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We give a unified description of the modular and quasi-modular functions usedin Viazovska's proof of the best packing bounds in dimension 8 and the proof byCohn, Kumar, Miller, Radchenko, and Viazovska of the best packing bound indimension 24. We show that necessarily modular forms have to be used to obtainthese results. We extend these constructions to arbitrary dimensions divisibleby 4.

Eigenfunctions of the Fourier Transform with specified zeros