A SIMPLE PROOF OF STOLARSKY%26apos;S INVARIANCE PRINCIPLE

作者：Brauchart Johann S^*; Dick Josef

来源：Proceedings of the American Mathematical Society, 2013, 141(6): 2085-2096.

DOI：10.1090/S0002-9939-2013-11490-5

• Invariance principle
• reproducing kernel Hilbert space
• sphere
• spherical cap discrepancy
• sum of distances
• worst-case numerical integration error

Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575-582] showed a beautiful relation that balances the sums of distances of points on the unit sphere and their spherical cap L-2-discrepancy to give the distance integral of the uniform measure on the sphere which is a potential-theoretical quantity (Bjorck [Ark. Mat. 3 (1956), 255-269]). Read differently it expresses the worst-case numerical integration error for functions from the unit ball in a certain Hilbert space setting in terms of the L-2-discrepancy and vice versa. In this note we give a simple proof of the invariance principle using reproducing kernel Hilbert spaces.

• 出版日期2013-6