A finite-matrix bound from the weighted Bernstein–Walsh estimate

Let $A$ be a positive definite matrix with spectrum $\Lambda(A)$, let $N$ denote its dimension, and let $E_n(\Lambda(A))$ be the relevant degree-$n$ approximation error. For a measure $\sigma$ with compact support and continuous potential, define

$$S(t)=\operatorname{supp}(\sigma/t-\nu_{t,\sigma}),$$

where $\nu_{t,\sigma}$ is the constrained energy minimizer, and let $g_{S(t)}(0,\infty)$ denote the associated Green function value. The conjecture concerns associating such a measure $\sigma$ with the spectrum of $A$.

Finite-matrix convergence conjecture. There is a (modest) constant $C\in\mathbb R$ and a technique of associating a measure $\sigma$ with compact support and continuous potential to the spectrum of a positive definite matrix $A$ such that, for all $n$ sufficiently small,

$$E_n(\Lambda(A)) \leq \exp\left( C - N \int_0^{n/N} g_{S(t)}(0,\infty)\\,dt \right).$$

The conjecture proposes a nonasymptotic analogue of the upper bound governing superlinear conjugate-gradient convergence for limiting eigenvalue distributions. The cited numerical evidence suggests such a bound may hold for a single matrix, but the existence of the measure-association technique and the stated estimate are not established here.