Linear combinations of frame generators in systems of translates

Abstract

A finitely generated shift invariant space V is a closed subspace of L2(Rd) that can be generated by the integer translates of a finite number of functions. A set of frame generators for V is a set of functions whose integer translates form a frame for V . In this note we give necessary and sucient conditions in order that a minimal set of frame generators can be obtained by taking linear combinations of the given frame generators. Surprisingly the results are very dierent from the recently studied case when the property to be a frame is not required.