Completeness of a collection generated by translating a set of functions (Q1293704)

The problem of completeness of a collection of functions generated from translates and dilates of a single function in a certain functional space is common both in approximation theory as well as in harmonic analysis. It has a long history and can be traced back to a work of N. Wiener in the 1930s. Motivated by Wiener's theorems, a natural question arises as to under what conditions the collection of functions \(\{f({\mathbf x}+{\mathbf c}):f\in {\mathbf A};{\mathbf c}\in {\mathbf S}\},\) where \(\mathbf A\) is a given set of functions defined on \(\mathbb R^n\) and \(\mathbf S\) is a given subset in \(\mathbb R^n,\) is complete in various function spaces. The aim of the present paper is to find sufficient conditions on the sets \(\mathbf A\) and \(\mathbf S\) such that the collection \(\{f({\mathbf x}+{\mathbf c})\}\) is complete in the functional space \(L_p(\mathbb R^n)\), \(C(\mathbb R^n)\), \(C_0(\mathbb R^n)\) and in the subspace of all entire functions \(C^\infty(\mathbb R^n)\).