## 6 Numerical Tests  
In the present section we perform a few numerical tests in order to assess the quality of the solution obtained by adopting the \(\omega\)-circulant regularization, in the setting of a singular interpolation operator. In all the following examples we employ the J-spline family [38]. We denote by \(\tilde{\mathbf{P}}^{0}\) and \(\tilde{\mathbf{P}}^{0}_{\omega}\) the control point vectors obtained as \(M_{n}^{\ddagger}\mathbf{U}^{(1)}\) (resp. \(\mathbf{M}_{n}^{\ddagger}\mathbf{U}^{(2)}\) in the Hermite case) and \(M_{n,\omega}^{\ddagger}\mathbf{U}^{(1)}\) (resp. \(\mathbf{M}_{n,\omega}^{\ddagger}\mathbf{U}^{(2)}\) in the Hermite case), respectively. The pseudo-inverses \(M_{n,\omega}^{\ddagger}\) and \(\mathbf{M}_{n,\omega}^{\ddagger}\) are given by  
\[(F_{n,\omega}\otimes I_{d})\mathcal{L}_{n,\omega}^{\ddagger}(F_{n,\omega}^{-1} \otimes I_{d}),\quad\text{with }(\mathcal{L}_{n,\omega}^{\ddagger})_{k}= \begin{cases} \frac{1}{(\mathcal{L}_{n,\omega})_{k}}&\text{, if }(\mathcal{L}_{n,\omega})_{k}\
eq 0,\\ 0&\text{, if }(\mathcal{L}_{n,\omega})_{k}=0,\end{cases} \]  
with \(d=1\) for \(M_{n,\omega}^{\ddagger}\) and \(d=2\) for \(\mathbf{M}_{n,\omega}^{\ddagger}\), and where \((\mathcal{L}_{n,\omega})_{k}\) is the \(k\)-th diagonal element in (3). For \(\omega=1\) we immediately get \(M_{n}^{\ddagger}\) and \(\mathbf{M}_{n}^{\ddagger}\). We compare the subdivision curves generated from the control point vectors \(\tilde{\mathbf{P}}^{0}\) and \(\tilde{\mathbf{P}}^{0}_{\omega}\); firstly in terms of interpolation, which is the main goal. With this purpose, consider the residuals \(M_{n}\tilde{\mathbf{P}}^{0}-\mathbf{U}^{(1)}\), \(M_{n}\tilde{\mathbf{P}}^{0}-\mathbf{U}^{(2)}\), \(M_{n}\tilde{\mathbf{P}}^{0}_{\omega}-\mathbf{U}^{(1)}\) and \(\mathbf{M}_{n}\tilde{\mathbf{P}}^{0}_{\omega}-\mathbf{U}^{(2)}\) which are vectors of points. Notice that the last two residuals are independent from the systems of equations \(M_{n,\omega}\tilde{\mathbf{P}}^{0}_{\omega}=\mathbf{U}^{(1)}\), \(\mathbf{M}_{n,\omega}\tilde{\mathbf{P}}^{0}_{\omega}=\mathbf{U}^{(2)}\), but they are related to the interpolation problems (10) and (14) with the regularized solution. By reasoning in local terms, we choose the norms  
\[\left\|M_{n}\tilde{\mathbf{P}}^{0}_{\omega}-\mathbf{U}^{(1)}\right\|,\;\left\| \mathbf{M}_{n}\tilde{\mathbf{P}}^{0}_{\omega}-\mathbf{U}^{(2)}\right\|\]  
with \(\left\|A\right\|:=\sup_{j}\left\|(A)_{j}\right\|_{2}\), \(A_{j}\) the \(j\)-row of matrix \(A\) and \(\|\cdot\|_{2}\) the vector 2-norm. This way we measure how far the interpolated information in \(\mathbf{U}^{(d)}\), \(d=1,2\) is from the approximations in \(M_{n,\omega}\tilde{\mathbf{P}}^{0}_{\omega}\), and \(\mathbf{M}_{n,\omega}\tilde{\mathbf{P}}^{0}_{\omega}\) by the maximum distance. On the other hand, reasoning in global terms we consider the relative error with Frobenius matrix norms  
\[\frac{\left\|M_{n}\widehat{\mathbf{P}}_{\omega}^{0}-\mathbf{U}^{(1)}\right\|_{ \text{fro}}}{\left\|\mathbf{U}^{(1)}\right\|_{\text{fro}}},\;\;\frac{\left\| \mathbf{M}_{n}\widehat{\mathbf{P}}_{\omega}^{0}-\mathbf{U}^{(2)}\right\|_{ \text{fro}}}{\left\|\mathbf{U}^{(2)}\right\|_{\text{fro}}}.\]  
In order to evaluate the quality of each curve, proper fairness measures could be used [46; 47], penalizing artifacts such as loops (see Figure 5a). In each figure, the curve obtained with the least square solution \(\widehat{\mathbf{P}}^{0}\) is portrayed in solid green lines, while the one obtained with the regularized solution \(\widehat{\mathbf{P}}_{\omega}^{0}\) is represented by dashed red lines. The solution computed with the \(\omega\)-circulant matrix interpolates the given points even if a non singular matrix is used. Indeed, in Figure 6, where we set \(\omega=\exp(5\times 10^{-3}\text{i})\), we only see one line because the two solutions \(\widehat{\mathbf{P}}^{0}\) and \(\widehat{\mathbf{P}}_{\omega}^{0}\) visually match. We get similar results with values of \(\omega\) close to \(\exp(5\times 10^{-3}\text{i})\).