# Adapted B-Spline Quasi-Interpolation for Approximating Piecewise Smooth Functions
## 2 Preliminaries
### Theoretical Considerations of the Adapted Approximation  
The following propositions quantify the overshoots produced by the symmetric quasi-interpolation operator and establish that the adapted quasi-interpolation operator eliminates such overshoots.  
Let \(f\) be a non-smooth function defined by  
\[f(x)=\left\{\begin{array}{ll}f_{1}(x),&x\in[-1,0),\\ f_{2}(x),&x\in[0,1],\end{array}\right\}, \tag{19}\]  
where \(f_{1}\in C^{4}([-1,0])\), \(f_{2}\in C^{4}([0,1])\). Let \(D\) denote the size of the jump discontinuity at \(x=0\), defined by \(D=|f^{+}(0)-f^{-}(0)|\). Consider the quasi-interpolation approximations \(Q^{Sym}f\) and \(Q^{Adapt}f\), constructed using the sampled data \(\{f(ih)\}\).  
Proposition 3: _For any \(h<0.2\), there exist points \(x_{1}\in(-2h,0)\) and \(x_{2}\in(0,2h)\) such that_  
\[|Q^{Sym}f(x_{2})-Q^{Sym}f(x_{1})|\geq(1+0.113)D. \tag{20}\]  
Proof: Since \(Q^{Sym}\) is a linear operator, it suffices to prove the result for the function  
\[f(x)=\begin{cases}0,&x<0,\\ 1,&x\geq 0.\end{cases},\]  
which corresponds to the case \(f_{1}\equiv 0\) and \(f_{2}\equiv 1\), and yields \(D=1\).  
From Equation (15), it follows that \(w_{i}^{Sym}=0\) for \(i<-1\) and \(w_{i}^{Sym}=1\) for \(i>0\), while \(w_{-1}^{Sym}=-1/6\) and \(w_{0}^{Sym}=7/6\). It follows that \(Q^{Sym}f(x)=f(x)\) for \(|x|\geq 2h\). In the region \(|x|<2h\), the approximation \(Q^{Sym}f(x)\) is a piecewise cubic function, attaining a minimum value of approximately \(-0.056935\) at \(x_{1}<0\) and a maximum value of approximately \(1.056935\) at \(x_{2}>0\). Therefore, the result in (20) follows.  
**Proposition 4**.: _Let \(f\) be defined as in (19). Then, the adapted quasi-interpolation approximation \(Q^{Adapt}f\) for \(f\) defined by (19) is globally \(C^{2}\), and satisfies the error estimate_  
\[|f(x)-Q^{Adapt}f(x)|\leq Ch^{4},\ \ |x|\geq 2h, \tag{21}\]  
_where \(C\) is a constant independent of \(h\). Moreover, the approximation \(Q^{Adapt}f\) does not exhibit any Gibbs-like overshoot near the discontinuity at \(x=0\)._  
Proof.: First, assume \(D>0\), and \(h\) to be sufficiently small. Then, by construction, each of the weights \(w_{i}\) selected for the adapted quasi-interpolation operator uses only function values from one side of the jump discontinuity at \(x=0\). Error estimate (21) follows by considering the support of \(B_{3}\) and the reproduction of cubic polynomials by \(Q^{Adapt}\) for \(|x|\geq 2h\).  
Next, we observe that each of the candidate weights, \(w_{i}^{1}=w_{i}^{Sym}\), \(w_{i}^{2}=w_{i}^{F}\), and \(w_{i}^{3}=w_{i}^{B}\), satisfies the approximation property  
\[w_{i}^{k}=f(ih)+O(h^{2}),\ \ \ \text{as}\ \ h\to 0, \tag{22}\]  
provided that the corresponding weights are constructed using only function values from one side of the jump discontinuity. This is readily verified in the case of \(w_{i}^{Sym}\), as it is symmetric and centered and reproduces linear polynomials. For \(k=2,3\), the approximation property in (22) also holds, as a direct consequence of the defining relation (11). Thus, in particular in \((-2h,2h)\), we have  
\[\begin{split} Q^{Adapt}f(x)=\sum_{i}w_{i}B_{3}(\frac{x}{h}-i)& =\sum_{i}(f(ih)+O(h^{2}))B_{3}(\frac{x}{h}-i)\\ &=\sum_{i}f(ih)B_{3}(\frac{x}{h}-i)+O(h^{2}).\end{split} \tag{23}\]  
As noted in Section 2, the Schoenberg approximation (6) avoids overshoots and oscillations, thereby preventing Gibbs-like artifacts. From (23), we observe that \(Q^{Adapt}f\) differs from the Schoenberg approximation by only a \(O(h^{2})\) perturbation. Therefore, the conclusion of the proposition follows directly.