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A bounded scalar Laurent operator on ${\ell }^2(\mathbb{Z})$ commutes with the bilateral lattice shift. Its bi-infinite matrix has constant diagonals, formally $(Lu{)}_m={\sum }_j{a}_j{u}_{m+j}$. Under the discrete Fourier transform, it becomes multiplication by an essentially bounded symbol $\ell (\theta )$, andThe boundedness assumption is essential; a formal constant-diagonal matrix need not define a bounded map on ${\ell }^2$.

For a finite difference operator with finitely many nonzero coefficients, $\ell (\theta )={\sum }_j{a}_j{e}^{ij\theta }$. The neighbor-sum operator $L{u}_m=u_{m-1}+u_{m+1}$ has symbol $2\cos \theta$ and operator norm two. Compression to a one-sided sequence space yields a Toeplitz operator, for which the boundary prevents the same full-lattice diagonalization.

Use the forward Euler method in time and evaluate both spatial finite difference operators at the old time level:With $\nu =\Delta t/\Delta x$ and $\mu =\Delta t/(\Delta x{)}^2$, the explicit update isEvery value on the new time level is computed directly from three old values. The sum of the stencil weights is one, as required to preserve constant solutions of the advection-diffusion equation.

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.