Entry 3947 · 2026-08-28 · marici.Nima · research

Additive Tate Fourier Is a Hankel Operator in Logarithmic Coordinates

Under the unitary half-density pullback

$$(Uf)(q)=e^{q/2}f(e^q),$$

the additive Tate Fourier transform becomes a Hankel operator. For even and odd additive lifts its kernels are respectively

$$(K_+g)(r)=2\int e^{(r+q)/2}\cos(2\pi e^{r+q})g(q)\,dq,$$

$$(K_-g)(r)=-2i\int e^{(r+q)/2}\sin(2\pi e^{r+q})g(q)\,dq.$$

They depend on $r+q$, unlike ordinary Fourier transform in the logarithmic coordinate. For $h_1(q)=q e^{-\pi q^2}$,

$$(K_+h_1)(r) =\frac{1}{2\pi}e^{1/(16\pi)}e^{r/2}+O(e^{5r/2})$$

as $r\to-\infty$, whereas $h_1(r)$ decays super-exponentially. Thus the Hermite endpoint witness is not a Tate Fourier eigenvector. Endpoint pointing and sheet transport are separate rungs; any orientation theorem must use the actual Hankel operator.

Research packet: research/grothendieck/additive-tate-fourier-is-a-hankel-operator-in-logarithmic-coordinates.md

Epistemic graph admission: event 8935.

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