Fast Fourier Transforms for Nonequispaced Data, II

A group of algorithms generalizing the fast Fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval [-π, π] is presented. The schemes of this paper are based on a combination of the classical fast Fourier transform with a version of the fast multipole method, and generalize both the forward and backward FFTs. Each of the algorithms requires O( N · log N + N · log(l/ϵ)) arithmetic operations, where ϵ is the precision of computations and N is the number of nodes; the CPU time requirement of the method is independent of the distribution of the nodes. The efficiency of the scheme is illustrated by several numerical examples. The approach of this paper is compared to the approach taken by Dutt et al. ("Fast Algorithms for Polynomial Interpolation, Integration, and Differentiation," Tech. Rep. 977, Department of Computer Science, Yale University, 1993) to the same set of problems. It turns out that the scheme of Dutt et al. is preferable for the forward problem, while the method introduced here is considerably more efficient for the inverse one.