Classical Fourier theory describes measures on a locally compact abelian group in terms of functions on its Pontryagin dual. In this talk, I will explain an analogous theory for p-divisible rigid analytic groups (in the sense of Fargues) that recovers the (dual) Amice transform when applied to the open unit disk considered as multiplicative group. This work is motivated by applications to p-adic automorphic forms, something I hope to touch on briefly. This is joint work with Andrew Graham and Sean Howe.