Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-25T14:58:31.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Eigenfunctions of the Fourier transform with specified zeros

Part of: Harmonic analysis in several variables Discontinuous groups and automorphic forms Geometry of numbers

Published online by Cambridge University Press:  15 February 2021

AHRAM S. FEIGENBAUM ,
PETER J. GRABNER  and
DOUGLAS P. HARDIN
AHRAM S. FEIGENBAUM
Affiliation:
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN37240, U.S.A. e-mail: ahram.s.feigenbaum@Vanderbilt.Edu
PETER J. GRABNER
Affiliation:
Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24, 8010Graz, Austria. e-mail: peter.grabner@tugraz.at
DOUGLAS P. HARDIN
Affiliation:
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN37240, U.S.A. e-mail: doug.hardin@Vanderbilt.Edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Eigenfunctions of the Fourier transform with prescribed zeros played a major role in the proof that the E8 and the Leech lattice give the best sphere packings in respective dimensions 8 and 24 by Cohn, Kumar, Miller, Radchenko and Viazovska. The functions used for a linear programming argument were constructed as Laplace transforms of certain modular and quasimodular forms. Similar constructions were used by Cohn and Gonçalves to find a function satisfying an optimal uncertainty principle in dimension 12. This paper gives a unified view on these constructions and develops the machinery to find the underlying forms in all dimensions divisible by 4. Furthermore, the positivity of the Fourier coefficients of the quasimodular forms occurring in this context is discussed.

MSC classification

Primary: 11F03: Modular and automorphic functions 11H31: Lattice packing and covering 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Secondary: 11F11: Holomorphic modular forms of integral weight
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 2 , September 2021 , pp. 329 - 367
Copyright
© Cambridge Philosophical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

The research of these authors was supported, in part, by the U. S. National Science Foundation under grant DMS-1516400.

‡

This author is supported by the Austrian Science Fund FWF project F5503 part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”.

References

REFERENCES

Aste, T. and Weaire, D.. The Pursuit of Perfect Packing, second ed. (Taylor & Francis, New York, 2008).Google Scholar
Boas, R. P. Jr. Entire Functions (Academic Press Inc., New York, 1954).Google Scholar
Bourgain, J., Clozel, L. and Kahane, J. P.. Principe d’Heisenberg et fonctions positives. Ann. Inst. Fourier (Grenoble) 60 (2010), no. 4, 1215–1232.CrossRefGoogle Scholar
Bruinier, J. H., van der Geer, G., Harder, G. and Zagier, D.. The 1-2-3 of modular forms, Universitext (Springer-Verlag, Berlin, 2008), Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid, June 2004, Edited by Ranestad, Kristian.CrossRefGoogle Scholar
Choie, Y. J. and Lee, M. H.. Jacobi-Like Forms, Pseudodifferential Operators and Quasimodular Forms, Monographs in Mathematics (Springer International Publishing, 2019).CrossRefGoogle Scholar
Cohen, H. and Strömberg, F.. Modular forms, Graduate Studies in Mathematics, vol. 179 (American Mathematical Society, Providence, RI, 2017), A classical approach.CrossRefGoogle Scholar
Cohn, H.. New upper bounds on sphere packings II. Geom. Topol. 6 (2002), 329–353.CrossRefGoogle Scholar
Cohn, H.. A conceptual breakthrough in sphere packing. Notices Amer. Math. Soc. 64 (2017), no. 2, 102–115.CrossRefGoogle Scholar
Cohn, H. and Elkies, N.. New upper bounds on sphere packings. I Ann. of Math. (2) 157 (2003), no. 2, 689–714.CrossRefGoogle Scholar
Cohn, H. and Gonçalves, F.. An optimal uncertainty principle in twelve dimensions via modular forms. Invent. Math. 217 (2019), no. 3, 799–831.CrossRefGoogle Scholar
Cohn, H., Kumar, A., Miller, S. D., Radchenko, D. and Viazovska, M.. The sphere packing problem in dimension 24. Ann. of Math. (2) 185 (2017), no. 3, 1017–1033.CrossRefGoogle Scholar
Cohn, H., Kumar, A., Miller, S. D., Radchenko, D. and Viazovska, M.. Universal optimality of the E 8 and Leech lattices and interpolation formulas. https://arxiv.org/abs/1902.05438, Feb 2019.Google Scholar
de Laat, D. and Vallentin, F.. A breakthrough in sphere packing: the search for magic functions, Nieuw Arch. Wiskd. (5) 17 (2016), no. 3, 184–192, Includes an interview with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Maryna Viazovska.Google Scholar
Deligne, P.. La conjecture de Weil. Publ. Math. I.H.E.S. 43 (1974), 273–307.CrossRefGoogle Scholar
Diamond, F. and Shurman, J.. A first course in modular forms Graduate Texts in Mathematics, vol. 228 (Springer-Verlag, New York, 2005).Google Scholar
Fejes, L.. Über die dichteste Kugellagerung. Math. Z. 48 (1943), 676–684.CrossRefGoogle Scholar
Gonçalves, F., e Silva, D. Oliveira and Steinerberger, S.. Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots. J. Math. Anal. Appl. 451 (2017), no. 2, 678–711.CrossRefGoogle Scholar
Grabner, P. J.. Quasimodular forms as solutions of modular differential equations. Int. J. Number Theory (2020), to appear, available at https://arxiv.org/abs/2002.02736.Google Scholar
Hales, T. C.. A proof of the Kepler conjecture. Ann. of Math. (2) 162 (2005), no. 3, 1065–1185.CrossRefGoogle Scholar
Iwaniec, H.. Topics in classical automorphic forms. Graduate Studies in Mathematics, vol. 17 (American Mathematical Society, Providence, RI, 1997).Google Scholar
Jenkins, P. and Rouse, J.. Bounds for coefficients of cusp forms and extremal lattices. Bull. London Math. Soc. 43 (2011), no. 5, 927–938.CrossRefGoogle Scholar
Kabatjanskiĭ, G. A. and Levenšteĭn, V. I.. Bounds for packings on the sphere and in space. Problemy Peredači Informacii 14 (1978), no. 1, 3–25.Google Scholar
Kaneko, M. and Koike, M.. On extremal quasimodular forms. Kyushu J. Math. 60 (2006), no. 2, 457–470.CrossRefGoogle Scholar
Kaneko, M., Nagatomo, K. and Sakai, Y.. The third order modular linear differential equations. J. Algebra 485 (2017), 332–352.CrossRefGoogle Scholar
Kaneko, M. and Zagier, D.. A generalied Jacobi theta function and quasimodular forms, The moduli space of curves (Texel Island, 1994) Progr. Math. vol. 129 (Birkhäuser Boston, Boston, MA, 1995), pp. 165–172.Google Scholar
Paley, R. E. A. C. and Wiener, N.. Fourier transformations in the complex domain, AMS Colloquium Publications, vol. XIX (American Mathematical Society, New York, 1934).Google Scholar
Radchenko, D. and Viazovska, M.. Fourier interpolation on the real line. Publ. Math. Inst. Hautes Études Sci. 129 (2019), 51–81.CrossRefGoogle Scholar
Rolen, L. and Wagner, I.. A note on Schwartz functions and modular forms. Arch. Math. 115 (2020), 35–51.CrossRefGoogle Scholar
Royer, E.. Quasimodular forms: an introduction. Ann. Math. Blaise Pascal 19 (2012), no. 2, 297–306.CrossRefGoogle Scholar
Shimura, G.. Modular forms: basics and beyond. Springer Monographs in Mathematics (Springer, New York, 2012).CrossRefGoogle Scholar
Stein, W.. Modular forms, a computational approach. Graduate Studies in Mathematics, vol. 79 (American Mathematical Society, Providence, RI, 2007). With an appendix by Paul E. Gunnells.Google Scholar
Viazovska, M. S.. The sphere packing problem in dimension 8. Ann. of Math. (2) 185 (2017), no. 3, 991–1015.CrossRefGoogle Scholar
Widder, D. V.. The Laplace Transform, Princeton Mathematical Series, v. 6 (Princeton University Press, Princeton, N. J., 1941).Google Scholar
Yamashita, T.. On a construction of extremal quasimodular forms of depth two. Master’s thesis, Tsukuba University, 2010, Japanese.Google Scholar
Zagier, D.. Modular forms and differential operators. Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 1, 57–75, K. G. Ramanathan memorial issue.CrossRefGoogle Scholar
Zagier, D., Elliptic modular forms and their applications. in The 1-2-3 of modular forms [4], pp. 1–103.Google Scholar