= Solution
Consider a general constant-coefficient linear difference scheme
$$
\sum_{r,j}a_{rj}u_{m+j}^{n+r}=0.
$$
A spatial Fourier mode has the form
$$
u_m^n=\widehat u^{,n}e^{im\theta}.
$$
If one time step multiplies its amplitude by $G$, then $\widehat u^{,n+r}=G^r\widehat u^{,n}$, and substitution gives the <amplification polynomial of a multilevel finite difference scheme>
$$
\boxed{\sum_{r,j}a_{rj}G^r e^{ij\theta}=0}.
$$
For a one-step method this equation directly determines
$$
G(\theta)=\frac{\widehat u^{,n+1}}{\widehat u^{,n}}.
$$
By Parseval's identity, <von Neumann stability analysis> gives $2$-norm stability when $|G(\theta)|\leq1$ for every Fourier mode. For a multilevel method, every root must obey the corresponding root condition.
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