Local Bernstein theory, and lower bounds for Lebesgue constants

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.