Fourier integrals

The Fourier transformation on the real line is analogous to the Fourier transformation of locally integrable periodic functions, where the exponentials :

are replaced by the continuous family of exponentials :

and where integration over a period interval is replaced by integration over as a whole.

Besides, a physicist would say that a function defined on is a periodic function of infinite period (!), and we can give a unified presentation of Fourier series and integrals in the abstract framework of locally compact abelian groups. The fact remains that, in the case of Fourier series, the basic group is the compact group of reals modulo 2π, whereas in the case of Fourier integrals, the basic group is the compact group of reals. This, as we shall see, is a major difference; even if,...