Nearly resonant Fourier mode

Fix a direction $v$ , a frequency range $\mathcal M\subset\mathbb Z^n\setminus\{0\}$ , and a tolerance $\delta>0$ . A mode $m\in\mathcal M$ is nearly resonant at tolerance $\delta$ for $v\cdot\nabla$ if

0<|v\cdot m|\le\delta.

The positive lower inequality excludes an exact resonance. Other normalizations may compare $|v\cdot m|$ to a frequency-dependent threshold; the convention must be specified.

Effect on inversion

Solving the mode equation divides the forcing coefficient by $2\pi i(v\cdot m)$ , which may amplify it strongly. “Nearly resonant” without a scale or tolerance is only a qualitative description. A Diophantine lower bound controls how small these nonzero divisors can be as frequency increases.