Abstract
This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Hölder class Lip α , α>1, by a polynomial SO(N) loop of degree ≤n is of order \(\mathcal{O}(n^{-\alpha+\epsilon})\) for n≥k, where k=k(Q) is determined by topological properties of the loop and ε>0 is arbitrarily small. The convergence rate is therefore ε-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects.
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Communicated by Erik Koelink.
The work was done while at the University of Cambridge.
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Shingel, T. Trigonometric Approximation of SO(N) Loops. Constr Approx 32, 597–618 (2010). https://doi.org/10.1007/s00365-010-9107-6
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DOI: https://doi.org/10.1007/s00365-010-9107-6
Keywords
- Nonlinearly constrained trigonometric approximation
- Jackson-type inequality
- Polynomial loops
- Lie groups
- Higher-order exponential splitting