Linear combinations of frame generators in systems of translates.

CARLOS CABRELLI, CAROLINA MOSQUERA, VICTORIA PATERNOSTRO

Mar del Plata, Buenos Aires

Congreso; CIMPA 2013 "New Trends in Applied Harmonic Analysis, Sparse Representation, Compressed Sensign and Multifractal Analysis"; 2013

 A finitely generated shift invariant space $V$ is a closed subspace of $L^2(R^d)$ that is generated by the integer translates of a finite number of functions. A set of frame generators of $V$ is a set of functions whose integer translates form a frame of $V$ . In this work we show necessary and sufficient conditions in order that a minimal set of frame generators can be obtained by taking linear combinations of the given frame generators. We also obtain a complete characterization in the case of Riesz basis and orthonormal basis of translates. Surprisingly the results are very different to the recently studied case when the property to be a frame is not required.