Eigenfunctions of the Fourier Transform with specified zeros

Ahram Feigenbaum, Peter Grabner, Douglas Hardin

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)329-367
Number of pages39
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number2
Publication statusPublished - Sept 2021


  • 11F03
  • 11F11
  • 11H31
  • 2020 Mathematics Subject Classification:
  • 42B10
  • 2020 Mathematics Subject Classification: 11F03 11H31 42B10 11F11

ASJC Scopus subject areas

  • Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing


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