arXiv subjects

Douglas P. Hardin

Publications and source records attributed to Douglas P. Hardin.

12 recordsLinked to original sources

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↗

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↗

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

Eigenfunctions of the Fourier Transform with specified zeros

We give a unified description of the modular and quasi-modular functions used in Viazovska's proof of the best packing bounds in dimension 8 and the proof by Cohn, Kumar, Miller, Radchenko, and Viazovska of the best packing bound in dimension 24. We show that necessarily modular forms have to be used to obtain these results. We extend these constructions to arbitrary dimensions divisible by 4.

math.MG↗

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↗

arXiv2016

Energy bounds for codes and designs in Hamming spaces

We obtain universal bounds on the energy of codes and for designs in Hamming spaces. Our bounds hold for a large class of potential functions, allow unified treatment, and can be viewed as a generalization of the Levenshtein bounds for maximal codes.

math.MG↗

arXiv2015

Next order energy asymptotics for Riesz potentials on flat tori

Let $Λ$ be a lattice in ${\bf R}^d$ with positive co-volume. Among $Λ$-periodic $N$-point configurations, we consider the minimal renormalized Riesz $s$-energy $\mathcal{E}_{s,Λ}(N)$. While the dominant term in the asymptotic expansion of $\mathcal{E}_{s,Λ}(N)$ as $N$ goes to infinity in the long range case that $0 0$ they are of the form $C_{s,d}|Λ|^{-s/d}N^{1+s/d}$ and $-\frac{2}{d}N\log N+\left(C_{\log,d}-2ζ'_Λ(0)\right)N$ where we show that the constant $C_{s,d}$ is independent of the lattice $Λ$.

math-ph↗

arXiv2015

A Sharp Balian-Low Uncertainty Principle for Shift-Invariant Spaces

A sharp version of the Balian-Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators $\{f_k\}_{k=1}^K \subset L^2(\mathbb{R}^d)$ are translated along a lattice to form a frame or Riesz basis for a shift-invariant space $V$, and if $V$ has extra invariance by a suitable finer lattice, then one of the generators $f_k$ must satisfy $\int_{\mathbb{R}^d} |x| |f_k(x)|^2 dx = \infty$, namely, $\widehat{f_k} \notin H^{1/2}(\mathbb{R}^d)$. Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space $H^{d/2+ε}(\mathbb{R}^d)$; our results provide an absolutely sharp improvement with $H^{1/2}(\mathbb{R}^d)$. Our results are sharp in the sense that $H^{1/2}(\mathbb{R}^d)$ cannot be replaced by $H^s(\mathbb{R}^d)$ for any $s<1/2$.

math.FA↗

arXiv2014

Wavelets centered on a knot sequence: theory, construction, and applications

We develop a general notion of orthogonal wavelets `centered' on an irregular knot sequence. We present two families of orthogonal wavelets that are continuous and piecewise polynomial. We develop efficient algorithms to implement these schemes and apply them to a data set extracted from an ocelot image. As another application, we construct continuous, piecewise quadratic, orthogonal wavelet bases on the quasi-crystal lattice consisting of the $τ$-integers where $τ$ is the golden ratio. The resulting spaces then generate a multiresolution analysis of $L^2(\mathbf{R})$ with scaling factor $τ$.

math.NA↗

arXiv2013

Inverse Bernstein inequalities and min-max-min problems on the unit circle

We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle. The inequality is used to show that equally-spaced points solve a min-max-min problem for the logarithmic potential of such polynomials. Using techniques recently developed for polarization (Chebyshev-type) problems, we show that this optimality also holds for a large class of potentials, including the Riesz potentials $1/r^s$ with $s>0.$

math.MG↗