Linear combinations of frame generators in systems of translates

TL;DR

This paper investigates conditions under which a minimal set of frame generators for a shift-invariant space can be derived through linear combinations of existing generators, highlighting differences from non-frame cases.

Contribution

It provides necessary and sufficient conditions for constructing minimal frame generator sets via linear combinations in shift-invariant spaces.

Findings

Characterization of when linear combinations yield minimal frame generators

Contrasts with non-frame generator cases

Conditions differ significantly from previously studied non-frame scenarios

Abstract

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 for $V$ is a set of functions whose integer translates form a frame for $V$. In this note we give 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. Surprisingly the results are very different to the recently studied case when the property to be a frame is not required.