Abstract
We consider an interpolatory quadrature formula having as nodes the zeros of the nth degree Chebyshev polynomial of the second kind, on which the Fejér formula of the second kind is based, and the additional points \(\pm \tau _{c}=\pm \cos \frac{\pi }{2(n+1)}\). The new formula is shown to have positive weights given by explicit formulae. Furthermore, we determine the precise degree of exactness, and we obtain optimal error bounds for this formula either by Peano kernel methods or by Hilbert space techniques for analytic functions and \(1\le n\le 40\). In addition, the convergence of the quadrature formula is shown not only for Riemann integrable functions on \([-1,1]\), but also for functions having monotonic singularities at \(\pm \)1. The new formula has essentially the same rate of convergence as, and it is therefore an alternative to, the well-known Clenshaw-Curtis formula.
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Notaris, S.E. On a corrected Fejér quadrature formula of the second kind. Numer. Math. 133, 279–302 (2016). https://doi.org/10.1007/s00211-015-0750-5
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DOI: https://doi.org/10.1007/s00211-015-0750-5