Fourier integrals

The Fourier transformation on the real line $ℝ$ is analogous to the Fourier transformation of locally integrable periodic functions, where the exponentials : $e_n\left(t\right)=\exp \left(\mathrm{i}\mathit{nt}\right)\left(n\mathrm{\hspace{0.17em}}\mathbf{entier}\right)$

are replaced by the continuous family of exponentials : $e_x\left(t\right)=\exp \left(\mathrm{i}\mathrm{ }x\mathrm{ }t\right)\left(x\mathrm{\hspace{0.17em}}\mathrm{réel}\right),$

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

Besides, a physicist would say that a function defined on $ℝ$ 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, in both cases, convolution is transformed into ordinary multiplication, which is a powerful tool for solving partial differential equations, the phenomena are often quite different; for example, there is no longer always uniqueness for the heat equation with initial data, or the orthonormal bases that come into play have nothing similar: e _n exponential basis in the case of Fourier series, Hermite function basis in the case of Fourier integrals, and so on.

Consequently, despite the similarities between the two theories, it seems...