{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Finite normalized tight frames - Advances in Computational...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 infinite frames. In this paper the theory of finite 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 classification: 42C99, 42C40, 65F, 74G65 1. Introduction We shall develop the theory of finite 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 finite sequence { x n : n = 1 , . . . , N } K d for which the Euclidean norm x n 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 n = 1 y, x n x n . (1.1) The author gratefully acknowledges support from DARPA Grant MDA 972011003 and the General Research Board of the University of Maryland.
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 definition of an A -FNTF because of the definition 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 Duffin 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 significant 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-
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern