On the convergence of formulas for mechanical quadratures and some related issues

Abstract

T. J. Stieltjes was one of the first mathematicians who posed from a very general point of view the question of the convergence of formulas of the so-called mechanical quadratures. Recently, a very simple and elegant proof of Stieltjes' result for the case of continuous functions was given by J. V. Uspenskii, who uses Weierstrass's main theorem regarding the approximation of continuous functions using polynomials. However, in view of the importance of the question and some of the consequences that follow from this, it seems not uninteresting, in our opinion, to interpret the question without the help of the above-mentioned Weierstrass theorem and without a preliminary study, a quite complex one, which will be the subject of the first paragraph of this work.