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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.

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  • $\begingroup$ Could you provide a definition of the subject of your interest? (maybe a Wikipedia one, or on mathworld.wolfram... at least to get understood). $\endgroup$ – metamorphy Aug 21 '18 at 9:50
  • $\begingroup$ I edit my post, i search definition for convolution of function with distribution and convlution of two distributions, with somes properties. $\endgroup$ – erika Aug 21 '18 at 9:53
  • $\begingroup$ Is it this what you're seeking for details on? (Or is it from some other field?) $\endgroup$ – metamorphy Aug 21 '18 at 10:04
  • $\begingroup$ no, not probability distrubtions. I ask for the second item who purpose. $\endgroup$ – erika Aug 21 '18 at 10:17
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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 $f*\delta$.

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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 $T$.

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  • $\begingroup$ there is an other book in english? Please $\endgroup$ – erika Aug 21 '18 at 11:07

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