The Fourier transform has no hair

There are two standard stories we tell about the Fourier transform, to different audiences with different backgrounds, and it is rarely made precise how they are related. One is analytic: start with a Gaussian and let it degenerate, so that the oscillatory kernel emerges as the boundary trace of holomorphic or heat-kernel regularization. The other is arithmetic: start with a finite or discrete Fourier transform and let the sampling lattice densify, so that the continuum transform appears as the limit of finer and finer exact discrete models.

In this talk I will argue that these are, in a precise sense, two boundary degenerations of the same underlying object. The meeting place is the Theta Grassmannian, whose points consist of a positive complex polarization together with a self-dual lattice in the Heisenberg group. Such a pair determines a theta object obtained by convolving the Gaussian vacuum with the lattice comb.

The main result is that the Shilov boundary of the resulting theta-coefficient algebra is purely Lagrangian: in rank one, and in higher rank up to a single remaining constant-term identification, the additional arithmetic lattice data does not produce new boundary components. Thus both degenerations land on the same Lagrangian Fourier boundary: the Fourier transform has no hair.