# Adapted B-Spline Quasi-Interpolation for Approximating Piecewise Smooth Functions
## 2 Preliminaries  
In this paper, we present an approximation method for a non-smooth function \(f:\Omega\to\mathbb{R}\), defined on a domain \(\Omega\subset\mathbb{R}^{d}\), based on its known values at a uniform grid \(\{f(a)\}_{a\in A}\), where \(A\subset\Omega\). The proposed approximation is constructed using a quasi-interpolation approach, expressed as  
\[Qf(x)=\sum_{a\in A}f(a)q_{a}(x), \tag{1}\]  
where \(\{q_{a}\}_{a\in A}\) denotes the basis functions of the quasi-interpolation operator. In [14], we introduced a method for constructing basis functions for quasi-interpolation approximation based on Box-Splines in \(\mathbb{R}^{d}\). In [14], we outlined two key properties that the basis functions for quasi-interpolation should satisfy:  
1. Polynomial reproduction: \[\sum_{a\in A}p(a)q_{a}(x)=p(x),\ \ \ \forall x\in\Omega,\ \ \ p\in\Pi_{m},\] (2) where \(\Pi_{m}\) is the space of polynomials of total degree \(\leq m\) in \(\mathbb{R}^{d}\). 2. Compact support: \[supp(q_{a})\subset B(a,Rh).\] (3) Here, \(B(a,r)\) denotes the open ball of radius \(r\) centered at \(a\), and \(h\) is the fill-distance of the point-set \(A\), that is, the radius of the largest open ball centered in \(\Omega\) that does not contain any point from \(A\). \(R\) is independent of \(a\in A\) and \(h\). In [14], we presented the following proposition. Proposition 1: _Let \(f\in C^{m+1}(\Omega)\), i.e., all partial derivatives of \(f\) of total order \(\leq m+1\) are continuous on \(\Omega\). If \(Q\) is a quasi-interpolation operator satisfying properties (2) and (3), then_  
\[\|f-Qf\|_{\infty\Omega}\leq C\cdot M_{m+1,\Omega}(f)h^{m+1}, \tag{4}\]_where_  
\[M_{k,\Omega}(f)=sup_{x\in\Omega}\{\ |\partial_{k}f(x)|\ :\ \partial_{k}=\partial_{x_{ 1}^{i_{1}}}\cdots\partial_{x_{\frac{i}{k}^{i}}}f\,\ \sum_{j=1}^{d}i_{j}=k\ \}. \tag{5}\]  
In the one-dimensional setting, [PERSON] in [15] presented a comprehensive analysis of spline quasi-interpolation operators on uniform grids. The simplest quasi-interpolation operator is the Schoenberg operator defined by  
\[Sf(x)=\sum_{i}f(ih)B_{m}(x/h-i), \tag{6}\]  
where \(B_{2m-1}\) denotes the centered B-spline of degree \(2m-1\), supported on the interval \((-mh,mh)\) with nodes \(-mh,-(m-1)h,\ldots,(m-1)h,mh\). The Schoenberg operator achieves an approximation order of \(O(h^{2})\) for smooth functions. When applied to functions with jump discontinuities, it produces a smooth transition between the continuous regions without introducing overshoots or oscillations, thereby avoiding Gibbs-like phenomena. An improved approximation order of \(O(h^{4})\), can be achieved using the following quasi-interpolation operator based on cubic B-splines on a uniform grid:  
\[Qf(x)=\sum_{i}\biggl{[}-\frac{1}{6}B_{3}\Bigl{(}\frac{x}{h}-i-1\Bigr{)}+ \frac{4}{3}B_{3}\Bigl{(}\frac{x}{h}-i\Bigr{)}-\frac{1}{6}B_{3}\Bigl{(}\frac{x }{h}-i+1\Bigr{)}\biggr{]}f(ih). \tag{7}\]  
This operator can also be expressed in form (1) using the basis functions  
\[q_{i}(x)=-\frac{1}{6}B_{3}\Bigl{(}\frac{x}{h}-i-1\Bigr{)}+\frac{4}{3}B_{3} \Bigl{(}\frac{x}{h}-i\Bigr{)}-\frac{1}{6}B_{3}\Bigl{(}\frac{x}{h}-i+1\Bigr{)}. \tag{8}\]  
Thanks to its locality and its exact reproduction of polynomials of a degree up to three, the operator achieves an approximation order of \(O(h^{4})\) for functions that are \(C^{4}\)-smooth. However, when applied to functions with jump discontinuities, it tends to introduce oscillations and Gibbs-like overshoots near the discontinuity. The term Gibbs-like overshoot refers to spurious oscillations in the approximation near a jump, characterized by the persistence of their magnitude even as \(h\to 0\); see [16]. This paper aims to identify quasi-interpolation operators that achieve high-order accuracy in smooth regions while remaining free of overshoots near jump discontinuities.