Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs
SIAM Journal on Numerical Analysis
Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere
SIAM Journal on Numerical Analysis
Minimizing the Condition Number of a Gram Matrix
SIAM Journal on Optimization
Quadrature nodes meet stippling dots
SSVM'11 Proceedings of the Third international conference on Scale Space and Variational Methods in Computer Vision
Numerical integration with polynomial exactness over a spherical cap
Advances in Computational Mathematics
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In this paper we first establish a new variational characterisation of spherical designs: it is shown that a set X"N={x"1,...,x"N}@?S^d, where S^d:={x@?R^d^+^1:@?"j"="1^dx"j^2=1}, is a spherical L-design if and only if a certain non-negative quantity A"L","N(X"N) vanishes. By combining this result with a known ''sampling theorem'' for the sphere, we obtain the main result, which is that if X"N@?S^d is a stationary point set of A"L","N whose ''mesh norm'' satisfies h"X"""N