Local Bernstein theory, and lower bounds for Lebesgue constants

I’ve just uploaded to the arXiv my paper “Local Bernstein theory, and lower bounds for Lebesgue constants“. This paper was initially motivated by a problem of Erdős on Lagrange interpolation, but in the course of solving that problem, I ended up modifying some very classical arguments of Bernstein and his contemporaries to obtain “local” versions of these classical “Bernstein-type inequalities”.

Bernstein proved many estimates concerning the derivatives of polynomials, trigonometric polynomials, and entire functions of exponential type. The first main result of the paper is to obtain localized versions of these estimates. Roughly speaking, these estimates assert that if a function is holomorphic on a wide thin rectangle and is “locally of exponential type”, then it can be bounded with small errors on the real line. The proof proceeds by a modification of the Duffin–Schaeffer argument, together with the two-constant theorem of Nevanlinna (provided to me by ChatGPT).

Once one localizes this “Bernstein theory”, it becomes suitable for the analysis of polynomials of a high degree, which is relevant in the theory of Lagrange interpolation. Erdős and Turán asked if certain lower bounds held for general intervals. This is shown in our paper, obtaining the sharp constant in the main term.

I also shared the story of playing around with AI tools. AlphaEvolve numerically confirmed certain inequalities, while ChatGPT Pro recognized an approximation problem and gave me a duality-based proof. While I eventually used pen and paper for the final rigorous steps involving the residue theorem, AI tools were valuable for quickly confirming numerical plausibility and recognizing the right techniques for isolated subproblems.