4. Fourier transform

We now turn to the Fourier transform, the very basis of modern harmonic analysis. We are still working in the space ${ℝ}^N$ , although in practice we often have N = 1; our notations, however, are designed to make very little difference to this simpler case (see the paragraph on notations): for there is no additional difficulty in the general case.

The Fourier transformation makes extensive use of the complex exponential, so it's worth recalling its basic properties. The exponential of a complex number c = a + ib can be defined from the sum of a series, or more simply using the real exponential, cosine and sine functions, by the formula :