Some Eigenfunction Methods for Computing a Numerical Fourier Transform

A number of methods for calculating the Fourier transform ofa function given numerically are studied. These methods exploitthe fact that the Hermite functions are eigen-functions of theFourier transform. The transforms of four types of functionsare considered: (i) functions of the form p(x) exp (–x²/2),where p(x) is a polynomial, (ii) functions with bounded support.(iii) rapidly decreasing functions, and (iv) functions whosetransform has bounded support. In each case algorithms for calculatingthe transformed function are derived. Error estimates are madein two of the cases and results of numerical experiments presentedin an appendix.