Optimal rules for numerical integration round the unit circle
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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 circleC1 : ∣z∣=1. In Theorem 2 we obtain explicitly a family of optimal rules forεc1f(z)∣dz∣, withf analytic onC1; interestingly, in general, the optimal nodes do not lie onC1. For functionsf(1/2(z +z−1)), Theorem 2 gives a family of optimal quadrature formulas for integration over [−1,1].
KeywordsHilbert Space Computational Mathematic Unit Circle Numerical Approximation Quadrature Formula
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