Chapter 7 - Fast Translations: Basic Theory and O(p3) Methods

Abstract

This chapter focuses on O(p3) translation methods, which have a low asymptotic constant and is considered as the best available translation methods for low and moderate frequencies. O(p3) matrix-based translation methods are stable to errors and do not require additional data filtering, which is an important stabilization step for high-frequency translation methods based on diagonal forms of translation operators. The chapter considers some of the basics of representation of translation operators, including matrix and integral representations, and shows that rotational–coaxial translation decomposition with recursive computation of the rotation and translation matrix elements results in O(p3) translation algorithm with relatively small asymptotic constant. This method is faster than O(p4) method based on direct product of the translation matrix with the vector of expansion coefficients. The chapter also introduces another 0(p3) translation method based on sparse matrix decomposition of the translation operator, where the sparse matrices are differentiation matrices commutative with translation matrices.