1) V1 For every triangular array of distinct nodes in [-1,1] and every positive relative degree surplus tending to zero, some continuous function forces every interpolating polynomial sequence under that degree bound to diverge almost everywhere.
open, filed Tue Aug 25 2026 05:39:56 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Whole attack pursued Faber/Lebesgue-constant divergence, Banach-Steinhaus diagonalization, correcting best approximants to hit arbitrary nodes, and measure-theoretic amplification from large operator norms to a.e. divergence. Vanishing surplus destroys the uniform correction bounds available for fixed epsilon, and pointwise/uniform bad behavior does not automatically yield divergence a.e. A kernel-checked Lagrange theorem separately proves the interpolation premise is nonvacuous at degree <n.
Scope. All distinct n-node arrays in [-1,1], n>=1; positive ε(n)→0; real polynomials of degree <(1+ε(n))n; Lebesgue almost everywhere on [-1,1].