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This Demonstration computes an improved fast Fourier transform that we call XFT to distinguish it from the usual FFT algorithms. The XFT is given by the product ... , where ... is a diagonal matrix with ... diagonal element given by ... , ... , ... is the standard discrete Fourier transform, and ... . Therefore, the XFT is as fast as the FFT algorithm used to compute the discrete Fourier transform. However, the output of the XFT is more accurate than the output of the FFT because it comes from an algorithm to compute the fast fractional Fourier transform based on a convergent quadrature formula. This Demonstration approximates the special case of the continuous Fourier transform defined as ... ... evaluated at the points ... , ... .
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