On the convergence of mechanical quadrature formulas and on some related issues
Abstract
T. J. Stieltjes was one of the first mathematicians to address, from a very general point of view, the problem of the convergence of the so-called mechanical quadrature formulas. Recently, a very simple and elegant proof of Stieltjes' result for the case of continuous functions was given by J. V. Uspensky, who employs Weierstrass' fundamental theorem on the approximation of continuous functions by polynomials. However, in view of the importance of the question and some of the consequences arising from it, it seems to us not without interest to treat the problem without the aid of the aforementioned Weierstrass theorem and without preliminary study of a rather complex nature, and this will be the subject of the first section of this work.
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References
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