In a mathematical physics exercise, some discrete fields can conveniently be expressed using finite difference schemes as follows: \begin{align} A_6 &= \phi_{i+1}+\phi_{i-1}+4\phi_i \, , \\ A_7 &= \phi_{i+2}+\phi_{i-2}-2\phi_i \, , \\ A_8 &= \phi_{i+2}-\phi_{i-2} \, , \end{align} wherein $\phi_i \equiv \phi(x_i)$ such that $x_i = x_0 + ih$ are the node positions. Moreover, $h$ represents a uniform grid spacing. The goal is to approximate these groups by continuous functions following a standard discrete-to-continuum approach.
Here is what I have tried: \begin{align} A_6 &= \phi_{i+1}+\phi_{i-1}-2\phi_i + 6\phi_i \approx h^2 \phi''(x)+6\phi(x) \, , \\ A_7 &= \phi_{i+2}-4\phi_{i+1}+6\phi_i-4\phi_{i-1}+\phi_{i+2} +4 \left( \phi_{i+1}+\phi_{i-1}-2\phi_i \right) \approx h^2 \left( h^2\phi''''(x)+4 \phi''(x) \right) \, , \\ A_8 &= 2 \left( \frac{\phi_{i+2}}{2}-\phi_{i-1}+\phi_{i+1}- \frac{\phi_{i-2}}{2} +\phi_{i-1}-\phi_{i+1} \right) \approx 2 h\left( h^2\phi'''(x) -2\phi'(x)\right) \, . \end{align}
I'd like to know whether these approximations are unique and/or whether a better accuracy can be reached using different approaches.
Any help or comment is very welcome.
Here is a link to the wiki article on finite difference coefficients