Abstract
The construction of optimal linear rules of numerical approximation by Davis' method has already been discussed for functions analytic in circles and in certain ellipses. In the present paper, introducing an appropriate Hilbert space, we discuss optimal linear rules for functions analytic in a circular annulus. We then consider the construction of optimal rules for numerical integration round the unit circleC 1 : ∣z∣=1. In Theorem 2 we obtain explicitly a family of optimal rules forεc 1 f(z)∣dz∣, withf analytic onC 1; interestingly, in general, the optimal nodes do not lie onC 1. For functionsf(1/2(z +z −1)), Theorem 2 gives a family of optimal quadrature formulas for integration over [−1,1].
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References
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Chawla, M.M., Kaul, V. Optimal rules for numerical integration round the unit circle. BIT 13, 145–152 (1973). https://doi.org/10.1007/BF01933486
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DOI: https://doi.org/10.1007/BF01933486