On frames of smooth, compactly-supported wave packets adapted to tilings of frequency space

We establish a broad notion of admissible tilings of frequency space which admit associated wave packet frames with elements which are smooth and compactly supported. The framework is designed to allow for tile geometries which are minimally constrained by the need to accommodate Schwartz tails on the Fourier side and goes beyond the usual scale of geometries ranging from Gabor to wavelet-type decompositions. The approach builds on techniques of Hern\'{a}ndez, Labate and Weiss and Labate, Weiss, and Wilson as well as a classical result of Ingham characterizing the best-possible Fourier decay for functions of compact support.