Question

9. Assume that the Radon transform $Rf$ in $\mathbb{R}^3$ is defined as above. Suppose that given $F \in S(\mathbb{R} \times S^2)$, a function $R^(F) : \mathbb{R}^3 \to \mathbb{C}$ satisfies $\int_{\mathbb{R} \times S^2} Rf(t, \gamma) \overline{F(t, \gamma)} dt d\sigma(\gamma) = \int_{\mathbb{R}^3} f(x) \overline{R^(F)(x)} dx$. (a) Express $R^(F)$ in terms of $F$, (b) Prove that $R^R(f)(x) = \int_{S^2} \int_{\mathbb{R}} \hat{f}(s\gamma) e^{2\pi is(x,\gamma)} ds d\sigma(\gamma)$ (c) Prove that $R^R(f)(x) = \int_{\mathbb{R}^3} \frac{\hat{f}(\xi)}{|\xi|^2} e^{2\pi i \xi \cdot x} d\xi + \int_{\mathbb{R}^3} \frac{\hat{f}(-\xi)}{|\xi|^2} e^{-2\pi i \xi \cdot x} d\xi$ (d) Use (c) with (0.1) above to express $(-\Delta)R^R(f)$ in terms of $f$.

          9. Assume that the Radon transform $Rf$ in $\mathbb{R}^3$ is defined as above. Suppose that
given $F \in S(\mathbb{R} \times S^2)$, a function $R^*(F) : \mathbb{R}^3 \to \mathbb{C}$ satisfies
$\int_{\mathbb{R} \times S^2} Rf(t, \gamma) \overline{F(t, \gamma)} dt d\sigma(\gamma) = \int_{\mathbb{R}^3} f(x) \overline{R^*(F)(x)} dx$.
(a) Express $R^*(F)$ in terms of $F$,
(b) Prove that $R^*R(f)(x) = \int_{S^2} \int_{\mathbb{R}} \hat{f}(s\gamma) e^{2\pi is(x,\gamma)} ds d\sigma(\gamma)$
(c) Prove that $R^*R(f)(x) = \int_{\mathbb{R}^3} \frac{\hat{f}(\xi)}{|\xi|^2} e^{2\pi i \xi \cdot x} d\xi + \int_{\mathbb{R}^3} \frac{\hat{f}(-\xi)}{|\xi|^2} e^{-2\pi i \xi \cdot x} d\xi$
(d) Use (c) with (0.1) above to express $(-\Delta)R^*R(f)$ in terms of $f$.

9. Assume that the Radon transform Rf in ℝ^3 is defined as above. Suppose that
given F ∈ S(ℝ× S^2), a function R^*(F) : ℝ^3 →ℂ satisfies
∫ℝ× S^2 Rf(t, γ) F(t, γ) dt dσ(γ) = ∫ℝ^3 f(x) R^*(F)(x) dx.
(a) Express R^*(F) in terms of F,
(b) Prove that R^*R(f)(x) = ∫S^2∫ℝf̂(sγ) e^2π is(x,γ) ds dσ(γ)
(c) Prove that R^*R(f)(x) = ∫ℝ^3(f̂(ξ))/(|ξ|^2) e^2π i ξ· x dξ + ∫ℝ^3(f̂(-ξ))/(|ξ|^2) e^-2π i ξ· x dξ
(d) Use (c) with (0.1) above to express (-Δ)R^*R(f) in terms of f.

Added by Hector P.

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