There is hardly a phase of modern astrophysics to which Fourier techniques do not lend some insight or practical advantage. Fourier concepts prove useful in the context of line absorption coefficients, the analysis of line profiles, spectrograph resolution, telescope diffraction, and the study of noise. In these and other applications, convolutions appear in the physics of the situation. Usually it is much easier to visualize a product of functions in place of their convolution and this can be done with Fourier transforms through the convolution theorem.

This chapter forms an introduction to Fourier transforms for those unfamiliar with them and a useful refresher for those who have studied them in past years. The treatment is highly abbreviated, but covers all the concepts used in the remainder of the book. Those wishing to learn the material in a more rigorous and extensive way are referred to the books dealing specifically with Fourier transforms, for instance, the books of Jennison (1961), Bracewell (1965), and Gaskill (1978).

The definition

The Fourier transform of a function is a specification of the amplitudes and phases of sinusoidals which, when added together, reproduce the function. Only one-dimensional functions are treated here. Expansion to two or higher dimensions can be done by the reader with modest effort.