Distributions - Convolution and Fourier transform

In the first article , we introduced the main operations on distributions and discussed the fundamental notion of the derivative of a distribution.

This second article deals more specifically with the convolution product of distributions and their Fourier transform.

Used together, the convolution product and the Fourier transform are two very effective tools for solving certain differential equations. For example, solve :

$-\mathrm{\hspace{0.17em}}\frac{1}{{\omega }^2}{g}^{″}+g=f$

Formally, and using the properties of the convolution product and the Fourier transform , we can write :

$-\mathrm{\hspace{0.17em}}\frac{1}{{\omega }^2}{\left(2\mathrm{ }\mathrm{i}\mathrm{ }\mathrm{\pi }t\right)}^2\widehat{g}\left(t\right)+\widehat{g}\left(t\right)=\widehat{f}\left(t\right)$

or :

$\widehat{g}\left(t\right)\left[1+\frac{4{\mathrm{ \pi }}^2\mathrm{ }{t}^2}{{\omega }^2}\right]=\widehat{f}\left(t\right)$

As the function $[1+\frac{4{\mathrm{ \pi }}^2}{}$