Uniqueness of Gaussian quadrature formula for computed tomography (Q619057)

The Radon transform \({\mathcal R}(f;t,\theta)\), \(t\in(-1,1)\), \(\theta\in[0,\pi)\) of a function \(f\), defined on the unit ball \(B:=\{(x,y):x^2+y^2\leq 1\}\), is given by the integral of \(f\) along the line segment \(I(t,\theta):=\{(x,y):x\cos\theta+y\sin\theta=t\}\cap B\), namely, \[ {\mathcal R}(f;t,\theta):=\int_{I(t,\theta)}f(x,y)\,ds= \int_{-\sqrt{1-t^2}}^{\sqrt{1-t^2}}f(t\cos\theta-s\sin\theta,t\sin\theta+s\cos\theta)\,ds. \] The quadrature formulas of type \[ \int_{B}f(x,y)U_{n}(x\cos\theta+y\sin\theta)\,dx\,dy\approx\sum_{j=1}^{n+1}b_{j}{\mathcal R}(f;\xi_{j},\theta), \] with nodes \(\xi_{j}\) and coefficients \(b_{j}\), with the algebraic degree of precision \(3n+1\) are called Gaussian. It is proved that the following formula \[ \int_{B}f(x,y)U_{n}(x\cos\theta+y\sin\theta)\,dx\,dy\approx \frac{\pi}{2n+2}\sum_{j=1}^{n+1}(-1)^{j-1}{\mathcal R}\left( f;\frac{\cos(2j-1)\pi}{2n+2},\theta\right) \] is the only Gaussian formula among formulas of type \[ \int_{B}f(x,y)U_{n}(x\cos\theta+y\sin\theta)\,dx\,dy\approx\sum_{j=1}^{n+1}b_{j}{\mathcal R}(f;\xi_{j},\theta_{j}), \] where the Radon transforms can be taken not along parallel lines.