Summary
We consider the problem of optimal quadratures for integrandsf: [−1,1]→ℝ which have an analytic extension\(\bar f\) to an open diskD r of radiusr about the origin such that\(\left| {\bar f} \right|\)≦1 on\(\bar D_r \). Ifr=1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degreen rather than at optimal points, tends to infinity withn. In particular there is an “infinite” penalty for using Gauss quadrature. On the other hand, ifr>1, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings.
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This research was supported in part by the National Science Foundation under Grants MCS-8203271 and MCS-8303111
This research was supported in part by the National Science Foundation under Grant MCS-8923676
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Kowalski, M.A., Werschulz, A.G. & Woźniakowski, H. Is Gauss quadrature optimal for analytic functions?. Numer. Math. 47, 89–98 (1985). https://doi.org/10.1007/BF01389877
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DOI: https://doi.org/10.1007/BF01389877