An estimate of goodness of cubatures for the unit circle in 𝑅²

Maeztu, J. I.

doi:10.1090/S0025-5718-1986-0856709-X

An estimate of goodness of cubatures for the unit circle in $\textbf {R}^ 2$
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Math. Comp. 47 (1986), 651-658 Request permission

Abstract:

The Sarma-Eberlein estimate ${s_E}$ is an estimate of goodness of cubature formulae for n-cubes defined as the integral of the square of the formula truncation error, over a function space provided with a measure. In this paper, cubature formulae for the unit circle in ${{\mathbf {R}}^2}$ are considered and an estimate of the above type is constructed with the desirable property of being compatible with the symmetry group of the circle.

References

H. S. M. Coxeter, Introduction to geometry, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0346644
Derek John Scott Robinson, A course in the theory of groups, Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York-Berlin, 1982. MR 648604, DOI 10.1007/978-1-4684-0128-8
V. L. N. Sarma, A generalisation of Eberlein’s integral over function space, Trans. Amer. Math. Soc. 121 (1966), 52–61. MR 193494, DOI 10.1090/S0002-9947-1966-0193494-5
V. L. N. Sarma, Eberlein measure and mechanical quadrature formulae. I. Basic theory, Math. Comp. 22 (1968), 607–616. MR 253559, DOI 10.1090/S0025-5718-1968-0253559-1
V. L. N. Sarma and A. H. Stroud, Eberlein measure and mechanical quadrature formulae II. Numerical results, Math. Comp. 23 (1969), 781–784. MR 258282, DOI 10.1090/S0025-5718-1969-0258282-6
A. H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0327006