Widom factors for Chebyshev and residual polynomials on semi-regular subsets of $\mathbb{R}$

What the math gives to ML

The paper gives a quantitative way to design and assess minimax polynomials when the admissible spectral set is disconnected or contains isolated, irregular points. The transferable asset is the separation between the dominant capacity term and a finite correction, exp[IR(E,x_*)], which predicts how isolated spectral components inflate worst-case polynomial response. This suggests replacing interval-only Chebyshev acceleration in neural-network optimization with residual polynomials explicitly constrained on estimated spectral clusters and outliers. The first practical test should use low-degree Hessian-polynomial preconditioning and compare ordinary interval Chebyshev polynomials against minimax polynomials on the full empirical spectrum.