Multivariate Padé-Bergman approximants (Q1864501)

Slightly modifying a previous definition [\textit{P. Guillaume, A. Huard} and \textit{V. Robin}, J. Approximation Theory 95, 203-214 (1998; Zbl 0916.41015)], the authors define mutivariate Padé-Bergman (or \(A_\rho^2\)) approximants to a function \(f\) on a polydisc via minimizing the Bergman space norm of projections of \(fS-R\) (where \(S\), \(R\) are polynomials) and taking the ratio \(R/S\). This approach simplifies some proofs with respect to the paper cited above. Existence, uniqueness and consistency of these approximants is analyzed. Finally, convergence is studied using tools from both algebraic geometry and complex analysis. In particular, a Montessus de Ballore type theorem is established.