Joint Rational Approximations of Laurent Series

The paper proposes a new method for approximating Laurent series by rational functions. The concept of a generalized polynomial is introduced and, based on it, the corresponding Padé problem is posed and solved. A constructive solution to this problem made it possible to determine rational functions that are considered as Padé approximations of the Laurent series. It is established that in the simplest case, the constructed Padé approximations of the Laurent series behave in the same way as the classical Padé approximations of a power series: they localize singular points of a function that is the sum of a Laurent series. All the obtained results are generalized to systems of k Laurent series. In particular, rational Hermite–Laurent approximations are constructed, which are analogs of Hermite–Padé approximations of power series. Under the conditions of the classical Fabry theorem ‘‘on the relation’’ it is established that rational Hermite–Laurent approximations localize singular points of functions that are sums of the corresponding Laurent series.