Finite normalized tight frames

Finite normalized tight frames - Advances in Computational...

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Advances in Computational Mathematics 18: 357–385, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands. Finite normalized tight frames John J. Benedetto a , and Matthew Fickus b a Department of Mathematics, University of Maryland, College Park, MD 20742, USA E-mail: [email protected] b Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853, USA E-mail: ±[email protected] Received 14 September 2001; accepted 20 January 2002 Communicated by C.A. Micchelli Frames are interesting because they provide decompositions in applications where bases could be a liability. Tight frames are valuable to ensure fast convergence of such decom- positions. Normalized frames guarantee control of the frame elements. Finite frames avoid the subtle and omnipresent approximation problems associated with the truncation of in±nite frames. In this paper the theory of ±nite normalized tight frames (FNTFs) is developed. The main theorem is the characterization of all FNTFs in terms of the minima of a potential en- ergy function, which was designed to measure the total orthogonality of a Bessel sequence. Examples of FNTFs abound, e.g., in R 3 the vertices of the Platonic solids and of a soccer ball are FNTFs. Keywords: tight frames, potential energy, Lagrange multipliers, equidistribution, equilibrium AMS subject classifcation: 42C99, 42C40, 65F, 74G65 1. Introduction We shall develop the theory of ±nite normalized tight frames (FNTFs) for real and complex Euclidean space K d , K = R , C , respectively; and we shall characterize all FNTFs for K d in terms of a potential energy function which we designate as the frame potential , see theorem 7.1. FNTFs are developed in section 2, but, for the sake of this preliminary discussion, an A -FNTF for K d is a ±nite sequence { x n : n = 1 ,...,N } K d for which the Euclidean norm k x n k is 1 for each x n , i.e., { x n } is normalized, and for which A> 0 such that y K d ,y = 1 A N X n = 1 ± y,x n i x n . (1.1) The author gratefully acknowledges support from DARPA Grant MDA 972011003 and the General Research Board of the University of Maryland.
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358 J.J. Benedetto, M. Fickus / Finite normalized tight frames It is elementary to see that if { x n } N n = 1 is an A -FNTF then A = 1 if and only if { x n } N n = 1 is an orthonormal basis (ONB) for R d , e.g., theorem 2.1. (Naturally we choose the terminology “normalized” in the de±nition of an A -FNTF because of the de±nition of an ONB, but see remark 1.1(b) below.) Of course, it is reasonable to ask why anyone would want to develop a theory of FNTFs. First, the theory of frames was initiated by Duf±n and Schaeffer [9] in 1952 as part of an on-going development of non-harmonic Fourier series, in which it is an im- portant, fundamental theory. The background for [9] and its relationship to non-uniform sampling theory is the subject of chapter 1 in [2]. Further, there is a signi±cant role in signal processing for the theory of frames. This was initiated by Daubechies et al. [8] in 1986, and since then there has been a plethora of activity in the area including a land-
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Finite normalized tight frames - Advances in Computational...

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