Finite interpolation with minimum uniform norm in C^n

Abstract

Given a finite sequence $a:=a_1,...,a_N$ in a domain $\Omega \subset C^n$, and complex scalars $v:=v_1,...,v_N$, consider the classical extremal problem of finding the smallest uniform norm of a holomorphic function verifying $f(a_j)=v_j$ for all $j$. We show that the modulus of the solutions to this problem must approach its least upper bound along a subset of the boundary of the domain large enough to contain the support of a measure whose hull contains a subset of the original $a$ large enough to force the same minimum norm. Furthermore, all the solutions must agree on a variety which also contains this hull. An example is given to show that the inclusions can be strict.