= Scalar Fourier power criterion
{title2=$\|S_{h,k}^n\|_{2\to2}=\mathop{\mathrm{ess\,sup}}_\theta|G_{h,k}(\theta)|^n$}
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 $|G|\leq1$. Mesh-uniform fixed-time <stability>, on $nk\leq T$, permits $\sup|G|\leq1+Ck$: powers are bounded by $e^{CT}$. Conversely a uniform bound at $n=\lfloor T/k\rfloor$ gives this $1+O(k)$ restriction as $k\to0$. For <matrix> symbols, use the existing <uniform power bound for matrix Fourier symbols> rather than only <eigenvalue> moduli.
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