ABSTRACT
The idea of the low discrepancy method is simple: approximate the integral of a function F : S 2 → ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315195674/823ed023-dd6c-41bb-a456-22174b64a780/content/imath14_1.tif"/> by a mean of functional values at prescribed points. This is a reasonable approach if the data set is somehow “equidistributed” over the sphere and if the function shows “good” behavior. These statements can be made rigorous by the concept of equidistribution and the notion of discrepancy. The starting point of the theory of equidistributed point sets is the work of H. Weyl [1916]. Results for Euclidean spaces ℝ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315195674/823ed023-dd6c-41bb-a456-22174b64a780/content/imath14_2.tif"/> can be found in L. Kuipers, H. Niederreiter [1974], I.H. Sloan, S. Joe [1994], S.K. Zaremba [1968], and many others. An extension of the Koksma–Hlawka formula to the spherical context has been proved by W. Freeden [1978c]. More detailed work about equidistribution on the sphere is due to E. Hlawka [1981], W. Freeden, M. Schreiner [1995], J. Cui, W. Freeden [1997].
