On the Bézout equation in the ring of periodic distributions

The aim of this short note is to study some algebraic and topological questions associated with the “Bézout equation” b1a1 + · · · + bNaN = e, where bi , ai (1 ≤ i ≤ N) are elements of the commutative unital topological ring (DA(R), +, *, TD′A(Rd)), defined below, and e denotes the identity element (which will be the locally finite sum of Dirac distributions placed at a lattice formed by the periods in A, as explained below). The Bézout equation in rings of distributions arises in problems of robust filtering, image processing, etc, see for example [2].