Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLocal properties of Riesz minimal energy configurations and equilibrium measures
89
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We investigate separation properties of $N$-point configurations that minimize discrete Riesz $s$-energy on a compact set $Asubset mathbb{R}^p$. When $A$ is a smooth $(p-1)$-dimensional manifold without boundary and $sin [p-2, p-1)$, we prove that the order of separation (as $Nto infty$) is the best possible. The same conclusions hold for the points that are a fixed positive distance from the boundary of $A$ whenever $A$ is any $p$-dimensional set. These estimates extend a result of Dahlberg for certain smooth $(p-1)$-dimensional surfaces when $s=p-2$ (the harmonic case). Furthermore, we obtain the same separation results for `greedy $s$-energy points. We deduce our results from an upper regularity property of the $s$-equilibrium measure (i.e., the measure that solves the continuous minimal Riesz $s$-energy problem), and we show that this property holds under a local smoothness assumption on the set $A$.
rate research
Read More
For a compact $ d $-dimensional rectifiable subset of $ mathbb{R}^{p} $ we study asymptotic properties as $ Ntoinfty $ of $N$-point configurations minimizing the energy arising from a Riesz $ s $-potential $ 1/r^s $ and an external field in the hypersingular case $ sgeq d$. Formulas for the weak$ ^* $ limit of normalized counting measures of such optimal point sets and the first-order asymptotic values of minimal energy are obtained. As an application, we derive a method for generating configurations whose normalized counting measures converge to a given absolutely continuous measure supported on a rectifiable subset of $ mathbb{R}^{p} $. Results on separation and covering properties of discrete minimizers are given. Our theorems are illustrated with several numerical examples.
For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where s>0. For a large class of manifolds A having finite, positive d-dimensional Hausdorff measure, we show that such minimizing configurations have asymptotic limit distribution (as N tends to infinity with s fixed) equal to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we obtain an explicit formula for the dominant term in the minimum energy. Our results are new even for the case of the d-dimensional sphere.
For $sgeqslant d$, we obtain the leading term as $Nto infty$ of the maximal weighted $N$-point Riesz $s$-polarization (or Chebyshev constant) for a certain class of $d$-rectifiable compact subsets of $mathbb{R}^p$. This class includes compact subsets of $d$-dimensional $C^1$ manifolds whose boundary relative to the manifold has $mathcal{H}_d$-measure zero, as well as finite unions of such sets when their pairwise intersections have $mathcal{H}_d$-measure zero. We also explicitly find the weak$^*$ limit distribution of asymptotically optimal $N$-point polarization configurations as $Nto infty$.
The Riesz transform is a natural multi-dimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness, decay, and vanishing moments. In this paper, we derive necessary conditions in terms of these three properties that guarantee the decay of the Riesz transform and its variants, and as an application, we show how the decay of the popular Simoncelli wavelets can be improved by appropriately modifying their Fourier transforms. By applying the Riesz transform to these new wavelets, we obtain steerable frames with rapid decay.
Combining the ideas of Riesz $s$-energy and $log$-energy, we introduce the so-called $s,log^t$-energy. In this paper, we investigate the asymptotic behaviors for $N,t$ fixed and $s$ varying of minimal $N$-point $s,log^t$-energy constants and configurations of an infinite compact metric space of diameter less than $1$. In particular, we study certain continuity and differentiability properties of minimal $N$-point $s,log^t$-energy constants in the variable $s$ and we show that in the limits as $srightarrow infty$ and as $srightarrow s_0>0,$ minimal $N$-point $s,log^t$-energy configurations tend to an $N$-point best-packing configuration and a minimal $N$-point $s_0,log^t$-energy configuration, respectively. Furthermore, the optimality of $N$ distinct equally spaced points on circles in $mathbb{R}^2$ for some certain $s,log^t$ energy problems was proved.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
