Numerical approximation of Fourier-transforms

Conclusion

A method has been presented for the numerical evaluation of the integrals occuring in Fourier transformation which is based upon the approximation of the transform as a function of its variable co. The numerical information necessary for the construction of the approximation is gathered by the formation of alternating trapezoidal and rectangular sums without the use of trigonometric functions The case of a polynomial approximation has been elaborated in detail and numerical results have been presented.

It is clear that using the same principle other types of approximating functions may be employed. In cases where the Fourier transform decreases faster than any power an exponential approximation may be effective. Further analysis and experimentation will serve to improve this seemingly powerful method.