Shift invariant spaces and BMO
A family of integer translates of a square integrable function forms a basic sequence if and only if the corresponding periodization function satisfies the Muckenhoupt condition. We develop a study of the periodization function from the point of view of the theory of weights. There is a well-known connection of such weights with the BMO space, and periodization functions that correspond to bounded functions via the BMO connections, are precisely the one that generate Riesz bases of translates. We are particularly interested in the ones which do not have that property, since they give us an additional insight into a class of conditional Schauder bases of translates. We employ the Garnett-Jones distance to identify some of the subclasses in this family. This is joint work with Hrvoje Sikic, University of Zagreb, Croatia.