A note on Schwartz functions and modular forms

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 the constructions that lead to the best 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.