Cauchy-Weil formula, Schur-Agler type classes, new Hardy spaces of the polydisk and interpolation problems

Abstract

Given n rational inner functions Φ=(Φ1,...,Φn) defining a 0-dimensional (hence finite) complete intersection in the polydisk Dn and a Weil polyhedron ΔrΦ relatively compact in Dn, we combine together the Cauchy-Weil representation formulas respectively in Dn (considered as a Weil polyhedron subordinated to the coordinate functions) and ΔrΦ in order to provide integral representations formulas involving a positive kernel provided Φ satisfies some Schur-Agler type conditions. We then study interpolation problems in H2(Dn) along the finite set Φ−1({0_}). One of the main objectives of this paper is to emphasize the important role played here by Cauchy-Weil representation formula (as in computational polynomial geometry), together with its intimate connection with multivariate residue calculus.