Tutorial/Artical didactique: A high-accuracy mathematical and numerical method for Fourier transform, integral, derivatives, and polynomial splines of any order

Abstract

From few simple and relatively well-known mathematical tools, an easily understandable, though powerful, method has been devised that gives many useful results about numerical functions. With mere Taylor expansions, Dirac delta functions and Fourier transform with its discrete counterpart, the DFT, we can obtain, from a digitized function, its integral between any limits, its Fourier transform without band limitations and its derivatives of any order. The same method intrinsically produces polynomial splines of any order and automatically generates the best possible end conditions. For a given digitized function, procedures to determine the optimum parameters of the method are presented. The way the method is structured makes it easy to estimate fairly accurately the error for any result obtained. Tests conducted on nontrivial numerical functions show that relative as well as absolute errors can be much smaller than 10-100, and there is no indication that even better results could not be obtained. The method works with real or complex functions as well; hence, it can be used for inverse Fourier transforms too. Implementing the method is an easy task, particularly if one uses symbolic mathematical software to establish the formulas. Once formulas are worked out, they can be efficiently implemented in a fast compiled program. The method is relatively fast; comparisons between computation time for fast Fourier transform and Fourier transform computed at different orders are presented. Accuracy increases exponentially while computation time increases quadratically with the order. So, as long as one can afford it, the trade-off is beneficial. As an example, for the fifth order, computation time is only ten times greater than that of the FFT while accuracy is 108 times better. Comparisons with other methods are presented.PACS Nos.: 02.00 and 02.60