A Stable and Accurate Butterfly Sparse Fourier Transform

Recently, the butterfly approximation scheme has been proposed for computing Fourier transforms with sparse and smooth sampling in frequency and spatial domain. We present a rigorous error analysis which shows how the local expansion degree depends on the target accuracy and the nonharmonic bandwidth. Moreover, we show that the original scheme becomes numerically unstable if a large local expansion degree is used. This problem is removed by representing all approximations in a Lagrange type basis instead of the previously used monomial basis. All theoretical results are illustrated by numerical experiments.