The s-energy of spherical designs on S2 |
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Authors: | Kerstin Hesse |
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Affiliation: | (1) School of Mathematics and Statistics, The University of New South Wales, Sydney, New South Wales, 2052, Australia |
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Abstract: | This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S 2. A spherical n-design is a point set on S 2 that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy E s (X) of a point set of m distinct points is the sum of the potential for all pairs of distinct points . A sequence Ξ = {X m } of point sets X m ⊂S 2, where X m has the cardinality card(X m )=m, is well separated if for each pair of distinct points , where the constant λ is independent of m and X m . For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy E s (X m(n)) for well separated sequences Ξ = {X m(n)} of spherical n-designs X m(n) with card(X m(n))=m(n). |
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Keywords: | Acceleration of convergence Energy Equal weight cubature Equal weight numerical integration Orthogonal polynomials Sphere Spherical design Well separated point sets on sphere |
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