Fourier transform of invariant differential operators on a locally- compact abelian group (Q1905269)

Let \(X\) be an LCA group and \(G\) its dual group. A function \(p : G \to \mathbb{C}\) is called a polynomial if its restriction \(p|_H\) to each compactly generated closed subgroup \(H \subset G\) can be represented as an ordinary polynomial of a finite collection of real characters of \(H\). The author proves for \(X\) a group analog of the following well known result: The Fourier transform turns any differential operator with constant coefficients on \(\mathbb{R}^n\) into an operator of multiplication by a polynomial.