# Adapted B-Spline Quasi-Interpolation for Approximating Piecewise Smooth Functions
## 4 Adapted Quasi Interpolation Operators for Non-Smooth Functions in \(\mathbf{R^{2}}\)
### Theoretical Results for the 2D Adapted Approximation  
Similarly to one-dimensional Proposition 2, we obtain the following proposition for the two-dimensional case. **Proposition 5**. : _Let \(f\in C^{4}(\mathbb{R}^{2})\), \((x_{i},y_{j})=(ih,jh)\), and let \(Q\) be the operator in (27) where the weights \(\{w_{ij}\}\) are arbitrarily chosen from any of the nine options in (28). Then, we have_  
\[|f(x,y)-Qf(x,y)|=O(h^{4}),\ \ \text{as}\ h\to 0. \tag{34}\]  
**Proposition 6**. : _Let \(f\) be a piecewise \(C^{4}\) function defined as in (26), with a singularity curve \(\Gamma\). Then, the symmetric quasi-interpolation approximation \(Q^{S_{x}S_{y}}f\) defined above is globally \(C^{2}\), and it satisfies the error estimate_  
\[|f(x,y)-Q^{S_{x}S_{y}}f(x,y)|\leq Ch^{4},\ \ dist((x,y),\Gamma)\geq 3\sqrt{2}h, \tag{35}\]  
_where \(C\) is a constant independent of \(h\). Near the discontinuity at \(\Gamma\), the approximation \(Q^{S_{x}S_{y}}f\) exhibits some Gibbs-like overshoots._  
Proof. : Observe that the support of the basis function \(B_{3}(x)B_{3}(y)\) is the Cartesian product of the supports of \(B_{3}(x)\) and \(B_{3}(y)\), each of which spans an interval of length \(4\). Consequently, the diameter of the two-dimensional support is \(4\sqrt{2}\). Moreover, the weights \(w_{ij}\) in the symmetric scheme involve function values located at a distance up to \(\sqrt{2}\) from the grid point \((ih,jh)\). Depending on the local orientation of the singularity curve \(\Gamma\), it follows that at points \((x,y)\) satisfying the condition in (35), the symmetric approximation accesses only values from one side of \(\Gamma\). Thus, the error estimate given in (35) follows as a direct consequence of the support structure and the localized nature of the approximation. The occurrence of Gibbs-like overshoots can be demonstrated using arguments analogous to those in the univariate case, as outlined in Proposition 3. In particular, near portions of \(\Gamma\) that are aligned with one of the coordinate axes, the overshoots are identical to those observed in the univariate setting. The following proposition shows that the adapted quasi-interpolation operator extends the region over which optimal approximation accuracy is attained, while eliminating Gibbs-like overshoots near discontinuities. **Proposition 7**. : _Let \(f\) be a piecewise \(C^{4}\) function defined as in (26), with a smooth singularity curve \(\Gamma\). Then, the adapted quasi-interpolation approximation \(Q^{Adapt}f\) defined above is globally \(C^{2}\), and, for sufficiently small \(h\), it satisfies the error estimate_  
\[|f(x,y)-Q^{Adapt}f(x,y)|\leq Ch^{4},\ \ dist((x,y),\Gamma)\geq 2\sqrt{2}h, \tag{36}\]_where \(C\) is a constant independent of \(h\). Moreover, the approximation \(Q^{Adapt}f\) does not exhibit any Gibbs-like overshoots near the discontinuity at \(\Gamma\)._  
Proof. : First, assume a non-zero jump across \(\Gamma\), and \(h\) is sufficiently small. Then, by construction, each of the weights \(w_{i,j}\) selected for the adapted quasi-interpolation operator uses only function values from one side of the jump discontinuity at \(\Gamma\). Error estimate (36) follows by considering the support of \(B_{3}(x)B_{3}(y)\) and the reproduction of cubic polynomials by \(Q^{Adapt}\). Next, we observe that each of the nine candidate weights, \(\{w_{i,j}^{k}\}_{i=1}^{k}\), satisfies the approximation property  
\[w_{i,j}^{k}=f(ih,jh)+O(h^{2}),\quad\text{as}\;\;h\to 0. \tag{37}\]  
This follows directly from the fact that each weight is constructed as a tensor product of two univariate weights, each of which satisfies the one-dimensional approximation property given in (22). Thus, we have  
\[Q^{Adapt}f(x,y)=\sum_{i,j}w_{i,j}B_{3}(\frac{x}{h}-i)B_{3}(\frac {y}{h}-j) \tag{38}\] \[=\sum_{i,j}(f(ih,jh)+O(h^{2}))B_{3}(\frac{x}{h}-i)B_{3}(\frac{y} {h}-j)\] \[=\sum_{i}f(ih,jh)B_{3}(\frac{x}{h}-i)B_{3}(\frac{y}{h}-j)+O(h^{2}).\]  
As noted in Section 2, the Schoenberg approximation (6) avoids overshoots and oscillations, thereby preventing Gibbs-like artifacts.