From asymptotic distribution and vague convergence to uniform convergence, with numerical applications
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Let {Λ_n={λ_1,n,…,λ_d_n,n}}_n be a sequence of finite multisets of real numbers such that d_n→∞ as n→∞, and let f:Ω⊂ℝ^d→ℝ be a Lebesgue measurable function defined on a domain Ω with 0<μ_d(Ω)<∞, where μ_d is the Lebesgue measure in ℝ^d. We say that {Λ_n}_n has an asymptotic distribution described by f, and we write {Λ_n}_n∼ f, if lim_n→∞1/d_n∑_i=1^d_nF(λ_i,n)=1/μ_d(Ω)∫_Ω F(f( x)) d x (*) 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∼_λ f. In the case where d=1, Ω is a bounded interval, Λ_n⊆ 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,…,λ_d_n,n] and the vector of samples [f(x_1,n),…,f(x_d_n,n)], i.e., lim_n→∞ max_i=1,…,d_n|f(x_i,n)-λ_τ_n(i),n|=0, (**) where x_1,n,…,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≥1 and Ω is a Peano–Jordan measurable set (i.e., a bounded set with μ_d(∂Ω)=0). See the rest of the abstract in the manuscript.
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