For $h>0$, let \begin{equation} \phi(x) = \begin{cases} h+x, \quad \text{if} \quad -h\le x\le 0,\\ h-x, \quad \text{if} \quad \hspace{5mm}0\le x\le h,\\ 0, \quad \hspace{8mm}\text{otherwise}. \end{cases} \end{equation} It is not hard to prove that $(u\ast\phi)''(x)=u(x+h)-2\,u(x)+u(x-h)$, where \begin{equation} (u\ast\phi)(x)=\int_{-\infty}^{\infty} u(z)\,\phi(x-z)\,dz. \end{equation} and $u\in C^2(\mathbb{R})$.
I am quite curious about the following question. In the study of difference equations, has any reference used the convolution operator with Gaussian-like kernel $\phi(x)$ defined as above to replace the second order difference operator $u(x+h)-2\,u(x)+u(x-h)$? I am not sure about this. Any reference, suggestion, idea, or comment is welcome, thanks!