arXiv subjects

Edward B. Saff

Publications and source records attributed to Edward B. Saff.

At least 19 recordsLinked to original sources

arXiv2024

Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere?

We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard--Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which dimension reduction occurs in the sense that the support of the equilibrium measure becomes a sphere. We also briefly discuss the relation between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk--Hecke formula, and the calculus of hypergeometric functions.

math.CA↗

arXiv2024

On a problem of E. Meckes for the unitary eigenvalue process on an arc

We study the problem originally communicated by E. Meckes on the asymptotics for the eigenvalues of the kernel of the unitary eigenvalue process of a random $n \times n$ matrix. The eigenvalues $p_{j}$ of the kernel are, in turn, associated with the discrete prolate spheroidal wave functions. We consider the eigenvalue counting function $|G(x,n)|:=\#\{j:p_j>Ce^{-x n}\}$, ($C>0$ here is a fixed constant) and establish the asymptotic behavior of its average over the interval $x \in (λ-\varepsilon, λ+\varepsilon)$ by relating the function $|G(x,n)|$ to the solution $J(y)$ of the following energy problem on the unit circle $S^{1}$, which is of independent interest. Namely, for given $θ$, $0<θ< 2 π$, and given $q$, $0<q<1$, we determine the function $J(q) =\inf \{I(μ): μ\in \mathcal{P}(S^{1}), μ(A_θ) = q\}$, where $I(μ):= \iint \log\frac{1}{|z - ζ|} dμ(z) dμ(ζ)$ is the logarithmic energy of a probability measure $μ$ supported on the unit circle and $A_θ$ is the arc from $e^{-i θ/2}$ to $e^{i θ/2}$.

math.PR↗

arXiv2023

Asymptotics of $k$-nearest neighbor Riesz energies

We obtain new asymptotic results about systems of $ N $ particles governed by Riesz interactions involving $ k $-nearest neighbors of each particle as $N\to\infty$. These results include a generalization to weighted Riesz potentials with external field. Such interactions offer an appealing alternative to other approaches for reducing the computational complexity of an $ N $-body interaction. We find the first-order term of the large $ N $ asymptotics and characterize the limiting distribution of the minimizers. We also obtain results about the $ Γ$-convergence of such interactions, and describe minimizers on the 1-dimensional flat torus in the absence of external field, for all $ N $.

math.CA↗

arXiv2023

Threshold condensation to singular support for a Riesz equilibrium problem

We compute the equilibrium measure in dimension d=s+4 associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that the equilibrium measure can be a mixture of a continuous part and a singular part. Depending on the value of the power, a threshold phenomenon occurs and consists of a dimension reduction or condensation on the singular part. In particular, in the logarithmic case s=0 (d=4), there is condensation on a sphere of special radius when the power of the external field becomes quadratic. This contrasts with the case d=s+3 studied previously, which showed that the equilibrium measure is fully dimensional and supported on a ball. Our approach makes use, among other tools, of the Frostman or Euler-Lagrange variational characterization, the Funk-Hecke formula, the Gegenbauer orthogonal polynomials, and hypergeometric special functions.

math.PR↗

arXiv2022

On the solution of a Riesz equilibrium problem and integral identities for special functions

The aim of this note is to provide a full space quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz s-kernels. We address the case s=d-3 for arbitrary dimension d, in particular the logarithmic kernel in dimension 3. The equilibrium measure for this full space external field problem turns out to be a radial arcsine distribution supported on a ball with a special radius. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functions. It seems that these identities are not found in the existing tables for series and integrals, and are not recognized by advanced mathematical software. Among other ingredients, our proofs involve the Euler-Lagrange variational characterization, the Funk-Hecke formula, the Weyl regularity lemma, the maximum principle, and special properties of hypergeometric functions.

math.PR↗

arXiv2022

Riesz energy problems with external fields and related theory

In this paper, we investigate Riesz energy problems on unbounded conductors in $\R^d$ in the presence of general external fields $Q$, not necessarily satisfying the growth condition $Q(x)\to\infty$ as $x\to\infty$ assumed in several previous studies. We provide sufficient conditions on $Q$ for the existence of an equilibrium measure and the compactness of its support. Particular attention is paid to the case of the hyperplanar conductor $\R^{d}$, embedded in $\R^{d+1}$, when the external field is created by the potential of a signed measure $ν$ outside of $\R^{d}$. Simple cases where $ν$ is a discrete measure are analyzed in detail. New theoretic results for Riesz potentials, in particular an extension of a classical theorem by de La Vallée-Poussin, are established. These results are of independent interest.

math.CA↗

arXiv2022

Point Source Equilibrium Problems with Connections to Weighted Quadrature Domains

We explore the connection between supports of equilibrium measures and quadrature identities, especially in the case of point sources added to the external field $Q(z)=|z|^{2p}$ with $p \in \mathbb{N}$. Along the way, we describe some quadrature domains with respect to weighted area measure $|z|^{2p}dA_z$ and complex boundary measure $|z|^{-2p}dz$.

math.CV↗

arXiv2021

Unconstrained polarization (Chebyshev) problems: basic properties and Riesz kernel asymptotics

We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an $N$-point configuration that maximizes the minimum value of its potential over a set $A$ in $p$-dimensional Euclidean space. This problem is compared to the constrained problem in which the points are required to belong to the set $A$. We find that for Riesz kernels $1/|x-y|^s$ with $s>p-2$ the optimum unconstrained configurations concentrate close to the set $A$ and based on this fundamental fact we recover the same asymptotic value of the polarization as for the more classical constrained problem on a class of $d$-rectifiable sets. We also investigate the new unconstrained problem in special cases such as for spheres and balls. In the last section we formulate some natural open problems and conjectures.

math.CA↗

arXiv2021

Asymptotic properties of short-range interaction functionals

We describe a framework for extending the asymptotic behavior of a short-range interaction from the unit cube to general compact subsets of $ \mathbb R^d $. This framework allows us to give a unified treatment of asymptotics of hypersingular Riesz energies and optimal quantizers. We further obtain new results about the scale-invariant nearest neighbor interactions, such as the $ k $-nearest neighbor truncated Riesz energy. Our generalized approach has applications to methods for generating distributions with prescribed density: strongly-repulsive Riesz energies, centroidal Voronoi tessellations, and a popular meshing algorithm due to Persson and Strang.

math.CA↗

arXiv2020

On the Search for Tight Frames of Low Coherence

We introduce a projective Riesz $s$-kernel for the unit sphere $\mathbb{S}^{d-1}$ and investigate properties of $N$-point energy minimizing configurations for such a kernel. We show that these configurations, for $s$ and $N$ sufficiently large, form frames that are well-separated (have low coherence) and are nearly tight. Our results suggest an algorithm for computing well-separated tight frames which is illustrated with numerical examples.

math.MG↗

arXiv2020

Dynamics of Particles on a Curve with Pairwise Hyper-singular Repulsion

We investigate the large time behavior of $N$ particles restricted to a smooth closed curve in $\mathbb{R}^d$ and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz $s$-energy with $s>1.$ We show that regardless of their initial positions, for all $N$ and time $t$ large, their normalized Riesz $s$-energy will be close to the $N$-point minimal possible. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.

math.DS↗

arXiv2019

Perturbations of Christoffel-Darboux kernels. I: detection of outliers

Two central objects in constructive approximation, the Christoffel-Darboux kernel and the Christoffel function, are encoding ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel-Darboux kernel in C^d, with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri)potential theory are widely used in the derivation of our perturbation formulas.

math.CV↗

arXiv2018

Inverse Potential Problems for Divergence of Measures with Total Variation Regularization

We study inverse problems for the Poisson equation with source term the divergence of an $\mathbf{R}^3$-valued measure, that is, the potential $Φ$ satisfies $$ ΔΦ= \text{div} \boldsymbolμ, $$ and $\boldsymbolμ$ is to be reconstructed knowing (a component of) the field grad $Φ$ on a set disjoint from the support of $\boldsymbolμ$. Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We develop methods for recovering $\boldsymbolμ$ based on total variation regularization. We provide sufficient conditions for the unique recovery of $\boldsymbolμ$, asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable. Numerical examples are provided to illustrate the main theoretical results.

math.OC↗

arXiv2018

Asymptotic Linear Programming Lower Bounds for the Energy of Minimizing Riesz and Gauss Configurations

Utilizing frameworks developed by Delsarte, Yudin and Levenshtein, we deduce linear programming lower bounds (as $N\to \infty$) for the Riesz energy of $N$-point configurations on the $d$-dimensional unit sphere in the so-called hypersingular case; i.e, for non-integrable Riesz kernels of the form $|x-y|^{-s}$ with $s>d.$ As a consequence, we immediately get (thanks to the Poppy-seed bagel theorem) lower estimates for the large $N$ limits of minimal hypersingular Riesz energy on compact $d$-rectifiable sets. Furthermore, for the Gaussian potential $\exp(-α|x-y|^2)$ on $\mathbb{R}^p,$ we obtain lower bounds for the energy of infinite configurations having a prescribed density.

math-ph↗

arXiv2017

Large deviation principles for hypersingular Riesz gases

We study $N$-particle systems in R^d whose interactions are governed by a hypersingular Riesz potential $|x-y|^{-s}$, $s>d$, and subject to an external field. We provide both macroscopic results as well as microscopic results in the limit as $N\to \infty$ for random point configurations with respect to the associated Gibbs measure at scaled inverse temperature $β$. We show that a large deviation principle holds with a rate function of the form `$β$-Energy +Entropy', yielding that the microscopic behavior (on the scale $N^{-1/d}$) of such $N$-point systems is asymptotically determined by the minimizers of this rate function. In contrast to the asymptotic behavior in the integrable case $s d$ is strongly $β$-dependent.

math-ph↗

arXiv2017

Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces

We prove that the covering radius of an $N$-point subset $X_N$ of the unit sphere $S^d \subset R^{d+1}$ is bounded above by a power of the worst-case error for equal weight cubature $\frac{1}{N}\sum_{\mathbf{x} \in X_N}f(\mathbf{x}) \approx \int_{S^d} f \, \mathrm{d} σ_d$ for functions in the Sobolev space $\mathbb{W}_p^s(S^d)$, where $σ_d$ denotes normalized area measure on $S^d.$ These bounds are close to optimal when $s$ is close to $d/p$. Our study of the worst-case error along with results of Brandolini et al. motivate the definition of Quasi-Monte Carlo (QMC) design sequences for $\mathbb{W}_p^s(S^d)$, which have previously been introduced only in the Hilbert space setting $p=2$. We say that a sequence $(X_N)$ of $N$-point configurations is a QMC-design sequence for $\mathbb{W}_p^s(S^d)$ with $s > d/p$ provided the worst-case equal weight cubature error for $X_N$ has order $N^{-s/d}$ as $N \to \infty$, a property that holds, in particular, for a sequence of spherical $t$-designs in which each design has order $t^d$ points. For the case $p = 1$, we deduce that any QMC-design sequence $(X_N)$ for $\mathbb{W}_1^s(S^d)$ with $s > d$ has the optimal covering property; i.e., the covering radius of $X_N$ has order $N^{-1/d}$ as $N \to \infty$. A significant portion of our effort is devoted to the formulation of the worst-case error in terms of a Bessel kernel, and showing that this kernel satisfies a Bernstein type inequality involving the mesh ratio of $X_N$. As a consequence we prove that any QMC-design sequence for $\mathbb{W}_p^s(S^d)$ is also a QMC-design sequence for $\mathbb{W}_{p^\prime}^s(S^d)$ for all $1 \leq p < p^\prime \leq \infty$ and, furthermore, if $(X_N)$ is a quasi-uniform QMC-design sequence for $\mathbb{W}_p^s(S^d)$, then it is also a QMC-design sequence for $\mathbb{W}_p^{s^\prime}(S^d)$ for all $s > s^\prime > d/p$.

math.NA↗

arXiv2017

Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere --- the Exceptional Case

We consider the minimal discrete and continuous energy problems on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field due to finitely many localized charge distributions on $\mathbb{S}^d$, where the energy arises from the Riesz potential $1/r^s$ ($r$ is the Euclidean distance) for the critical Riesz parameter $s = d - 2$ if $d \geq 3$ and the logarithmic potential $\log(1/r)$ if $d = 2$. Individually, a localized charge distribution is either a point charge or assumed to be rotationally symmetric. The extremal measure solving the continuous external field problem for weak fields is shown to be the uniform measure on the sphere but restricted to the exterior of spherical caps surrounding the localized charge distributions. The radii are determined by the relative strengths of the generating charges. Furthermore, we show that the minimal energy points solving the related discrete external field problem are confined to this support. For $d-2\leq s<d$, we show that for point sources on the sphere, the equilibrium measure has support in the complement of the union of specified spherical caps about the sources. Numerical examples are provided to illustrate our results.

math-ph↗

arXiv2017

Covering and separation of Chebyshev points for non-integrable Riesz potentials

For Riesz $s$-potentials $K(x,y)=|x-y|^{-s}$, $s>0$, we investigate separation and covering properties of $N$-point configurations $ω^*_N=\{x_1, \ldots, x_N\}$ on a $d$-dimensional compact set $A\subset \mathbb{R}^\ell$ for which the minimum of $\sum_{j=1}^N K(x, x_j)$ is maximal. Such configurations are called $N$-point optimal Riesz $s$-polarization (or Chebyshev) configurations. For a large class of $d$-dimensional sets $A$ we show that for $s>d$ the configurations $ω^*_N$ have the optimal order of covering. Furthermore, for these sets we investigate the asymptotics as $N\to \infty$ of the best covering constant. For these purposes we compare best-covering configurations with optimal Riesz $s$-polarization configurations and determine the $s$-th root asymptotic behavior (as $s\to \infty$) of the maximal $s$-polarization constants. In addition, we introduce the notion of "weak separation" for point configurations and prove this property for optimal Riesz $s$-polarization configurations on $A$ for $s>\text{dim}(A)$, and for $d-1\leqslant s < d$ on the sphere $\mathbb{S}^d$.

math.CA↗