From asymptotic distribution and vague convergence to uniform convergence, with numerical applications

Abstract

Let $\{{\Lambda }_n=\{{\lambda }_{1,n},\dots ,{\lambda }_{d_n,n}\}{\}}_n$ be a sequence of finite multisets of real numbers such that $d_n\to \infty$ as $n\to \infty$, and let $f:\Omega \subset {\mathbb{R}}^d\to \mathbb{R}$ be a Lebesgue measurable function defined on a domain $\Omega$ with $0<{\mu }_d(\Omega )<\infty$, where ${\mu }_d$ is the Lebesgue measure in ${\mathbb{R}}^d$. We say that $\{{\Lambda }_n{\}}_n$ has an asymptotic distribution described by $f$, and we write $\{{\Lambda }_n{\}}_n\sim f$, if \[ \lim_{n\to\infty}\frac1{d_n}\sum_{i=1}^{d_n}F(λ_{i,n})=\frac1{μ_d(Ω)}\int_ΩF(f({\boldsymbol x})){\rm d}{\boldsymbol x}\qquad\qquad(*) \] for every continuous function $F$ with bounded support. If ${\Lambda }_n$ is the spectrum of a matrix $A_n$, we say that $\{A_n{\}}_n$ has an asymptotic spectral distribution described by $f$ and we write $\{A_n{\}}_n{\sim }_{\lambda }f$. In the case where $d=1$, $\Omega$~is a bounded interval, ${\Lambda }_n\subseteq f(\Omega )$ for all $n$, and $f$ satisfies suitable conditions, Bogoya, Böttcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to $0$ of the difference between the properly sorted vector $[{\lambda }_{1,n},\dots ,{\lambda }_{d_n,n}]$ and the vector of samples $[f(x_{1,n}),\dots ,f(x_{d_n,n})]$, i.e., \[ \lim_{n\to\infty}\,\max_{i=1,\ldots,d_n}|f(x_{i,n})-λ_{τ_n(i),n}|=0, \qquad\qquad(**) \] where $x_{1,n},\dots ,x_{d_n,n}$ is a uniform grid in $\Omega$ and ${\tau }_n$ is the sorting permutation. We extend this result to the case where $d\ge 1$ and $\Omega$ is a Peano--Jordan measurable set (i.e., a bounded set with ${\mu }_d(\partial \Omega )=0$). See the rest of the abstract in the manuscript.