The following equation may be meaningful, but how can we make it well-defined
$$\delta(x-a)\cdot\delta(x-b)=0$$
**Question**: How do we defined this equation? Or more broadly define product between generalized functions with certain restrictions. Whether this definition satisfies the product rule [$D(fg)=Df\cdot g+f\cdot Dg$].




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**Added**: Maybe theory of hyperfunction can explain this, which I am not familiar with. Should its product satisfy the product rule?



Product of Dirac delta function
You can, of course, if you wish, consult the highly sophisticated treatise of Hörmander to verify that your result holds (for distinct values of $a$ and $b$—there is no serious text which claims this for the case $a=b$). Or you can refer to work which precedes this by decades and is completely elementary, in order to verify it as follows, without the use of functional analysis (to simplify the notation, we assume that $a=0$ and $b=1$). Then it is clear that the product exists and is equal to $0$ on each of the intervals $]-\infty,\frac 2 3[$ and $]\frac 1 3,\infty[$. We now piece these two distributions together (trivial case of “recollement des morceuses”) to get the zero distribution on the line.


We remark that the very elementary theory which justifies these simple manipulations is easily available online (reference below), where you will also find explicitly the following generalisations—just as easily proved:


1. if $f$ is a distribution on the line which equals a smooth function on a neighbourhood of $0$, then $f\delta\_0$ and indeed $f\delta\_0^{(n)}$ exist and are equal to zero if $f$ vanishes there.
2. If $a$ and $b$ are distinct, then $\delta^{(n)}(x-a)\delta^{(m)}(x-b)=0$.


The reference is to the site <https://jss100.campus.ciencias.ulisboa.pt> of the late portuguese mathematician J. Sebastião e Silva where you will find his text “Theory of Distributions” (the above results are in the fourth chapter “[Multiplication and Change of Variables](http://jss100.campus.ciencias.ulisboa.pt/Publicacoes/Textos-Didaticos-/Ens-JSS/TexDida_V3_III.1%20Theory%20of%20Distributions%20Chapter%204.pdf)”) under the "[Textos Didáticos](http://jss100.campus.ciencias.ulisboa.pt/Publicacoes/publicacoes.html)".


In response to the above request for an explicit formula for the product of a $C^n$ function and a distribution of order $n$, I can’t comment there but it is:


$$fg=\sum\_{k=0}^{n} \binom n k D^{n-k}(f^{(k)}G)
$$
where $f$ is $C^n$ and $g$ is a distribution of order $n$ of the form $D^n G $ with continuous $G$ ($D$ is the distributional derivative).


This is 4.1.2 in the reference given here. The most used version, where $g=\delta^{(n)}$, is 4.2.3.


