From asymptotic distribution and vague convergence to uniform convergence, with numerical applications
Let $\{Λ_n=\{λ_{1,n},\ldots,λ_{d_n,n}\}\}_n$ be a sequence of finite multisets of real numbers such that $d_n\to\infty$ as $n\to\infty$, and let $f:Ω\subset\mathbb R^d\to\mathbb R$ be a Lebesgue measurable function defined on a domain $Ω$ with $0<μ_d(Ω)<\infty$, where $μ_d$ is the Lebesgue measure in $\mathbb R^d$. We say that $\{Λ_n\}_n$ has an asymptotic distribution described by $f$, and we write $\{Λ_n\}_n\sim f$, 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 $Λ_n$ 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 $\{A_n\}_n\sim_λf$. In the case where $d=1$, $Ω$~is a bounded interval, $Λ_n\subseteq f(Ω)$ 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 $[λ_{1,n},\ldots,λ_{d_n,n}]$ and the vector of samples $[f(x_{1,n}),\ldots,f(x_{d_n,n})]$, 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 $x_{1,n},\ldots,x_{d_n,n}$ is a uniform grid in $Ω$ and $τ_n$ is the sorting permutation. We extend this result to the case where $d\ge1$ and $Ω$ is a Peano--Jordan measurable set (i.e., a bounded set with $μ_d(\partialΩ)=0$). See the rest of the abstract in the manuscript.