Zsuzsanna Nagy-Csiha | Searcharxiv

arXiv2020

Minimal energy point systems on the unit circle and the real line

In this paper, we investigate discrete logarithmic energy problems in the unit circle. We study the equilibrium configuration of $n$ electrons and $n-1$ pairs of external protons of charge $+1/2$. It is shown that all the critical points of the discrete logarithmic energy are global minima, and they are the solutions of certain equations involving Blaschke products. As a nontrivial application, we refine a recent result of Simanek, namely, we prove that any configuration of $n$ electrons in the unit circle is in stable equilibrium (that is, they are not just critical points but are of minimal energy) with respect to an external field generated by $n-1$ pairs of protons.

math.CA↗

arXiv2019

On the existence of an extremal function in the delsarte extremal problem

This paper is concerned with a Delsarte type extremal problem. Denote by $\mathcal{P}(G)$ the set of positive definite continuous functions on a locally compact abelian group $G$. We consider the function class, which was originally introduced by Gorbachev, \begin{multline*} \mathcal{G}(W, Q)_G = \left\{ f \in \mathcal{P}(G) \cap L^1(G) ~ : \right. ~ \left. f(0) = 1, ~ \text{supp}f_+ \subseteq W,~ \text{supp}\hat{f} \subseteq Q \right\} \end{multline*} where $W\subseteq G$ is closed and of finite Haar measure and $Q\subseteq \hat{G}$ is compact. We also consider the related Delsarte type problem of finding the extremal quantity \begin{equation*} \mathcal{D}(W,Q)_G = \sup \left\{ \int_{G} f(g) dλ_G(g) ~ : ~ f \in \mathcal{G}(W,Q)_G\right\}. \end{equation*} The main objective of the current paper is to prove the existence of an extremal function for the Delsarte type extremal problem $\mathcal{D}(W,Q)_G$. The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where $G=\mathbb{R}^d$. So the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way our result provides a far reaching generalization of the former work of Berdysheva and Révész.

math.CA↗