= Semidiscrete Fourier transform
{title2=$\widehat W(\theta)=\sum_{j\in\mathbb Z}W_je^{-ij\theta}$}
For a sequence indexed by the <integers>, its semidiscrete Fourier transform is the displayed <Fourier series> in $\theta\in[-\pi,\pi]$. For square-summable data the series is interpreted in the <L2 norm>. Its inversion and <Parseval identity> are
$$
W_j=\frac1{2\pi}\int_{-\pi}^{\pi}\widehat W(\theta)e^{ij\theta}d\theta,\qquad
h\sum_{j\in\mathbb Z}|W_j|^2=\frac h{2\pi}\int_{-\pi}^{\pi}|\widehat W(\theta)|^2d\theta.
$$
The physical-frequency convention uses $\theta=kh$ and $\widetilde W(k)=h\widehat W(kh)$ on $[-\pi/h,\pi/h]$. This is distinct from the <discrete Fourier transform> of a finite vector. Translation-invariant <finite difference operators> become <Fourier multipliers>, making <Fourier stability analysis> possible.
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