Constructing Finite Frames with a Given Spectrum
Broadly speaking, frame theory is the study of how to produce well-conditioned frame operators, often subject to nonlinear application-motivated restrictions on the frame vectors themselves. In this chapter, we focus on one particularly well-studied type
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Constructing Finite Frames with a Given Spectrum Matthew Fickus, Dustin G. Mixon, and Miriam J. Poteet
Abstract Broadly speaking, frame theory is the study of how to produce wellconditioned frame operators, often subject to nonlinear application-motivated restrictions on the frame vectors themselves. In this chapter, we focus on one particularly well-studied type of restriction: having frame vectors of prescribed lengths. We discuss two methods for iteratively constructing such frames. The first method, called Spectral Tetris, produces special examples of such frames, and only works in certain cases. The second method combines the idea behind Spectral Tetris with the classical theory of majorization; this method can build any such frame in terms of a sequence of interlacing spectra, called eigensteps. Keywords Tight frames · Schur-Horn · Majorization · Interlacing
2.1 Introduction Although we work over the complex field for the sake of generality, the theory presented here carries over verbatim to the real-variable setting. The syntheN M N sis operator of a sequence of vectors Φ = {ϕm }M m=1 in C is Φ : C → C , M Φy := m=1 y(m)ϕm . That is, Φ is the N × M matrix whose columns are the ϕm ’s. Note that we make no notational distinction between the vectors themselves and the synthesis operator they induce. Φ is said to be a frame for CN if there exist frame bounds 0 < A ≤ B < ∞ such that Ax2 ≤ Φ ∗ x2 ≤ Bx2 for all x ∈ CN . The optimal frame bounds A and B of Φ are the least and greatest eigenvalues of the
M. Fickus () · M.J. Poteet Department of Mathematics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA e-mail: [email protected] D.G. Mixon Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA P.G. Casazza, G. Kutyniok (eds.), Finite Frames, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8373-3_2, © Springer Science+Business Media New York 2013
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frame operator ΦΦ ∗ =
M
∗ ϕm ϕm ,
(2.1)
m=1 ∗ is the linear functional ϕ ∗ : CN → C, ϕ ∗ x := x, ϕ . In respectively. Here, ϕm m m m particular, Φ is a frame if and only if the ϕm ’s span CN , which necessitates N ≤ M. Frames provide numerically stable methods for finding overcomplete decompositions of vectors, and as such are useful tools in various signal processing applications [26, 27]. Indeed, if Φ is a frame, then any x ∈ CN can be decomposed as
x = Φ Φ˜ ∗ x =
M
x, ϕ˜m ϕm ,
(2.2)
m=1
˜∗ where Φ˜ = {ϕ˜m }M m=1 is a dual frame of Φ, meaning it satisfies Φ Φ = Id. The most often-used dual is the canonical dual, namely the pseudoinverse Φ˜ = (ΦΦ ∗ )−1 Φ. Computing a canonical dual involves inverting the frame operator. As such, when designing a frame for a given application, it is important to control over the spectrum ∗ {λn }N n=1 of ΦΦ . Here and throughout, such spectra are arranged in nonincreasing order, with the optimal frame bounds A and B being λN and λ1 , respectively. Of particular interest are tight frames, namely frames f
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