Dual Radon transform

Remarks

The pair $(\theta, x \cdot \theta)$ describes the hyperplane with normal $\theta$ through $x$, so $R^* g(x)$ integrates $g$ over all hyperplanes through $x$. For $f$ continuous with compact support and $g$ continuous, Fubini's theorem gives the duality $$\int_{S^{n-1}} \int_\R (R f)(\theta, s)\, g(\theta, s)\, ds\, d\theta = \int_{\R^n} f(x)\, (R^* g)(x)\, dx,$$ where $R$ is the Radon transform.

Example: $g(\theta, s) = s^2$. Then $R^* g(x) = \int_{S^{n-1}} (x \cdot \theta)^2\, d\theta = \frac{\omega_{n-1}}{n} \abs{x}^2$, where $\omega_{n-1}$ is the area of $S^{n-1}$, because the $n$ integrals $\int \theta_j^2\, d\theta$ are equal and add up to $\omega_{n-1}$.

Non-example: $R^*$ is not an inverse of $R$. For $n = 2$ and $f$ continuous with compact support, polar coordinates give $R^* R f(x) = 2 \int_{\R^2} \frac{f(x + z)}{\abs{z}}\, dz$, a smoothed version of $f$, not $f$ itself.

Other notations are $R^\#$ and $R^t$; some authors divide by the area of the sphere.