Formulas for Rational Interpolation and Remainders

This paper deals with the existence of interpolating rational functions serving as Thiele continued fraction convergents and also presents the expression for the remainder for such rational interpolations. These problems are similar to multipoint Padé approximations. For the limit case, the expression for the remainder in diagonal Padé approximations at zero is obtained; also a sufficiently simple expression for the exact value of the remainder in the case of the function $\sqrt{1+z}$ is derived.