Splitting methods for SU(N) loop approximation

Abstract

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 Hölder-Zygmund class Lip α , α > 1/2, by a polynomial SU(N )-loop of degree ≤ n is of the order O(n −α/(1+α) ) as n → ∞. Although this approximation rate is not considered final, to our knowledge it is the first general, nontrivial result of this type.

Trigonometric Approximation of SO(N) Loops

“…This approximation rate was significantly improved when employing higher-order splitting methods [8], at least for α > 1. In short, the approach to the problem was as follows: by suitable factorization, the problem was reduced to studying U(t) = e X(t) , where X(t) ∈ Lip α (T → su(N)); X(t) was approximated componentwise by linear methods, and then a splitting method for the exponential map was applied to obtain a polynomial SU(N )-valued loop.…”

“…The main result of the paper is the following Jackson-type estimate for the Hölder classes Lip α (T → SO(N )) of loops: In a previous paper [7] we proved that the approximation of an SU(N )-loop belonging to a Hölder-Zygmund class Lip α , α > 1/2, by a polynomial SU(N )-loop of degree ≤n is of order O(n −α/(1+α) ) as n → ∞. This approximation rate was significantly improved when employing higher-order splitting methods [8], at least for α > 1.…”

“…It is natural to aim to reduce the problem to the case of Lie algebra loops, for which we can apply the results of classical approximation theory for linear spaces. This was a routine step in the algorithm in [7], since the loop space C(T → SU(N )) is connected. In the real setting, there exist SO(N ) loops which cannot be contracted to a point, hence the exponential representation is not always feasible.…”

“…Any element in the principal component can be represented as a product of exponentials of the algebra loops using, for example, a simple homotopy argument (cf. [8]). As the group theory suggests, any connected component of a group is a coset of the identity component.…”

“…Here we continue the study of asymptotic properties of trigonometric approximation of loops initiated in [7,8]. The motivation for this work is mainly theoretical.…”