I was interested in how
- $1_{\hat{ℤₚ}} : ℚₚ ⭢ ℂ$ is Fourier dual to itself
- $e^{-πx^2} : ℝ ⭢ ℝ$ is Fourier dual to itself
- $e^{-2πz\overline{z}} : ℂ ⭢ ℂ$ is Fourier dual to itself.
One question I have is:
Is $1_{\hat{ℤ}} : ⭢ ℂ$ Fourier dual to itself?
I thought there would be an equation involving $1_{\hat{ℤ}},1_{\hat{ℤₚ}}$, and $e^{-πx^2}$, much like $Π νₚ(x) = ||x||$.
How does $\hat{ℤ} ⊗ ℚ$ relate to ℝ? If the Adeles are $\hat{ℤ} ⊗ ℚ$, then where does the factor of $ℝ$ come from?