A New Algorithm for the Nonequispaced Fast Fourier Transform on the Rotation Group

Abstract

We develop an approximate algorithm to efficiently calculate the discrete Fourier transform on the rotation group $\text{SO}(3)$. Our method needs $\mathcal{O}(L^3 \log L \log (1/\varepsilon) + \log^3 (1/\varepsilon)\,Q)$ arithmetic operations for a degree-$L$ transform at $Q$ nodes free of choice and with desired accuracy $\varepsilon$. Our main contribution is a method that allows us to replace finite expansions in Wigner-$d$ functions of arbitrary orders with those of low orders. It is based on new insights into the structure of related semiseparable matrices. This enables us to employ an established divide-and-conquer algorithm for symmetric semiseparable eigenproblems together with the fast multipole method to achieve an efficient algorithm.