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2011

Every Hilbert space frame has a Naimark complement

Casazza, Peter G., Fickus, Matt, Mixon, Dustin et al.

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Naimark complements for Hilbert space Parseval frames are one of the most fundamental and useful results in the field of frame theory.

We will show that actually all Hilbert space frames have Naimark complements which possess all the usual properties for Naimark complements with one notable exception.
So these complements can be used for equiangular frames, RIP property, fusion frames etc.
Along the way, we will correct a mistake in a recent fusion frame paper where chordal distances for Naimark complements are computed incorrectly.

Every Hilbert space frame has a Naimark complement · Around

Built on

R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985
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J. H. Conway, R. H. Hardin, and N. J. A. Sloane, ÒPacking lines, planes, etc.: packings in Grassmannian spaces,Ó Experiment. Math., 5
1996
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O. Christensen, An introduction to frames and Riesz bases
2003
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P.G. Casazza and G. Kutyniok, Frames of subspaces
2004
Earlier work this paper cites.

Similar

E.J. Candes, T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory 44
2005
Cited alongside, same era.
P. G. Casazza, G. Kutyniok, and S. Li, Fusion frames and distributed processing
2008
Cited alongside, same era.
P. G. Casazza and M. Fickus, Minimizing fusion frame potential
2009
Cited alongside, same era.
G. Kutyniok, A. Pezeshki, A.R. Calderbank, and T. Liu, Robust Dimension Reduction, Fusion Frames, and Grassmannian Packings. Appl. Comput. Harmon. Anal. 26
2009
Cited alongside, same era.
J. Cahill and P. G. Casazza, The Paulsen problem in operator theory
Original
Cited in the paper.

Then

R. Calderbank, P.G. Casazza, A. Heinecke, G. Kutyniok and A. Pezeshki, Sparse fusion frames: Existence and construction
2011
Closest in time.
P.G. Casazza, M. Fickus, D. Mixon, Y. Wang and Z. Zhou, Constructing tight fusion frames
2011
Closest in time.
P.G. Casazza and G. Kutyniok, Introduction to finite frame theory
2012
Closest in time.
M. Fickus, D. Mixon, and J. Tremain, Steiner equiangular tight frames
2012
Closest in time.

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