Cours of convolution product in theory of distribution

Question

i search a good and complet cours of convolution product in theory of distribution who contains definition of convolution of an function with distribution and convolution of two distributions (for example how we calculate $\delta * \delta$ with $\delta$ is Dirac distribution, or how we calculate $(H * \delta)*\delta$ where $H$ is function of Heaviside) with properties. Can you give me some one please. Thank's in advance.

Answer 1

Two good sources are Gelfand, Shilov, Generalized functions, Volume 1, which gives a rigorous treatment and also has some very lucid explanations for the concepts, and Kanwal, Generalized Functions: Theory and Applications.

You need to impose some conditions on the supports of functionals $f$ and $g$ in order for $f*g$ to be well-defined. If $g$ is an ordinary locally integrable function, you take the functional induced by $g$. The definition will be $$(f*g, \phi) = (f, x \mapsto (g, y \mapsto \phi(x + y))),$$ from which it's easy to find $f*\delta$.

Answer 2

If you read French, then "Théorie des distributions" by Laurent Schwartz is the book of choice for you. Schwartz is the one who founded distribution theory, and as far as I can tell, his monograph is pretty much up to date.

By the way, $\delta$ is the identity with regard to convolution, so $\delta * T = T$ for any distribution $T$.