Reconstruction of polytopes from the modulus of the Fourier transform with small wave length [article]

Konrad Engel, Bastian Laasch
2020 arXiv   pre-print
Let 𝒫 be an n-dimensional convex polytope and 𝒮 be a hypersurface in ℝ^n. This paper investigates potentials to reconstruct 𝒫 or at least to compute significant properties of 𝒫 if the modulus of the Fourier transform of 𝒫 on 𝒮 with wave length λ, i.e., |∫_𝒫 e^-i1/λ𝐬·𝐱 𝐝𝐱| for 𝐬∈𝒮, is given, λ is sufficiently small and 𝒫 and 𝒮 have some well-defined properties. The main tool is an asymptotic formula for the Fourier transform of 𝒫 with wave length λ when λ→ 0. The theory of X-ray scattering of
more » ... oparticles motivates this study since the modulus of the Fourier transform of the reflected beam wave vectors are approximately measurable in experiments.
arXiv:2011.06971v1 fatcat:zfazq2fh7ndzpevatpf3o2ykje