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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alexeev, B., Cahill, J., Mixon, D.G.: Full spark frames, preprint
Balan, R.: Equivalence relations and distances between Hilbert frames. Proc. Am. Math. Soc. 127(8), 2353–2366 (1999)
Bodmann, B.G., Casazza, P.G.: The road to equal-norm Parseval frames. J. Funct. Anal. 258(2), 397–420 (2010)
Bodmann, B.G., Casazza, P.G., Paulsen, V.I., Speegle, D.: Spanning and independence properties of frame partitions. Proc. Am. Math. Soc. 40(7), 2193–2207 (2012)
Bodmann, B.G., Casazza, P.G., Kutyniok, G.: A quantitative notion of redundancy for finite frames. Appl. Comput. Harmon. Anal. 30, 348–362 (2011)
Bourgain, J.: Λ p -Sets in Analysis: Results, Problems and Related Aspects. Handbook of the Geometry of Banach Spaces, vol. I, pp. 195–232. North-Holland, Amsterdam (2001)
Cahill, J.: Flags, frames, and Bergman spaces. M.S. Thesis, San Francisco State University (2010)
Casazza, P.G.: Custom building finite frames. In: Wavelets, Frames and Operator Theory, College Park, MD, 2003. Contemp. Math., vol. 345, pp. 61–86. Am. Math. Soc., Providence (2004)
Casazza, P., Christensen, O., Lindner, A., Vershynin, R.: Frames and the Feichtinger conjecture. Proc. Am. Math. Soc. 133(4), 1025–1033 (2005)
Casazza, P.G., Fickus, M., Weber, E., Tremain, J.C.: The Kadison–Singer problem in mathematics and engineering—a detailed account. In: Han, D., Jorgensen, P.E.T., Larson, D.R. (eds.) Operator Theory, Operator Algebras and Applications. Contemp. Math., vol. 414, pp. 297–356 (2006)
Casazza, P.G., Kovačević, J.: Equal-norm tight frames with erasures. Adv. Comput. Math. 18(2–4), 387–430 (2003)
Casazza, P., Kutyniok, G.: A generalization of Gram-Schmidt orthogonalization generating all Parseval frames. Adv. Comput. Math. 18, 65–78 (2007)
Casazza, P.G., Kutyniok, G., Speegle, D.: A redundant version of the Rado-Horn theorem. Linear Algebra Appl. 418, 1–10 (2006)
Casazza, P.G., Kutyniok, G., Speegle, D., Tremain, J.C.: A decomposition theorem for frames and the Feichtinger conjecture. Proc. Am. Math. Soc. 136, 2043–2053 (2008)
Casazza, P.G., Peterson, J., Speegle, D.: Private communication
Casazza, P.G., Tremain, J.: The Kadison–Singer problem in mathematics and engineering. Proc. Natl. Acad. Sci. 103(7), 2032–2039 (2006)
Christensen, O., Lindner, A.: Decompositions of Riesz frames and wavelets into a finite union of linearly independent sets. Linear Algebra Appl. 355, 147–159 (2002)
Edmonds, J., Fulkerson, D.R.: Transversals and matroid partition. J. Res. Natl. Bur. Stand. Sect. B 69B, 147–153 (1965)
Horn, A.: A characterization of unions of linearly independent sets. J. Lond. Math. Soc. 30, 494–496 (1955)
Janssen, A.J.E.M.: Zak transforms with few zeroes and the tie. In: Feichtinger, H.G., Strohmer, T. (eds.) Advances in Gabor Analysis, pp. 31–70. Birkhäuser, Boston (2002)
Oxley, J.: Matroid Theory. Oxford University Press, New York (2006)
Rado, R.: A combinatorial theorem on vector spaces. J. Lond. Math. Soc. 37, 351–353 (1962)
Acknowledgements
The first author is supported by NSF DMS 1008183, NSF ATD 1042701, and AFOSR FA9550-11-1-0245.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Casazza, P.G., Speegle, D. (2013). Spanning and Independence Properties of Finite Frames. In: Casazza, P., Kutyniok, G. (eds) Finite Frames. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8373-3_3
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8373-3_3
Publisher Name: Birkhäuser, Boston
Print ISBN: 978-0-8176-8372-6
Online ISBN: 978-0-8176-8373-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)