ABSTRACT

To implement the Fourier transform directly, it is necessary to evaluate integrals involving highly oscillatory integrands. In rare cases this can be done analytically, such as for certain distributions or some rather simple functions. An exact analytical formula gives considerable insight into a problem, but the Fourier transform is far too useful to be restricted to just those few cases. In the more general case, we can employ a numerical integration method such as the extended Simpson’s rule ([2], p.605ff), but the rapid oscillation of etkx for large |fc| will necessitate many small subintervals and much work. Also, Simpson’s rule gives good approximations only when the function being transformed has four continuous derivatives.