How Well Does the Finite Fourier Transform Approximate the Fourier Transform?

We show that the answer to the question in the title is “very well indeed.” In particular, we prove that, throughout the maximum possible range, the finite Fourier coefficients provide a good approximation to the Fourier coefficients of a piecewise continuous function. For a continuous periodic function, the size of the error is estimated in terms of the modulus of continuity of the function. The estimates improve commensurately as the functions become smoother. We also show that the partial sums of the finite Fourier transform provide essentially as good an approximation to the function and its derivatives as the partial sums of the ordinary Fourier series. Along the way we establish analogues of the Riemann-Lebesgue lemma and the localization principle. c © 2004 Wiley Periodicals, Inc.