Spanning and independence properties of finite frames

Abstract

The fundamental notion of frame theory is redundancy. It is this property which makes frames invaluable in so many diverse areas of research in mathematics, computer science, and engineering, because it allows accurate reconstruction after transmission losses, quantization, the introduction of additive noise, and a host of other problems. This issue also arises in a number of famous problems in pure mathematics such as the Bourgain-Tzafriri conjecture and its many equivalent formulations. As such, one of the most important problems in frame theory is to understand the spanning and independence properties of subsets of a frame. In particular, how many spanning sets does our frame contain? What is the smallest number of linearly independent subsets into which we can partition the frame? What is the least number of Riesz basic sequences that the frame contains with universal lower Riesz bounds? Can we partition a frame into subsets which are nearly tight? This last question is equivalent to the infamous Kadison–Singer problem. In this section we will present the state of the art on partitioning frames into linearly independent and spanning sets. A fundamental tool here is the famous Rado-Horn theorem. We will give a new recent proof of this result along with some nontrivial generalizations of the theorem.