Abstract
In a finite-dimensional complex Euclidean space, a maximally equiangular frame is a tight frame which has a number of elements equal to the square of the dimension of the space, and in which the inner products of distinct elements are of constant magnitude. Though the general question of their existence remains open, many examples of maximally equiangular frames have been constructed as finite Gabor systems. These constructions involve number theory, specifically Schaar’s identity, which provides a reciprocity formula for quadratic Gauss sums. To be precise, Zauner used Schaar’s identity to compute the spectrum of a chirp-Fourier operator, the eigenvectors of which he conjectured to be well-suited for the construction of maximally equiangular Gabor frames. We provide two new characterizations of such frames, both of which further confirm the relevance of the theory of Gauss sums to this area of frame theory. We also show how the unique time-frequency properties of a particular cyclic chirp function may be exploited to provide a new, short and elementary proof of Schaar’s identity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Appleby, D.M.: Symmetric informationally complete-positive operator valued measures and the extended Clifford group. J. Math. Phys. 46, 052107 (2005)
Appleby, D.M., Dang, H.B., Fuchs, C.A.: Physical significance of symmetric informationally-complete sets of quantum states. Preprint (2007)
Armitage, V., Rogers, A.: Gauss sums and quantum mechanics. J. Phys. A Math. Gen. 33, 5993–6002 (2000)
Auslander, L., Tolimieri, R.: Is computing with the fast Fourier transform pure or applied mathematics? Bull. Am. Math. Soc. New Ser. 1, 847–897 (1979)
Benedetto, J.J., Donatelli, J.J.: Ambiguity function and frame theoretic properties of periodic zero autocorrelation waveforms. IEEE J. Sel. Top. Signal Process. 1, 6–20 (2007)
Berndt, B.C., Evans, R.J.: The determination of Gauss sums. Bull. Am. Math. Soc. New Ser. 5, 107–129 (1981)
Casazza, P.G., Fickus, M.: Fourier transforms of finite chirps. EURASIP J. Appl. Signal Process. 2006, 7 (2006)
Feichtinger, H.G., Hazewinkel, M., Kaiblinger, N., Matusiak, E., Neuhauser, M.: Metaplectic operators on ℂn. Q. J. Math. 59, 15–28 (2008)
Fiedler, H., Jurkat, W., Koerner, O.: Asymptotic expansions of finite theta series. Acta Arith. 32, 129–146 (1977)
Grassl, M.: On SIC-POVMs and MUBs in dimension 6. Preprint (2004)
Khatirinejad, M.: On Weyl-Heisenberg orbits of equiangular lines. J. Algebr. Comb. 28, 333–349 (2008)
Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171–2180 (2004)
Schur, I.: Über die Gaußschen Summen. Gött. Nachr. 147–153 (1921)
Siegel, C.L.: Über das quadratische Reziprozitätsgesetz in algebraischen Zahlkörpern. Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1, 1–16 (1960)
Strohmer, T., Heath, R.: Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14, 257–275 (2003)
Wootters, W.K.: Quantum measurements and finite geometry. Found. Phys. 36, 112–126 (2006)
Xia, X.G.: Discrete chirp-Fourier transform and its application to chirp rate estimation. IEEE Trans. Signal Process. 48, 3122–3133 (2000)
Zauner, G.: Quantendesigns: Grundzüge einer nichtkommutativen Designtheorie. Ph.D. thesis, Univ. Vienna, Austria (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Thomas Strohmer.
Rights and permissions
About this article
Cite this article
Fickus, M. Maximally Equiangular Frames and Gauss Sums. J Fourier Anal Appl 15, 413–427 (2009). https://doi.org/10.1007/s00041-009-9064-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-009-9064-2