Fractional discrete Laplacian versus discretized fractional Laplacian

We define and study some properties of the fractional powers of the discrete Laplacian $$(-\Delta_h)^s,\quad\hbox{on}~\mathbb{Z}_h = h\mathbb{Z},$$ for $h>0$ and $0<s<1$. A comparison between our fractional discrete Laplacian and the \textit{discretized} continuous fractional Laplacian as $h\to0$ is carried out. We get estimates in $\ell^\infty$ for the error of the approximation in terms of $h$ under minimal regularity assumptions. Moreover, we provide a pointwise formula with an explicit kernel and deduce H\"older estimates for $(-\Delta_h)^s$. A study of the negative powers (or discrete fractional integral) $(-\Delta_h)^{-s}$ is also sketched. Our analysis is mainly performed in dimension one. Nevertheless, we show certain asymptotic estimates for the kernel in dimension two that can be extended to higher dimensions. Some examples are plotted to illustrate the comparison in both one and two dimensions.