Remarks Concerning Completeness of Translates in Function Spaces

This note explains how to translate the author's old result on cyclic vectors of the multiple shift operator into the language of completeness theorems for integer translates. This translation, together with those results, turns out to be a source for many completeness theorems. In particular, there follows the existence of functions f whose positive integer translates f(x−k), where k ₊ are complete in the spaces C^l₀( ), L^p( ), W^l_p( ), 2<p<∞, l=0, 1, …, as well as in their weighted and/or vector-valued analogues.