An Identity Theorem for the Fourier–Laplace Transform of Polytopes on Nonzero Complex Multiples of Rationally Parameterizable Hypersurfaces

Abstract

AbstractA set $$\mathcal {S}$$ S of points in $$\mathbb {R}^n$$ R n is called a rationally parameterizable hypersurface if there is vector function $$\varvec{\sigma }:\mathbb {R}^{n-1}\rightarrow \mathbb {R}^n$$ σ : R n - 1 → R n having as components rational functions defined on some common domain D such that $$\mathcal {S}=\{\varvec{\sigma }(\textbf{t}):\textbf{t}\in D\}$$ S = { σ ( t ) : t ∈ D } . A generalized n-dimensional polytope in $$\mathbb {R}^n$$ R n is a union of a finite number of convex n-dimensional polytopes in $$\mathbb {R}^n$$ R n . The Fourier–Laplace transform of such a generalized polytope $$\mathcal {P}$$ P in $$\mathbb {R}^n$$ R n is defined by $$F_{\mathcal {P}}(\textbf{z})=\int _{\mathcal {P}}e^{\textbf{z}\cdot \textbf{x}}\,\textbf{dx}$$ F P ( z ) = ∫ P e z · x dx . Let $$\gamma $$ γ be a fixed nonzero complex number. We prove that $$F_{\mathcal {P}_1}(\gamma \varvec{\sigma }(\textbf{t}))=F_{\mathcal {P}_2}(\gamma \varvec{\sigma }(\textbf{t}))$$ F P 1 ( γ σ ( t ) ) = F P 2 ( γ σ ( t ) ) for all $$\textbf{t} \in O$$ t ∈ O implies $$\mathcal {P}_1=\mathcal {P}_2$$ P 1 = P 2 if O is an open subset of D satisfying some well-defined conditions and we present similar results for the null set of the Fourier–Laplace transform of $$\mathcal {P}$$ P . 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.