## 3 Error Analysis for Regularized Delta and Heaviside Functions
### Fourier Transforms  
Let us consider a finite element spatial discretization of the geometrical domain where the representative mesh size is equal to \(h\). In the case of regularized functions with compact support \(\ell_{\rho}\) we can write \(\ell_{\rho}=m\,h\) where \(m\) is a positive integer or non-integer number. [PERSON] and [PERSON] [19] proved that, whenever \(m\) is a non-integer number in the one-dimensional case, or, alternatively, the two-dimensional and the three-dimensional cases are considered, the error Equation (11) does not decrease for decreasing mesh size. This drawback was discussed by taking into consideration the discrete form of the first moment condition--also denoted as _mass condition_--whose satisfaction is necessary in order for the regularized formulation to preserve homogeneous fields. [PERSON] and [PERSON] [22] showed that, even in the one-dimensional case, simple compact support functions such as the hat function or the cosine function lead to a satisfying convergence in the one-dimensional case when \(m=1\) and half the support is taken equal to \(m\,h\). However, compact support functions fail to satisfy the mass condition for any non-integer \(m\). In the two-dimensional and the three-dimensional case, this limitation becomes more evident [19]. This is because the number of quadrature points within the support is not constant, but it changes depending on the discretization and the geometry of the interface with respect to the mesh. It should be emphasized that, though effective, we are not focusing on mesh adaptive schemes [3]. We consider instead meshes that do not necessarily conform to the interface position, in line with the spirit of the XFEM [24]. If we choose indeed compact support functions, the width of the support will depend on the position of the interface with respect to the mesh. Consequently, the mesh has to conform to the weight function. In order to discuss the accuracy of the quadrature procedure, we exploit the results recently obtained by [22] based on the Fourier transforms \(\hat{\varphi}\) of \(\varphi\)  
\[\hat{\varphi}(\kappa)=\int_{-\infty}^{\infty}\varphi(\xi)e^{-2\pi\,i\kappa \xi}\,d\xi \tag{13}\]Conversely, we have that  
\[\varphi(\xi)=\int_{-\infty}^{\infty}\hat{\varphi}(\kappa)e^{2\pi\,i\,\kappa\xi}\,d\kappa \tag{14}\]  
Under the hypothesis that the space is covered by a regular grid \(\{\mathbf{x}_{j}\}\), with \(j\in\mathbb{Z}\), of quadrature points, [PERSON] and [PERSON] [22] showed that the discrete version of the moment conditions (12) is related to the derivatives of the Fourier transform \(\hat{\varphi}\) as follows  
\[M_{r}(\delta_{\rho},h)=\frac{1}{(-2\pi\,i)^{r}}\frac{\partial^{r}\hat{\varphi} (m\kappa)}{\partial\kappa^{r}}\bigg{|}_{\kappa=0}+\frac{1}{(-2\pi\,i)^{r}} \sum_{\kappa\in\mathbb{Z},\kappa\
eq 0}e^{-2\pi\,i\kappa\,p}\frac{\partial^{r} \hat{\varphi}(m\kappa)}{\partial\kappa^{r}} \tag{15}\]  
Now, suppose that \(\hat{\varphi}(\kappa)\approx 0\) for \(\kappa\geq\beta\); consequently, the last term on the r.h.s. of Equation (15)  
\[e^{-2\pi\,i\kappa\,p}\frac{\partial^{r}\hat{\varphi}(m)}{\partial\kappa^{r}}+ e^{-2\pi\,i\kappa\,p}\frac{\partial^{r}\hat{\varphi}(2m)}{\partial\kappa^{r}}+\ldots \tag{16}\]  
turns out to vanish for any \(m\geq\lambda\). Hence if the conditions  
\[\frac{\partial^{r}}{\partial\kappa^{r}}\hat{\varphi}(\kappa)|_{\kappa=0}=1, \qquad\text{for}\ \ r=0,\qquad\frac{\partial^{r}}{\partial\kappa^{r}}\hat{\varphi}(\kappa)|_{ \kappa=0}=0\quad\text{for}\ \ 1\leq r<q \tag{17}\]  
are satisfied for \(r=0,1,\ldots,q-1\), then \(\delta_{\rho}\) satisfies \(q\) discrete moment conditions. In particular, if \(\hat{\varphi}(0)=1\), the mass condition  
\[\int_{-\infty}^{\infty}\varphi_{\rho}(x)\,dx=\hat{\varphi}(0)=1 \tag{18}\]  
is satisfied [19]. Moreover, the derivatives of the Fourier transform and the moment conditions on \(\varphi\) are related by the relationship \(\left.\frac{\partial^{r}\hat{\varphi}}{\partial\kappa^{r}}\right|_{\kappa=0} =(-i)^{p}\int_{-\infty}^{\infty}\varphi(\xi)\xi^{r}\,d\xi\).