Abstract
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 decompositions. 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 energy 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Balan, P.G. Casazza, C. Heil and Z. Landau, Deficits and excesses of frames, Adv. in Comput. Math. (2002).
J.J. Benedetto and P.J.S.G. Ferreira, eds., Modern Sampling Theory: Mathematics and Applications (Birkhäuser, Boston, MA, 2001).
J.J. Benedetto and S. Li, The theory of multiresolution frames and applications to filter banks, Appl. Comput. Harmonic Anal. 5 (1998) 389–427.
J.J. Benedetto and A. Teolis, A wavelet auditory model and data compression, Appl. Comput. Harmonic Anal. 1 (1993) 3–28.
J.J. Benedetto and O.M. Treiber, Wavelet frames: Multiresolution analysis and extension principles, in: Wavelet Transforms and Time-Frequency Signal Analysis, ed. L. Debnath (Birkhäuser, Boston, MA, 2001) pp. 3–36.
J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups (Springer, New York, 1999).
I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992).
I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27(5) (1986) 1271–1283.
R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366.
M. Fickus, Constructions of normalized tight frames.
V.K. Goyal, J. Kovačević and J.A. Kelner, Quantized frame expansions with erasures, Appl. Comput. Harmonic. Anal. 10(3) (2001) 203–233.
V.K. Goyal, M. Vetterli and N.T. Thao, Quantized overcomplete expansions in R n: Analysis, synthesis, and algorithms, IEEE Trans. Inform. Theory 44(1) (1998) 16–31.
M.L. Harrison, Frames and irregular sampling from a computational perspective, Ph.D. thesis, University of Maryland, College Park, MD (1998).
A.B.J. Kuijlaars and E.B. Saff, Asymptotics for minimal discrete energy on the sphere, Trans. Amer. Math. Soc. 350(2) (1998) 523–538.
E.B. Saff and A.B.J. Kuijlaars, Distributing many points on a sphere, Math. Intelligencer 19(1) (1997) 5–11.
N.J.A. Sloane, Home page http://www.research.att.com~njas (April 2001).
T. Strohmer, Computationally attractive reconstruction of bandlimited images from irregular samples, IEEE Trans. Image Processing 6(4) (1997) 540–548.
T. Strohmer, Numerical analysis of the non-uniform sampling problem, in: Numerical Analysis 2000, Vol. II: Interpolation and Extrapolation, J. Comput. Appl. Math. 122(1/2) (2000) 297–316.
G. Zimmermann, Normalized tight frames in finite dimensions, in: Recent Progress in Multivariate Approximation, eds. K.J.W. Haußmann and M. Reimer (Birkhäuser, Basel, 2001) pp. 249–252.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Benedetto, J.J., Fickus, M. Finite Normalized Tight Frames. Advances in Computational Mathematics 18, 357–385 (2003). https://doi.org/10.1023/A:1021323312367
Issue Date:
DOI: https://doi.org/10.1023/A:1021323312367