Abstract
Consider a Markov system of functions whose linear span is dense with respect to the uniform norm in the space of the continuous functions on a finite interval. Gaussian rules are those which correctly integrate as many successive basis functions as possible with the lesser number of nodes. In this paper we provide a simple proof of the fact that such rules converge for all bounded Riemann-Stieltjes integrable functions. The proposed proof is also valid for any sequence of quadrature rules with positive coefficients which converge for the basis functions. Taking the nodes of the Gaussian rules as nodes for Lagrange interpolation, we give a sufficient condition for the convergence in L 2-norm of such processes for bounded Riemann-Stieltjes integrable functions.
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Pérez-Acosta, F., Santos-Léon, J.C. Convergence of Gaussian Quadrature and Lagrange Interpolation in Haar Systems. BIT Numerical Mathematics 39, 579–584 (1999). https://doi.org/10.1023/A:1022378905061
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DOI: https://doi.org/10.1023/A:1022378905061