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首页> 外文期刊>Journal of inverse and ill-posed problems >The modulus of the Fourier transform on a sphere determines 3-dimensional convex polytopes
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The modulus of the Fourier transform on a sphere determines 3-dimensional convex polytopes

机译：The modulus of the Fourier transform on a sphere determines 3-dimensional convex polytopes

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Abstract Let ? and P′mathcal{P}^{prime} be 3-dimensional convex polytopes in R3mathbb{R}^{3} and S⊆R3Ssubseteqmathbb{R}^{3} be a non-empty intersection of an open set with a sphere. As a consequence of a somewhat more general result it is proved that ? and P′mathcal{P}^{prime} coincide up to translation and/or reflection in a point if |∫Pe-i⁢s⋅x⁢dx|=|∫P′e-i⁢s⋅x⁢dx|bigl{lvert}int_{mathcal{P}}e^{-imathbf{s}cdotmathbf{x}},mathbf{dx}bigr{rvert}=bigl{lvert}int_{mathcal{P}^{prime}}e^{-imathbf{s}cdotmathbf{x}},mathbf{dx}bigr{rvert} for all s∈Smathbf{s}in S . This can be applied to the field of crystallography regarding the question whether a nanoparticle modelled as a convex polytope is uniquely determined by the intensities of its X-ray diffraction pattern on the Ewald sphere.

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