{B},μ)$ be a standard probability space. We give new fundamental results determining solutions to the coboundary equation: \begin{eqnarray*} f = g - g \circ T \end{eqnarray*} where $f \in L^p$ and $T$ is ergodic invertible measure preserving on $(X, \mathcal{B}, μ)$. We extend previous results by showing for any measurable $f$ that is non-zero on a set of positive measure, the class of measure preserving $T$ with a measurable solution $g$ is meager (including the case where $\int_X f dμ= 0$). From this fact, a natural question arises: given $f$, does there always exist a solution pair $T$ and $g$? In regards to this question, our main results are: (i) Given measurable $f$, there exists an ergodic invertible measure preserving transformation $T$ and measurable function $g$ such that $f(x) = g(x) - g(Tx)$ for a.e. $x\in X$, if and only if $\int_{f > 0} f dμ= - \int_{f < 0} f dμ$ (whether finite or $\infty$). (ii) Given mean-zero $f \in L^p$ for $p \geq 1$, there exists an ergodic invertible measure preserving $T$ and $g \in L^{p-1}$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x \in X$. (iii) In some sense, the previous existence result is the best possible. For $p \geq 1$, there exist mean-zero $f \in L^p$ such that for any ergodic invertible measure preserving $T$ and any measurable $g$ such that $f(x) = g(x) - g(Tx)$ a.e., then $g \notin L^q$ for $q > p - 1$. Also, we show this situation is generic for mean-zero $f \in L^p$. Finally, it is shown that we cannot expect finite moments for solutions $g$, when $f \in L^1$. In particular, given any $ϕ: \mathbb{R} \to \mathbb{R}$ such that $\lim_{x\to \infty} ϕ(x) = \infty$, there exist mean-zero $f \in L^1$ such that for any solutions $T$ and $g$, the transfer function $g$ satisfies: \begin{eqnarray*} \int_{X} ϕ\big( | g(x) | \big) dμ= \infty. \end{eqnarray*}

math.DS↗
arXiv2018

Differentiating Orlicz spaces with rare bases of rectangles

In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space.

math.CA↗
arXiv2018

Optimal quantization for piecewise uniform distributions

Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using independent random variables and ergodic maps; these give asymptotically the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$. Secondly two piecewise uniform distributions are considered on $\mathbb R$: one with infinite number of pieces and one with finite number of pieces. For these two probability measures, we describe the optimal sets of $n$-means and the $n$th quantization errors for all $n\in \mathbb N$. It is seen that for a uniform distribution with infinite number of pieces to determine the optimal sets of $n$-means for $n\geq 2$ one needs to know an optimal set of $(n-1)$-means, but for a uniform distribution with finite number of pieces one can directly determine the optimal sets of $n$-means and the $n$th quantization errors for all $n\in \mathbb N$.

math.PR↗
arXiv2017

Joint coboundaries

We ask under what conditions on the function $f$, and a set of maps $\mathcal T$, it is the case that $f$ is a coboundary for some map in $\mathcal T$. We also consider for a function $f$, and a set of maps $\mathcal T$, when we have $f$ being a coboundary for all the maps in $\mathcal T$.

math.DS↗
arXiv2012

Multivariable averaging on sparse sets

Nonstandard ergodic averages can be defined for a measure-preserving action of a group on a probability space, as a natural extension of classical (nonstandard) ergodic averages. We extend the one-dimensional theory, obtaining L^1 pointwise ergodic theorems for several kinds of nonstandard sparse group averages, with a special focus on the group Z^d. Namely, we extend results for sparse block averages and sparse random averages to their analogues on virtually nilpotent groups, and extend Christ's result for sparse deterministic sequences to its analogue on Z^d. The second and third results have two nontrivial variants on Z^d: a "native" d-dimensional average and a "product" average from the 1-dimensional averages.

math.DS↗