Fourier amplification symbol ($G(\theta )={\sum }_j{A}_j{e}^{ij\theta }$)

A constant-coefficient translation-invariant grid update is diagonalized spatially by discrete Fourier modes. Its scalar or matrix multiplier is the displayed symbol. In a semidiscrete evolution, the analogous symbol is a continuous-time generator and its exponential propagates each mode. Discrete Parseval identity converts uniform modal bounds to discrete L2 norm bounds.

For a scalar constant-coefficient grid update, Fourier transformation turns $S$ into multiplication by its symbol $G$. Parseval identity proves the displayed norm equality; necessity is obtained by concentrating Fourier data near a maximizing frequency. Contractivity for every step is equivalent to $|G|\le 1$. Mesh-uniform fixed-time stability, on $nk\le T$, permits $\sup |G|\le 1+Ck$: powers are bounded by $e^{CT}$. Conversely a uniform bound at $n=\lfloor T/k\rfloor$ gives this $1+O(k)$ restriction as $k\to 0$. For matrix symbols, use the existing uniform power bound for matrix Fourier symbols rather than only eigenvalue moduli.

This uniform bound is the actual fixed-time stability requirement for a vector-valued periodic grid evolution. Eigenvalue moduli alone are insufficient for nonnormal matrices. A unit Jordan block with an order-one off-diagonal entry grows like $n$, while an entry proportional to $\Delta t$ can give bounded growth on $n\Delta t\le T$. Uniform eigenvector conditioning or a suitable discrete energy estimate can establish the bound. Mesh-dependent diagonalizations cannot be treated as automatically harmless.