Let \({\mathbf S}_m\) be the space of symmetric \(m\)-covariant tensors in \(\mathbb R^2\) and \(L_2({\mathbb D},{\mathbf S}_m)\) the space of \(L^2\) symmetric \(m\)-covariant tensor fields \({\mathbf a}^{(m)}\) on the unit disk \(\mathbb D\) in \(\mathbb R^2\). The authors extend the classical Radon transform \({\mathcal R}\) on \(\mathbb D\) to tensor fields \({\mathbf a}^{(m)}\) by applying it componentwise, and define the \(m\)-tensor Radon probe transform \({\mathcal R}_m\) on \(L_2({\mathbb D},{\mathbf S}_m)\) as follows:
\[ [{\mathcal R}_m{\mathbf a}^{(m)}](s,\phi)=\langle(\theta^\bot)^m,[{\mathcal R}{\mathbf a}^{(m)}](s,\phi)\rangle_{{\mathbf S}_m}, \]
where \(s\in[-1,1],\;\phi\in[0,2\pi)\), and \((\theta^\bot)^m\) is the probe tensor vector. Let \({\mathbf H}({\mathbb D},{\mathbf S}_m;\delta=0)\) denote the solenoidal subspace of \(L_2({\mathbb D},{\mathbf S}_m)\) according to the Helmholtz-Hodge decomposition of \(L_2({\mathbb D},{\mathbf S}_m)\).
In this paper, when \(m=1\) and \(m=2\), the authors reconstruct the solenoidal fields \({\mathbf a}_{sol}^{(m)}\in{\mathbf H}({\mathbb D},{\mathbf S}_m;\delta=0)\) from their Radon transforms \({\mathcal R}_m{\mathbf a}_{sol}^{(m)}(s,\phi)\). In this process the Fourier series expansions based on an orthogonal basis for \(L_2({\mathbb D},{\mathbf S}_m)\) are necessary. They obtain a new orthogonal polynomial basis for \(L_2({\mathbb D},{\mathbf S}_m)\) and for \({\mathbf H}({\mathbb D},{\mathbf S}_m;\delta=0)\) that is built with the help of bivariate Chebyshev ridge polynomials. The numerical results of a novel inversion algorithm based on their expansions are presented.