## 5 Interpolation Model with Scalar Subdivision Schemes
### Our Regularizing Strategy  
We have already observed that in some cases the matrix obtained by interpolating at integer parameters is singular and that sometimes the singularity cannot be avoided neither by a proper shift of the parameters. In those cases, independently of the interpolated points, the inverse of \(M_{n}\) or \(\mathbf{M}_{n}\) is not defined and an alternative solution has to be chosen. The first idea is using the pseudo-inverse of the matrix, providing a least-square solution [41]. Another approach is considering a _regularization_, which is an approximation of the _ill-posed_ problem by a family of neighboring _well-posed_ problems [40]. Among the regularization methods for solving an ill-posed problem whose discrete form is the linear system \(Ax=b\) we mention the _Tikhonov approach_, which consists in solving the family of problems  
\[\operatorname{argmin}_{x}\|Ax-b\|^{2}+\lambda\|Lx\|^{2},\]  
depending on the _regularization parameter_\(\lambda\). The latter controls the weight given to the residual norm and the regularization term. The optimal value for \(\lambda\) can be chosen by _discrepancy principle_, _generalized cross-validation_, or the _L-curve method_[43]. The operator \(L\), which can be taken as the _identity operator_ or a _differential operator_ (for instance, the first or second derivative operator), looks to alter the least square solution to enforce special features of the regularized approximations [40; 43]. As a necessary condition it is required that \(Ker(A)\ \cap\ Ker(L)=\{0\}\). This problem is equivalent to solve the normal equations \((A^{\top}A+\lambda L^{\top}L)x=A^{\top}b\) for each \(\lambda\). Since the quality of the obtained solution either by least-squares or Tikhonov regularization is not acceptable in many cases, in the following we study a new regularized problem depending on a parameter \(\omega\) in which a (possibly block) \(\omega\)-circulant matrix replaces the original circulant coefficient matrix. First we consider the basic interpolation case where \(M_{n,\omega}\) is the \(\omega\)-circulant counterpart of \(M_{n}\) with \(\omega=\exp(\mathrm{i}\psi)\). By Remark 3, the eigenvalues of the new matrix \(M_{n,\omega}\) are given by the standard uniform sampling of the function \(b(\theta+\frac{\psi}{n})\). In this manner, the spectrum of \(M_{n}\) is shifted. We observe that \(M_{n}\) and \(M_{n,\omega}\) are both singular if there exists at least two roots of \(b(\theta)\) in the grid \(\frac{2\pi\mathbb{N}}{n}\cap[0,2\pi]\) with distance \(\frac{\psi}{n}\). If the latter condition is not satisfied, then the initial system \(M_{n}\mathbf{P}^{0}=\mathbf{V}^{0}\) is singular, while its perturbation \(M_{n,\omega}\mathbf{P}^{0}=\mathbf{V}^{0}\) has a unique, complex solution, even though the original problem is defined in the real domain. However, for \(\psi\) small enough in modulus, the imaginary part of the solution becomes negligible and the perturbed system is close to the original one. Figure 5: Interpolating points and tangents directions with a cubic B-spline curve by shifting the parameters. Secondly, we consider the interpolation of points and associated tangent vectors. The block (20) in the corresponding \(\omega\)-\((2\times 2)\)-circulant matrix \(\mathbf{M}_{n,\omega}\) is  
\[(\mathcal{L}_{n,\omega})_{1,1} =\sum_{j=0}^{\left\lceil\frac{p}{2}\right\rceil}\mathbf{B}_{j}\; \omega^{j/n}+\sum_{j=1}^{\left\lceil\frac{p}{2}\right\rceil}\mathbf{B}_{-j}\; \omega^{(n-j)/n}\] \[= \begin{bmatrix}\beta_{0}^{0}&\beta_{1}^{0}\\ \beta_{0}^{1}&-\beta_{1}^{1}\end{bmatrix} +\sum_{j=1}^{\left\lceil\frac{p}{2} \right\rceil} \begin{bmatrix}\beta_{2j}^{0}&\beta_{2j-1}^{0}\\ \beta_{2j}^{1}&\beta_{2j-1}^{1}\end{bmatrix} \omega^{j/n}+\sum_{j=1}^{\left\lceil \frac{p}{2}\right\rceil} \begin{bmatrix}\beta_{2j}^{0}&\beta_{2j+1}^{0}\\ -\beta_{2j}^{1}&-\beta_{2j+1}^{1}\end{bmatrix} \omega^{(n-j)/n}\] \[=