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I'm trying to figure out the algebraic properties of the Cauchy product ($c_n=\sum_{k=0}^na_kb_{n-k}$). I'm doing it by myself and I feel like there should be some literature on it. I didn't find it on Google nor in jstor. It is a comutative product, most sequences have inverses, I'm starting to see a pattern in the inverse, but this is too basic and it should be written somewhere. Do any of you know where to find this algebraic properties, more on the formal series side, of the Cauchy product?

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  • $\begingroup$ The phrase "discrete convolution" might be helpful as a search term - as the wiki page states, the Cauchy product is basically a discrete analogue of convolution of functions. $\endgroup$ Commented Aug 18 at 1:42
  • $\begingroup$ @NoahSchweber this wiki page is pretty much like very other paper: they focus on convergence and baby examples, but I would like to see how the inverses look like and some applications of its algebra. The wiki mentions it is a convolution but doesn't even prove it is associative for example. $\endgroup$ Commented Aug 18 at 1:51
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    $\begingroup$ Again, I think looking at convolution will help. E.g. the proof of associativity for convlution is the same in the continuous and discrete (= Cauchy product) case; see e.g. here. In general, I'd expect most of the algebraic properties to be the same (and via the same proof as) those of continuous convolution, and for the latter to be treated in more detail in the literature. $\endgroup$ Commented Aug 18 at 1:54
  • $\begingroup$ In the context of formal power series the convolution corresponds to the product of these series. The existence of inverses is connected to the characterication of units in the ring of formal power series. $\endgroup$ Commented Aug 18 at 1:58
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    $\begingroup$ Fairly certain you're (really) asking about blind deconvolution. ui.adsabs.harvard.edu/abs/2018InvPr..34c5003L/abstract is a toe-hold to the literature. Maybe deconvolution is a slightly better fit. $\endgroup$ Commented Aug 18 at 3:34

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