Abstract
A classical theorem due to Chebyshev, Markov and Stieltjes states that the Gauss-Legendre quadrature of a generic function f is a Riemann sum of f. In this note we prove an analogue of this theorem for Romberg quadrature.
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Notes
All of the integration rules we shall discuss here satisfy \(\sum \limits _{i = 0}^{n_{\tau }} w_{\tau ,i} \ = \ b-a\), i.e., \(I_{\tau }\) integrates the constant function \(f \equiv 1\) exactly.
Note that, when m is odd, \(\alpha _{1,1,m,m} \ = \ \frac{4}{3} \left( \alpha _{0,{0},m,0} + \alpha _{0,{0},m,m} \right) \ = \ \frac{4}{3}\left( \frac{1}{2} + \frac{1}{2} \right) \ = \ \frac{4}{3} \ =\ \Gamma _{1}.\)
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We thank two anonymous referees for their helpful comments and constructive criticisms concerning earlier drafts of this manuscript.
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de Camargo, A.P. A Chebyshev-Markov-Stieltjes separation type theorem for classical Romberg quadrature. Numer Algor 92, 2365–2376 (2023). https://doi.org/10.1007/s11075-022-01393-w
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DOI: https://doi.org/10.1007/s11075-022-01393-w