Two problems in asymptotic analysis Padé-orthogonal approximation and Riesz polarization constants and configurations

Abstract

We investigate two subjects in asymptotic analysis. The first focuses on a class of rational functions called Padé-orthogonal approximants. We study the relation of the convergence of poles of row sequences of Padé-orthogonal approximants and the singularities of the approximated function. We prove both direct and inverse results for these row sequences. Thereby, we obtain analogues of the theorems of R. de Montessus de Ballore and E. Fabry. The second concerns the so-called maximal and minimal $N$-point Riesz $s$-polarization constants and associated configurations. First, we investigate basic asymptotic properties when $N$ fixed and $s$ varying of these constants and configurations. Next, we prove a conjecture of T. Erd'{e}lyi and E.B. Saff, concerning the dominant term as $N\to \infty$ of the maximal $N$-point Riesz $d$-polarization constant of an infinite compact subset $A$ of a $d$-dimensional $C^{1}$-manifold embedded in $\mathbb{R}^{m}$. Moreover, if we assume further that $\mathcal{H}_d(A)>0$, we show that the maximal $N$-point Riesz $d$-polarization configurations of $A$ distribute asymptotically uniformly on $A$ with respect to $\mathcal H_d|_A$. These results also hold for finite unions of such sets $A$ provided that their pairwise intersections have $\mathcal H_d$-measure zero. Finally, we determine the maximal and minimal $N$-point Riesz $s$-polarization configurations of the unit sphere $\mathbb{S}^{m}$ in $\mathbb{R}^{m+1}$ for certain values $s.$