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Is Gauss quadrature optimal for analytic functions? (1985)

Kowalski, M. A. ; Werschulz, A. G. ; Woźniakowski, H.

Springer

Numerische Mathematik 47 (1985), S. 89-98

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ISSN: 0945-3245

Keywords: AMS(MOS): 65D30 ; CR: G1.4

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: 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.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF01389877

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