## 5 Interpolation Model with Scalar Subdivision Schemes
### A Solution by Shifting Parameters  
Propositions 4 and 5 state that, for even-symmetric and odd-symmetric masks, the matrices \(M_{n}\) and \(\mathbf{M}_{n}\) are singular independently of the mask. In this section, we propose a different model for interpolating that avoids this inconvenience. What follows is motivated by [PERSON]'s work [44] on Hermite interpolation with B-spline parametric curves. Let us consider the dual subdivision schemes again. In Section 4, by interpolating at the parameters \(t_{j}=j,j=0,\ldots,n-1\), we obtained the first limit stencil as  
\[V_{j}^{0}=c(j)=\sum_{s\in\mathbb{Z}}\varphi(j-s)P_{s}^{0}=\sum_{s=-p}^{q} \varphi(s)P_{j-s}^{0},\qquad\text{with supp}\,\varphi=[-p,q].\]  
Here we consider the variant  
\[V_{j}^{0}=c(j+\sigma)=\sum_{s\in\mathbb{Z}}\varphi(j-s+\sigma)P_{s}^{0}=\sum_ {s=-p}^{q-1}\varphi(s+\sigma)P_{j-s}^{0}\]  
with \(\sigma\in(0,1)\). More specifically, given a positive integer \(\
u\), we investigate the method for \(\sigma\) varying in \(\frac{1}{\
u}\mathbb{Z}\cap(0,1)\). The evaluation of the basic function is done as in [35]. In the case of symmetric masks, as in (11) and (12), the following results hold. Proposition 6: _For an odd-symmetric subdivision scheme with mask \(\mathbf{a}=\{a_{-p},a_{-p+1},\ldots,a_{p}\}\) where \(p\in\mathbb{N}\), the stencil obtained with the set \(\varphi\Big{(}\frac{1}{2}+\mathbb{Z}\Big{)}\cap[-p,p]\) is even-symmetric. Analogously, for an even-symmetric subdivision scheme with mask \(\mathbf{a}=\{a_{-p},a_{-p+1},\ldots,a_{p+1}\}\) where \(p\in\mathbb{N}\), the stencil obtained with the set \(\varphi\Big{(}\frac{1}{2}+\mathbb{Z}\Big{)}\cap[-p,p+1]\) is odd-symmetric._  
Our proposal consists in modifying the interpolation models (10) and (14) to \(c^{(k)}\Big{(}\frac{1}{2}+dj\Big{)}=V_{j}^{k}\). For point interpolation, the symbol changes from (18) to (17) and this way we resolve the singularity in Proposition 4. Figure 3 portrays an example of even-symmetric subdivision scheme where after a \(\frac{1}{2}\)-shift the new matrix is nonsingular. However, when the new stencil leads to a trigonometric polynomial with roots in the grid \(\frac{1}{n}\mathbb{Z}\cap[0,1]\) it is not possible to avoid the singularity of the corresponding matrix. In such situations, one may consider the general approach \(V_{j}^{0}=c(j+\sigma)\), with \(\sigma\in(0,1)\) or treat the singularity as discussed in the next subsection. The use of shifted parameters for interpolation can be employed as a degree of freedom for the geometry of the interpolation curve (see Figure 4). Nevertheless, the symmetry provided by the subdivision scheme might be lost. Figure 3: Point interpolation with a dual subdivision scheme [39] considering the shifted parameters. Now let us consider the point and tangent interpolation with the interpolation at the parameters \(\frac{1}{2}+\mathbb{Z}\) for odd-symmetric schemes. The first block (20) in the factorization of \(\mathbf{M}_{n}\) becomes  
\[(\mathcal{L}_{n})_{1,1} =\sum_{j=-\lceil\frac{p-1}{2}\rceil}^{\left[\frac{q}{2}\right]} \begin{bmatrix}\varphi(2j+\frac{1}{2})&\varphi(2j-\frac{1}{2})\\ \varphi^{\prime}(2j+\frac{1}{2})&\varphi^{\prime}(2j-\frac{1}{2})\end{bmatrix} \] \[= \begin{bmatrix}\varphi(\frac{1}{2})+\sum_{j=1}^{\left[\frac{p-1} {2}\right]}(\varphi(2j-\frac{1}{2})+\varphi(2j+\frac{1}{2}))&\varphi(\frac{1}{ 2})+\sum_{j=1}^{\left[\frac{p-1}{2}\right]}(\varphi(2j-\frac{1}{2})+\varphi( 2j+\frac{1}{2}))\\ \varphi^{\prime}(\frac{1}{2})+\sum_{j=1}^{\left[\frac{p-1}{2}\right]}(\varphi ^{\prime}(2j+\frac{1}{2})-\varphi^{\prime}(2j-\frac{1}{2}))&-\varphi^{\prime} (\frac{1}{2})+\sum_{j=1}^{\left[\frac{p-1}{2}\right]}(\varphi^{\prime}(2j- \frac{1}{2})-\varphi^{\prime}(2j+\frac{1}{2}))\end{bmatrix} \] \[= \begin{bmatrix}\Sigma_{1}&\Sigma_{1}\\ \Sigma_{2}&-\Sigma_{2}\end{bmatrix}