Semidiscrete Fourier transform ($\hat{W}(\theta )={\sum }_{j\in \mathbb{Z}}{W}_j{e}^{-ij\theta }$)

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 areThe physical-frequency convention uses $\theta =kh$ and $\tilde{W}(k)=h\hat{W}(kh)$ 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.

Consider an implicit centered heat stencil of width $\Delta x$ weighted by $\theta$, combined with an explicit centered stencil of width $2\Delta x$ weighted by $1-\theta$. With $\mu =\kappa \Delta t/(\Delta x{)}^2$, its amplification factor is $G(\xi )=[1-\mu (1-\theta ){\sin }^2\xi ]/[1+4\mu \theta {\sin }^2(\xi /2)]$. For $0\le \theta <1/2$, contraction in the discrete ${\ell }^2$ norm holds exactly under the displayed condition. It is unconditional for $1/2\le \theta \le 1$. Put $v={\sin }^2(\xi /2)$: the condition $G\ge -1$ reduces to $4\mu v[(1-2\theta )-(1-\theta )v]\le 2$. Maximizing this concave quadratic proves the threshold; the Parseval identity converts the multiplier bound into contraction.