Sharp pointwise lower bounds for Lebesgue functions

Abstract

For each n, let λn be the Lebesgue function for polynomial interpolation at an arbitrary set of n distinct nodes in [−1, 1]. We prove that there are a fixed point x ∈ (−1, 1) and a constant C such that λn(x) > (2/π) log n − C for infinitely many n, and that lim supn→∞ λn(x) / log n ≥ 2/π for almost every x ∈ (−1, 1). The first conclusion answers the bounded-loss question of Erdős Problem 1132 in the interpretation that the constant may depend on the array and the point; the second answers the almost-everywhere question. Both statements hold for every triangular array of distinct nodes, with no nesting assumption. The first proof combines Tao's local potential estimates with a local Riesz differentiation formula, an energy estimate for nodal derivative jumps, and a second moment argument. The second proof uses positive Cauchy transforms and harmonic measure, and is independent of Tao's local Bernstein theory. Status. Preprint, not yet refereed. As of 5 September 2026, Erdős Problem 1132 is listed as open on erdosproblems.com, with no proof claims submitted. Declaration of generative AI and AI-assisted technologies. GPT-6 Astra was used to generate the mathematical proofs and draft the manuscript. GPT-5.6 Sol and Claude Opus 5 were used for editorial review of the exposition. The author reviewed the final manuscript and takes full responsibility for its content.