100 years of improvements of bounding properties of Padé approximants to the Stieltjes functions: One-point, two-point and N-point Padé approximants

The inequalities between the convergents of the Stieltjes continued fractions (CF) allowing to prove their convergence to the Stieltjes functions were first introduced in the famous work of Stieltjes (1894) [26]. Of course, the name of Stieltjes was attributed to these objects more later, such as the identification of the convergents of CF with Padé approximants (PA). The present review relates the evolution of PA errors inequalities for the Stieltjes functions up to now. Alphonse Magnus and the author (Gilewicz and Magnus (1979) [13]) remarked that the original Stieltjes inequalities are not optimal, not order equilibrated. From this time up to now we have established the optimal inequalities for the errors of PA for the Stieltjes functions in the cases of the classical one-point PA, of two-point (0 and ∞) PA and of N-point ( N > 2 ) PA. The last case is presented for the first time in the present review.