Convolution: $(f * g)(x) = \int_{\mathbb{R}^d} f(y)\, g(x-y)\, dy$
Spectral Convolution Theorem: $\mathcal{F}[f * g] = \hat{f}\,\hat{g}$
Forward Transform: $\hat{f}(\omega) = \int_{-\infty}^{\infty} f(x)\, e^{-i \omega x}\, dx$
Inverse Transform: $f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{f}(\omega)\, e^{+i \omega x}\, d\omega$
Fourier Series on $[0,L]$: $f(x) = \sum_{k=-\infty}^{\infty} c_k\, e^{+i \omega_k x}$
Fourier Coefficients: $c_k = \frac{1}{L} \int_0^L f(x)\, e^{-i \omega_k x}\, dx$
Harmonic Frequencies: $\omega_k = \frac{2\pi k}{L}$
Fourier Series on $\mathcal{V}$: $f(\mathbf{x})=\sum_{\boldsymbol{\omega} \in \Omega} \hat{f}(\boldsymbol{\omega})\,e^{+i \boldsymbol{\omega}\cdot\mathbf{x}}$
Fourier Coefficients: $\hat{f}(\boldsymbol{\omega}) = \frac{1}{|\mathcal{V}|} \int_{\mathcal{V}} f(\mathbf{x})\,e^{-i \boldsymbol{\omega}\cdot\mathbf{x}}\, d\mathbf{x}$
Wave Vector: $\boldsymbol{\omega}$
Wave Number: $\|\boldsymbol{\omega}\|$
Component Wave Number: $\omega_i = \frac{2\pi}{\lambda_i} \in (-\infty, +\infty)$