Remarks concerning completeness of translates in function spaces

This note explains how to translate the author's old result on cyclic vecto rs of the multiple shift operator into the language of completeness theorem s for integer translates, This translation, together with those results, tu rns out to be a source for many completeness theorems. In particular, there follows the existence of functions fa hose positive integer translates f(x - k), where k is an element of Z(+) are complete in the spaces C-0(l)(R), L-p(R), W-p(l)(R), 2 < p < infinity, l = 0, 1,..., as well as in their weig hted and/or vector-valued analogues, (C) 1999 Academic Press.