  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.
