On a diagonal Padé approximation in two complex variables (Q1849718)

The motivation for the present paper is a special reconstruction algorithm of planar shapes [cf. \textit{B. Gustafsson, C. He, P. Milanfar}, and \textit{M. Putinar}, Inverse Prob. 16, No.~4, 1053-1070 (2000; Zbl 0959.44010)]. The reconstruction problem leads via the approximation by rational functions of transfer functions in linear control theory [\textit{C. Foiaş, A. E. Frazho}, The commutant lifting approach to interpolation problems. Birkhäuser Verlag, Basel, Boston (1990; Zbl 0718.47010)] to a Padé approximation approach of a class of power series in two variables. The problem is then reduced to a Padé approximation problem in one variable [\textit{G. A. Baker, jun.}, and \textit{P. Graves-Morris}, Padé approximants. Part I: Basic theory (1981; Zbl 0468.30032), Part II: Extensions and applications (1981; Zbl 0468.30033); \textit{A. Cuyt}, J. Comput. Appl. Math. 105, No.~1-2, 25-50 (1999; Zbl 0945.41012)]. The convergence of the approximation process is derived from operator theory which governs the whole argumentation.