## 4 Basic Ideas on Scalar Subdivision Schemes
This problem is modeled by the linear operator \(M_{n}\) such that  
\[M_{n}\mathbf{P}^{0}= \begin{bmatrix}\beta_{0}&\beta_{-1}&\beta_{-2}&\ldots& \beta_{2}&\beta_{1}\\ \beta_{1}&\beta_{0}&\beta_{-1}&\ldots&\beta_{3}&\beta_{2}\\ \ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\ \beta_{-1}&\beta_{-2}&\beta_{-3}&\ldots&\beta_{1}&\beta_{0}\end{bmatrix} \begin{bmatrix}P_{0}^{0}\\ P_{0}^{0}\\ \vdots\\ P_{n-1}^{0}\end{bmatrix} = \begin{bmatrix}V_{0}^{0}\\ V_{1}^{0}\\ \vdots\\ c(n-1)\end{bmatrix} . \tag{10}\]  
\(M_{n}\) is referred to as the matrix that represents the _point interpolation operator_ for linear subdivision schemes. Note that the structure of \(M_{n}\) is circulant, according to Definition 1, and the first row is the vector  
\[\hat{\rho}_{n}:= \begin{bmatrix}\beta_{0},\beta_{-1},\ldots,\beta_{-p},0, \ldots,0,\beta_{q},\ldots,\beta_{1}\end{bmatrix} \in\mathbb{R}^{m}.\]  
Depending on the symmetry of the stencils, some particular cases may be analysed. **Definition 4**. : _A subdivision scheme is called_  
* Odd-symmetric _if_ \(a_{-j}=a_{j}\)_;_
* Even-symmetric _if_ \(a_{1-j}=a_{j}\) _for_ \(j\in\mathbb{N}\)_._  
This definition contains particular cases of the primal [38] and dual [39] form of subdivision schemes, respectively, and the corresponding limit stencils show the same type of symmetries [33]. As a consequence, the odd-order limit stencil inherits the odd or even symmetry:  
\[\beta_{j}^{2d}= \begin{cases}\beta_{-j^{\prime}}^{2d},&\text{for odd-symmetric schemes},\\ \beta_{1-j^{\prime}}^{2d},&\text{for even-symmetric schemes},\end{cases}  \qquad d\in\mathbb{N}. \tag{11}\]  
Then, for the even-order limit stencil we get  
\[\beta_{j}^{2d+1}= \begin{cases}-\beta_{-j}^{2d+1}&\text{for odd-symmetric schemes},\\ -\beta_{1-j}^{2d+1}&\text{for even-symmetric schemes},\end{cases}  \qquad d\in\mathbb{N}. \tag{12}\]  
For \(d\geq 1\) it holds \(\sum\limits_{j\in\mathbb{Z}}\beta_{j}^{d}=0\) and \(\sum\limits_{j\in\mathbb{Z}}\beta_{j}^{0}=1\). Now, let us consider the interpolation of \(n\) points \(\mathbf{V}^{0}\) with a subdivision curve. It is natural to think of those points as a sampling of the curve at integer parameters \(V_{s}^{0}=c(s)\), \(s=0,\ldots,n-1\), as in (8). By doing so, we get the _inverse problem_ with respect to (10) (see Ref. [40]). In this setting, an interpolation problem is said to be _singular_ if the operator \(M_{n}\) is singular and therefore _ill-posed_ for the selected stencil. In Section 5, we show that in some cases the singularity depends on the value of \(n\). To solve a non-singular interpolation problem where \(M_{n}\) has a circulant structure, one can take advantage of the diagonalization through the Fourier matrix recalled in Section 2. However, in the singular case more strategy is needed. In the literature, in the context of curve and surface schemes those cases are treated by a fitting model [41] or by a fairness functional [42] while introducing more points as degrees of freedom. Another possible approach consists in the regularization in a Tikhonov sense [40,43]. Moreover, it is possible to consider a non-singular perturbation of the matrix \(M_{n}\). This is the strategy that we explore in Section 5.2, using a computationally convenient \(\omega\)-circulant matrix. The Block-Circulant Case for Hermite Interpolation  
Considering the case of interpolating points and associated derivatives up to the \((d-1)\)-th order, with tangent interpolation as the first instance, Equation (9) provides some insight.