Fast Gauss transforms with complex parameters using NFFTs

at the target knots yj ∈ [−14 , 1 4 ], j = 1, . . . ,M , where σ = a + ib, a > 0, b ∈ R denotes a complex parameter. Fast Gauss transforms for real parameters σ were developed, e.g., in [15, 8, 9]. In [12], we have specified a more general fast summation algorithm for the Gaussian kernel. Recently, a fast Gauss transform for complex parameters σ with arithmetic complexity O(N log N + M) was introduced by Andersson and Beylkin [1]. In this paper, we show how our general fast summation algorithm developed in [11, 12, 6] can be specified for the Gaussian kernel with complex parameter σ to obtain a fast Gauss transform with arithmetic complexity O(N +M). This results in a simpler algorithm than those in [1] with competitive performance in practice. We prove error estimates concerning the dependence of the computational speed on the desired accuracy and the parameters a and |σ|. The heart of our algorithm is the discrete Fourier transform for non equispaced knots (NDFT), i.e., the evaluation of