Chapter 8 - Asymptotically Faster Translation Methods

Abstract

This chapter presents various algorithms that can be used for reduction of complexity of translation. It shows some exact decompositions of the rotation, truncated coaxial translation, and truncated general translation matrices. Fast rotation algorithm has complexity 0(p2 log p) and can be used in various methods of translations. The other two algorithms relies on the translation matrix diagonalization via fast Legendre and fast spherical transforms, which can be performed with complexity 0(p2 log2p). It is the fastest complexity that can be currently achieved for a single translation with truncated translation matrices. Low- and high-frequency asymptotic expansions are also reviewed, which shows that for very low frequencies, an arbitrary translation can be performed with complexity 0(p2 log p). For higher frequencies, complete asymptotic representations that enable any order asymptotic expansions for coaxial translation operators, are obtained. These expansions, however, are derived for fixed sizes of the truncated translation matrices, and further research is needed to obtain uniform asymptotic representation of translation operators. The diagonal forms of the translation operators interpreted as a method of signature function achieves complexity 0(p2) for translations when implemented for the class of band-unlimited functions. Theoretically, this is the lowest complexity of translations and while the tests show that the method works and can be practical, its accuracy is limited at low frequencies and the algorithm is unstable for a sequence of translations. This instability can be damped by the application of the spherical filters, which control the function bandwidths.