A general theory of convolution productin my childhood, I learned about convolution products for function over R (1). For quite a while now, I have played with polynomial rings, where also, the product is sometime called a convolution product (2). I am now discovering convolution products for arithmetic function and multiplicative functions [Apostol, Introduction to Analytic Number Theory] (3).In case (1), the domain is R, an ordered abelian group. In case (2), seeing a ring of polynomial as a group ring Z⟨G⟩, the domain is an abelain group, without ordering. In case (3), the domain seems to be the monoid ( Z , × ), with an ordered by the divisibilty relation.I don't have anything very formal, but it seems to me that there should be a general theory of convolution I don't know yet about ? Is the monoid structure the most general domain, or maybe something less structured as an acyclic graph ? Would you have lectures notes on such a theory ?

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A general theory of convolution productin my childhood, I learned about convolution products for function over R (1). For quite a while now, I have played with polynomial rings, where also, the product is sometime called a convolution product (2). I am now discovering convolution products for arithmetic function and multiplicative functions [Apostol, Introduction to Analytic Number Theory] (3).In case (1), the domain is R, an ordered abelian group. In case (2), seeing a ring of polynomial as a group ring Z⟨G⟩, the domain is an abelain group, without ordering. In case (3), the domain seems to be the monoid   (      Z    ,  ×  ), with an ordered by the divisibilty relation.I don&#039;t have anything very formal, but it seems to me that there should be a general theory of convolution I don&#039;t know yet about ? Is the monoid structure the most general domain, or maybe something less structured as an acyclic graph ? Would you have lectures notes on such a theory ?

Marshall Flowers · Accepted Answer

Step 1&quot;Is the monoid structure the most general domain, or maybe something less structured as an acyclic graph ?&quot;Just one example: A locally finite partially ordered set is a partially ordered set in which between two comparable elements there are only finitely many others. On each such set there is an &quot;incidence algebra&quot;. Each function f assigning a scalar to each interval   [  a  ,  b  ]  =  {  x  :  a  ≤  x  ≤  b  } is a member of the incidence algebra. The multiplication in this algebra is a sort of convolution:  (  f  ∗  g  )  [  a  ,  b  ]  =      ∑          x            :            a      ≤      x      ≤      b        f  [  a  ,  x  ]  g  [  x  ,  b  ]  .Step 2The case where the partial ordering is divisibility of positive integers is well known in number theory.

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