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A note on Schwartz functions and modular forms

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Abstract

We generalize the recent work of Viazovska by constructing infinite families of Schwartz functions, suitable for Cohn–Elkies style linear programming bounds, using quasi-modular and modular forms. In particular, for dimensions \(d \equiv 0 \pmod {8}\), we give new constructions for obtaining sphere packing upper bounds via modular forms. In dimension 8 and 24, these exactly match the functions constructed by Viazovska and Cohn, Kumar, Miller, Radchenko, and Viazovska which resolved the sphere packing problem in those dimensions.

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Acknowledgements

The authors would like to thank Ken Ono for his thoughts on an earlier version of this work and Henry Cohn for his many helpful comments which improved this paper.

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  1. Vanderbilt University, Nashville, USA

    Larry Rolen & Ian Wagner

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  1. Larry Rolen
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  2. Ian Wagner
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Correspondence to Ian Wagner.

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Rolen, L., Wagner, I. A note on Schwartz functions and modular forms. Arch. Math. 115, 35–51 (2020). https://doi.org/10.1007/s00013-020-01459-y

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