Local Bernstein theory, and lower bounds for Lebesgue constants

TL;DR

This paper develops a localized Bernstein theory for functions in rectangles, deriving bounds that lead to new lower bounds for Lebesgue constants on shorter intervals, addressing questions posed by Erdős and Turán.

Contribution

It extends Bernstein theory to rectangular regions and establishes sharp lower bounds for Lebesgue constants on shorter intervals, answering longstanding questions.

Findings

Localized Bernstein bounds hold for functions in rectangles with controlled growth.

Derived lower bounds for Lebesgue constants on shorter intervals than [-1,1].

Provided asymptotically sharp lower bounds for integral variants of Lebesgue constants.

Abstract

Classical (or ``global'') Bernstein theory establishes sharp control on entire functions of exponential type that are bounded and real-valued on the real axis. We localize some of this theory to rectangular regions $\{ x+iy: x \in I, 0 \leq y \leq y_0 \}$, showing that Bernstein-type bounds with acceptable errors can continue to hold for functions holomorphic in such rectangles, bounded and real-valued on the lower edge of the rectangle, at most exponentially large on the upper edge, and at most double exponentially large on the vertical sides. As a consequence of these bounds, we are able to localize the Erd\H{o}s lower bound $\sup_{x \in [-1,1]} \lambda(x) \geq \frac{2}{\pi} \log n - O(1)$ on the Lebesgue constant of interpolation on $C([-1,1])$ to shorter intervals $I$ than $[-1,1]$, answering a question of Erd\H{o}s and Tur\'an. By using suitably weighted versions of the residue theorem, we also obtain the asymptotically sharp lower bound $\int_I \lambda(x)\ dx \geq \frac{4|I|}{\pi^2} \log n - o(\log n)$ for integral variants of such constants, answering a further question of Erd\H{o}s.