Spanning and Independence Properties of Finite Frames
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 lo
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Spanning and Independence Properties of Finite Frames Peter G. Casazza and Darrin Speegle
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. Keywords Spanning sets · Independent sets · Redundancy · Riesz sequence · Rado-Horn theorem · Spark · Maximally robust · Matroid · K-ordering of dimensions
P.G. Casazza () Department of Mathematics, University of Missouri, Columbia, MO 65211, USA e-mail: [email protected] D. Speegle Department of Mathematics and Computer Science, Saint Louis University, 221 N. Grand Blvd., St. Louis, MO 63103, USA e-mail: [email protected] P.G. Casazza, G. Kutyniok (eds.), Finite Frames, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8373-3_3, © Springer Science+Business Media New York 2013
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P.G. Casazza and D. Speegle
3.1 Introduction The primary focus of this chapter is the independence and spanning properties of finite frames. More specifically, we will be looking at partitioning frames into sets {Ak }K k=1 which are linearly independent, spanning, or both. Since increasing the number of sets in the partition makes it easier for each set to be independent, and harder to span, we will be looking for the smallest K needed to be able to choose independent sets, and the largest K allowed so that we still have each set of vecN tors spanning. In order to fix notation, let Φ = (ϕi )M i=1 be a set of vectors in H , not necessarily a frame. It is clear from dimension counting that if Ai is linearly independent for each 1 ≤ i ≤ K, then K ≥ M/N. It is also clear from dimension counting that if Ai spans HN for each 1 ≤ i ≤ K, then K ≤ M/N. So, in terms of linear independence and spanning properties, Φ is most “spread out” if it can be partitioned into K = M/N linearly independent sets, M/N of which are
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