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The product of two PDFs is not usually a PDF; but it is a (possibly zero) scalar multiple of another PDF.

I will call a set $S$ of PDFs over $\mathbb{R}$ closed under translation and multiplication if

  1. $f(x)\in F \Rightarrow f(x-a) \in F$ for all $a\in\mathbb{R}$;
  2. $f, g\in F \Rightarrow fg = \alpha h$ for some $h\in F$ and $\alpha \in \mathbb{R}$.

The set of all Gaussians is one notable nontrivial example of such an $S$. Are there any other notable examples? How rich is the space of such sets $S$?

user7530
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    Not sure I get your question. Isn't the set of all pdfs with compact support an example? – spaceisdarkgreen Jun 29 '17 at 19:21
  • @spaceisdarkgreen Sure; my question is, how rich is the set of examples? Other than the obvious examples (the empty set; the set of all PDFs) and the Gaussians, is it very easy to construct a large number of such $S$? Or are there only a few (or a few families) of $S$ with this property? – user7530 Jun 29 '17 at 20:52

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