Convex Bernstein limits and sharp recovery on expanding intervals

We determine when approximation by convex polynomials on expanding intervals converges to best approximation on the real line by convex entire functions of exponential type at most $σ>0$. For intervals of radius $(n-γ_n)/σ$, where $0\leγ_n<n$ and $γ_n=o(n)$, one margin sequence gives this limit for every convex function of polynomial growth if and only if $γ_n/[n^{1/3}(\log n)^{2/3}]\to\infty$. When this condition fails, a fixed uniformly strictly convex entire function has approximation errors with unbounded limsup. We apply the limit theorem to prove existence of the convex Bernstein constant for $|x|^λ$ for every $λ\ge1$, and identify it with an attained entire-function minimum. We also give a sufficient margin for growth bounded by $\exp(a|x|^α)$, $0<α<1$. For arbitrary subexponential growth on the real line, each target with finite entire-function error admits a suitable margin, whereas no single margin is sufficient for the whole class. Further results concern attainment, right continuity with respect to the exponential type, weighted growth conditions, and the endpoints of fixed-fraction approximation intervals.

Source: arXiv:2610.31546v1 - http://arxiv.org/abs/2610.31546v1 PDF: https://arxiv.org/pdf/2610.31546v1 Original Link: http://arxiv.org/abs/2610.31546v1