In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFT-algorithms with respect to roundoff errors and apply NDFT-algorithms for the fast computation of Bessel transforms.

This paper presents some new results on numerical stability for various fast trigonometric transforms. In a worst case study, we consider the numerical stability of the classical fast Fourier transform (FFT) with respect to different precomputation methods for the involved twiddle factors and show the strong influence of precomputation errors on the numerical stability of the FFT. The examinations are extended to fast algorithms for the computation of discrete cosine and sine transforms and to efficient computations of discrete Fourier transforms for nonequispaced data. Numerical tests confirm the theoretical estimates of numerical stability.