The Nonequispaced Fast Fourier Transform: Implementation and Error Analysis

The Fast Fourier Transform (FFT) reduces the number of computations required for the straightforward calculation of the discrete Fourier transform. For a one-dimensional signal of length N , this provides a reduction in complexity from O(N) to O(N log2 N) operations. Recently, fast algorithms have been introduced for discrete Fourier transforms on nonequispaced data sets (NDFT). These algorithms reduce the complexity to O(N log2 N) as well, but do so by providing approximations. In this thesis, we study the classical FFT, and provide algorithms for fast computation of the NDFT. We pay special attention to the NDFT with nonequispaced data in the time/space domain and equispaced data in the frequency domain, as this problem is equivalent to evaluating a trigonometric polynomial at scattered points. The algorithm in this case uses ideas from the theory of shift-invariant approximation. In addition to providing and implementing the algorithm, we provide reasonably sharp approximation error estimates.