Fast evaluation of real and complex exponential sums

TL;DR

This paper introduces a fast Fourier-Laplace transform method that combines butterfly and hierarchical approximations to efficiently evaluate integral transforms and polynomials in the complex plane, supported by numerical experiments.

Contribution

It presents a novel combination of butterfly and hierarchical approximation schemes for rapid evaluation of integral transforms and polynomials in the complex domain.

Findings

Efficient computation of oscillatory integral transforms.

Fast evaluation of polynomials at complex unit disk nodes.

Numerical experiments validate theoretical results.

Abstract

Recently, the butterfly approximation scheme and hierarchical approximations have been proposed for the efficient computation of integral transforms with oscillatory and with asymptotically smooth kernels. Combining both approaches, we propose a certain fast Fourier-Laplace transform, which in particular allows for the fast evaluation of polynomials at nodes in the complex unit disk. All theoretical results are illustrated by numerical experiments.