The multivariate Herglotz-Nevanlinna class: Rational approximation

We return to Takagi's variational principle, generalized after forty years to two complex variables by Pfister. Both isolating some extremal rational functions associated to a bounded holomorphic function in the unit disk, respectively the bidisk. The rational inner functions arising from the Takagi-Pfister skew eigenvectors lead to a Pade type approximation scheme. For these rational functions, we prove a Montessus de Ballore type convergence theorem, on the polydisk in any complex dimension. On the natural and more restrictive class of Agler holomorphic functions with non-negative real part, we show that Cayley rational inner functions match any finite section of the Taylor expansion at a prescribed point. We derive from the Hilbert space proof that the finite section coefficient set of Taylor series of the Agler functions in the Herglotz-Nevanlinna setting is semi-algbraic. The pole distribution of the Takagi-Pfister interpolation sequence is identified as a main open question on the subject.