A Fast Method for the Numerical Evaluation of Continuous Fourier and Laplace Transforms

The fast Fourier transform (FFT) is often used to compute numerical approximations to continuous Fourier and Laplace transforms. However, a straightforward application of the FFT to these problems often requires a large FFT to be performed, even though most of the input data to this FFT may be zero and only a small fraction of the output data may be of interest. In this note, the "fractional Fourier transform," previously developed by the authors, is applied to this problem with a substantial savings in computation.