On the inversion of generalized Radon transforms over \(n\)-dimensional paraboloids (Q6542816)

Let \(f\in C^{\infty}(\mathbb{R}^{n+1})\) have compact support in \(\mathbb{R}^{n+1}\). The author studies the problem of reconstruction the function \(f\) from given integrals\N\[\N(\mathcal{R}_1 f)(\xi,\eta)=\int_{\mathbb{R}^{n}} f(x,\eta+p|x-\xi|^2) \,dx\N\]\Nfor \(\xi\in\mathbb{R}^{n}\), \(\eta\in\mathbb{R}\) and a fixed \(p\in\mathbb{R}^{+}\).\N\NThe following inversion formulas are obtained:\N\N\begin{itemize}\N\item[(i)]\N If \(n\) is odd, then\N\[\N\begin{multlined} (\mathcal{F}f) (k,y)=\frac{p^{n/2}}{\pi^{(n+1)/2}} \left(-\frac{\partial}{\partial y}\right)^{(n+1)/2} \int_{y}^{\infty} (\mathcal{F}\mathcal{R}_1 f) (k,\eta )\times\\\N\times \frac{\cosh (|k|\sqrt{\eta-y}/\sqrt{p})}{\sqrt{\eta-y}} \, d\eta. \end{multlined}\N\]\N\item[(ii)]\N If \(n\) is even, then\N\[\N\begin{multlined} (\mathcal{F}f) (k,y)= \frac{p^{n/2}}{\pi^{(n+2)/2}} \left(-\frac{\partial}{\partial y}\right)^{(n+2)/2} \int_{y}^{\infty}(\mathcal{F}\mathcal{R}_1 f) (k,\eta )\times\\\N\times \int_{0}^{\pi} \exp (|k|\sqrt{\eta-y}\cos t/\sqrt{p}) \,dt \,d\eta. \end{multlined}\N\]\N\end{itemize}\N\NHere \(\mathcal{F}\) is the Fourier transform operator acting on the corresponding variable, i.e.\N\[\N(\mathcal{F}f) (k,y)= \int_{\mathbb{R}^{n}}f(x,y)e^{-i k\cdot x} \, dx,\quad k\in\mathbb{R}^{n}.\N\]\N\NIn addition, the paper establishes similar results for the integral transform\N\[\N(\mathcal{R}_2 g)(\xi,\eta)=\int_{|x-\xi|^2<\eta/p} g(x,\eta-p|x-\xi|^2) \,dx,\N\]\Nwhere \(p\in\mathbb{R}^{+}\) is fixed, \(\xi\in\mathbb{R}^{n}\), \(\eta\in\mathbb{R}^{+}\) and \(g\in C^{\infty}(\mathbb{R}^{n+1})\) has compact support in \(\mathbb{R}^{n}\times (0,\infty)\).\N\NThe proofs are based on the properties of the Fourier transform and reducing the reconstruction problem to solving a Volterra integral equation of the first kind.