An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect s...

This paper describes FFTW, a portable C package for computing the oneand multidimensional complex discrete Fourier transform (DFT). FFTW is typically faster than all other publicly available DFT software, including the well-known FFTPACK and the code from Numerical Recipes. More interestingly, FFTW is competitive with or better than proprietary, highly-tuned codes such as Sun’s Performance Libr...

This paper is devoted to some problems that appear in derivations of the discrete time Fourier transform from a formula for its continuous counterpart transformation into frequency domain as well those regarding inverse direction. In particular, latter ones remained so far an unresolved problem. It solved first here. Many detailed explanations accompanying solution found are presented. Finally,...

Among the different forms of spectroscopy, nuclear magnetic resonance (NMR) shows an intimate connection between theory and experiment. NMR therefore enjoys great popularity in the development of new methods. Following the rapid development in electronic and computer technology, NMR techniques evolved from singlechannel to multi-channel and multi-dimensional modes. The original continuous wave ...

Fourier transform is among the most widely used tools in computer science. Computing the Fourier transform of a signal of length N may be done in time Θ(N logN) using the Fast Fourier Transform (FFT) algorithm. This time bound clearly cannot be improved below Θ(N), because the output itself is of length N . Nonetheless, it turns out that in many applications it suffices to find only the signifi...