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A generalized n<\/jats:italic>-dimensional polytope in $$\\mathbb {R}^n$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a union of a finite number of convex n<\/jats:italic>-dimensional polytopes in\u00a0$$\\mathbb {R}^n$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The Fourier\u2013Laplace transform of such a generalized polytope $$\\mathcal {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$\\mathbb {R}^n$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is defined by $$F_{\\mathcal {P}}(\\textbf{z})=\\int _{\\mathcal {P}}e^{\\textbf{z}\\cdot \\textbf{x}}\\,\\textbf{dx}$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n P<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n \u222b<\/mml:mo>\n P<\/mml:mi>\n <\/mml:msub>\n \n e<\/mml:mi>\n \n z<\/mml:mi>\n \u00b7<\/mml:mo>\n x<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \n dx<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Let $$\\gamma $$<\/jats:tex-math>\n \u03b3<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be a fixed nonzero complex number. We prove that $$F_{\\mathcal {P}_1}(\\gamma \\varvec{\\sigma }(\\textbf{t}))=F_{\\mathcal {P}_2}(\\gamma \\varvec{\\sigma }(\\textbf{t}))$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n \n P<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:msub>\n \n (<\/mml:mo>\n \u03b3<\/mml:mi>\n \n \u03c3<\/mml:mi>\n <\/mml:mrow>\n \n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n F<\/mml:mi>\n \n P<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:msub>\n \n (<\/mml:mo>\n \u03b3<\/mml:mi>\n \n \u03c3<\/mml:mi>\n <\/mml:mrow>\n \n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for all $$\\textbf{t} \\in O$$<\/jats:tex-math>\n \n t<\/mml:mi>\n \u2208<\/mml:mo>\n O<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> implies $$\\mathcal {P}_1=\\mathcal {P}_2$$<\/jats:tex-math>\n \n \n P<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n =<\/mml:mo>\n \n P<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if O<\/jats:italic> is an open subset of D<\/jats:italic> satisfying some well-defined conditions and we present similar results for the null set of the Fourier\u2013Laplace transform of\u00a0$$\\mathcal {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres.<\/jats:p>","DOI":"10.1007\/s00454-022-00467-9","type":"journal-article","created":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T21:33:10Z","timestamp":1671485590000},"page":"209-231","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["An Identity Theorem for the Fourier\u2013Laplace Transform of Polytopes on Nonzero Complex Multiples of Rationally Parameterizable Hypersurfaces"],"prefix":"10.1007","volume":"69","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9396-5332","authenticated-orcid":false,"given":"Konrad","family":"Engel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,12,19]]},"reference":[{"key":"467_CR1","doi-asserted-by":"crossref","unstructured":"Barke, I., Hartmann, H., Rupp, D., Fl\u00fcckiger, L., Sauppe, M., Adolph,\u00a0M., Schorb,\u00a0S., Bostedt,\u00a0Ch., Treusch,\u00a0R., Peltz,\u00a0Ch., Bartling,\u00a0S., Fennel,\u00a0Th., Meiwes-Broer, K.-H., M\u00f6ller,\u00a0Th.: The 3D-architecture of individual free silver nanoparticles captured by X-ray scattering. 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