Numerical Quadrature of Fourier Transform

In some cases the function <£(£) or \pik) is given by a closed expression which is too complicated to permit a sufficiently accurate analytic evaluation of the integral for the entire range of the parameter x. In other cases 4>ik) or ^(&) may be available only in numerical form. The conventional methods of numerical quadrature (e.g., Simpson's rule) are not suitable for evaluation of the above integrals when x is large. There are two reasons for the failure of the standard methods in this case. First, because of the rapid oscillation of the trigonometric function when x is large the integrand cannot be accurately approximated by simple polynomials unless undesirably small intervals of integration are chosen. Secondly, the extremely strong cancellation between the contributions to the integral from regions where the trigonometric function is positive and regions where it is negative tends to accentuate the errors in the conventional integration procedures. Filon [1] and Luke [2] have suggested methods of integration which avoid the first difficulty by using polynomial approximations for ik) or \pik) rather than for the entire integrand. In this note we shall discuss a new scheme which attempts to alleviate the second difficulty as well. In this scheme all half cycles of the trigonometric functions are treated in an identical manner so that no cancellation errors arise. The integration scheme for the individual half cycles is based on a Gaussian type integration procedure [3] which minimizes the error in the final result for a given number of integration points per half cycle. In developing the integration formulas for the individual half cycles, it is imagined that the integrals in (la) and (lb) are evaluated by first performing the sum of the integrand at corresponding points in all half cycles and then integrating the result over a single half cycle. The sum over corresponding points in all the half cycles possesses certain properties which, particularly for large x, enable the final integral to be accurately evaluated with a small number of points per half cycle. In the actual numerical evaluation of the integral, the integrations are first performed over the individual half cycles, and then the total contributions of the various half cycles are summed. Although the integration formulas are not accurate for the individual half cycles, most of the error cancels when the sum over half cycles is performed (see section V). In performing the sum over half cycles, the standard techniques for accelerating the convergence of the sums of an oscillating series can be employed. This