COMPLETENESS OF INTEGER TRANSLATES IN FUNCTION-SPACES ON R

We show that each of the Banach spaces C-0(R) and L(p)(R), 2 < p < inf inity,contains a function whose integer translates are complete. This function can also be chosen so that one of the following additional co nditions hold: (1) Its non-negative integer translates are already com plete. (2) Its integer translates form an orthonormal system in L(2)(R ). (3) Its integer translates form a minimal system. A similar result holds for the corresponding Sobolev space, for certain weighted L(2) s paces, and in the multivariate setting. We also prove some results in the opposite direction. (C) 1996 Academic Press, Inc.