Let's say we didn't know about the Dirac Delta distribution.
How could we prove that, if $f$ is integrable over $\mathbb{R}^n$, there exists a tempered distribution $d$ such that when convoluted with $f,$ $(f*d)(t) = f(t)$?
2026-04-11 23:25:47.1775949947
How do we prove a distribution identity exists?
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The Dirac Delta distribution is DEFINED in a way that would satisfy your search. If by (f*d)(t) you are defining a convolution integral over the whole real line then the delta distribution picks the value of f at zero but in the equation you have highlighted, it could be shifted to the value ‘t’ so your equation by definition gives the value at ‘t’ i.e. the unit impulse is at t rather than 0.
If I am right, Laurent Schwarz (Fields Medallist and Alexander Grothendieck’s doctoral adviser) showed that such a delta could NOT exist as an ordinary ‘function’ but he generalised functions to ‘distributions’ so that your equation is satisfied by definition. In doing so he also provided a rigorous basis for Dirac’s Delta function which was more of an ad-hoc (but genius)construction.