Multirate systems and filter banks
Multirate systems and filter banks
Convergence and C1 analysis of subdivision schemes on manifolds by proximity
Computer Aided Geometric Design - Special issue: Geometric modelling and differential geometry
Notes: Close-to-optimal bounds for SU(N) loop approximation
Journal of Approximation Theory
| Hi-index | 0.00 |
The problem of finding the correct asymptotic rate of approximation by polynomial loops in dependence of the smoothness of the elements of a loop group seems not well-understood in general. For matrix Lie groups such as SU(N), it can be viewed as a problem of nonlinearly constrained trigonometric approximation. Motivated by applications to optical FIR filter design and control, we present some initial results for the case of SU(N)-loops, N=2. In particular, using representations via the exponential map and first order splitting methods, we prove that the best approximation of an SU(N)-loop belonging to a Holder-Zygmund class Lip"@a, @a1/2, by a polynomial SU(N)-loop of degree @?n is of the order O(n^-^@a^/^(^1^+^@a^)) as n-~. Although this approximation rate is not considered final, to our knowledge it is the first general, nontrivial result of this type.