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Gräf, Manuel; Potts, Daniel : Sampling sets and quadrature formulas on the rotation group

Gräf, Manuel ; Potts, Daniel : Sampling sets and quadrature formulas on the rotation group

Author(s):
Gräf, Manuel
Potts, Daniel
Title:
Sampling sets and quadrature formulas on the rotation group
Electronic source:
application/pdf
Preprint series:
Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 8, 2009
Mathematics Subject Classification:
65T40 [ Trigonometric approximation and interpolation ]
33C55 [ Spherical harmonics ]
42C10 [ Fourier series in special orthogonal functions ]
42C15 [ Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions ]
65D32 [ Quadrature and cubature formulas ]
Abstract:
In this paper we construct sampling sets over the rotation group SO(3). The proposed construction is based on a parameterization, which reflects the product nature S^2xS^1 of SO(3) very well, and leads to a spherical Pythagorean-like formula in the parameter domain. We prove that by using uniformly distributed points on S^2 and S^1 we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on S^2 and S^1 lead to quadratures on SO(3), as well. For scattered data on SO(3) we give a necessary condition on the mesh norm such that the sampling nodes possesses nonnegative quadrature weights. We confirm our theoretical results with examples and numerical tests.
Keywords:
rotation group SO(3), spherical harmonics, sampling sets, quadrature rule, scattered data
Language:
English
Publication time:
4 / 2009