[proofplan] We prove both directions of the equivalence $|H(\theta)| \leq 1$ for all $\theta$ iff the scheme is $\ell_2$-stable. The key tool is [Parseval's Identity](/theorems/434), which equates $\|U^n\|_{\ell_2}$ with $\|\hat{U}^n\|_*$. Since the scheme acts as pointwise multiplication in Fourier space — $\hat{U}^n(\theta) = [H(\theta)]^n \hat{U}^0(\theta)$ — the $\ell_2$ norm of $U^n$ is controlled by $\sup_\theta |H(\theta)|^n$. If $|H(\theta)| \leq 1$ everywhere, the Fourier amplitudes cannot grow, giving stability. Conversely, if $|H(\theta_0)| > 1$ at some point, we construct initial data concentrated near $\theta_0$ whose $\ell_2$ norm grows without bound. [/proofplan] [step:Reduce the $\ell_2$ stability condition to a pointwise condition on $|H(\theta)|$ via Parseval] The numerical scheme $\sum_k b_k U_{m+k}^{n+1} = \sum_k c_k U_{m+k}^n$ acts on the discrete Fourier transform as pointwise multiplication. Substituting the ansatz $U_m^n = \hat{U}^n(\theta) e^{im\theta}$ into the scheme equation: \begin{align*} \sum_k b_k \hat{U}^{n+1}(\theta) e^{i(m+k)\theta} = \sum_k c_k \hat{U}^n(\theta) e^{i(m+k)\theta}. \end{align*} Dividing by $e^{im\theta}$: \begin{align*} \hat{U}^{n+1}(\theta) \sum_k b_k e^{ik\theta} = \hat{U}^n(\theta) \sum_k c_k e^{ik\theta}. \end{align*} Defining the amplification factor $H(\theta) := \frac{\sum_k c_k e^{ik\theta}}{\sum_k b_k e^{ik\theta}}$ (assuming $\sum_k b_k e^{ik\theta} \neq 0$): \begin{align*} \hat{U}^{n+1}(\theta) = H(\theta) \hat{U}^n(\theta). \end{align*} By induction, $\hat{U}^n(\theta) = [H(\theta)]^n \hat{U}^0(\theta)$ for all $n \geq 0$. By [Parseval's Identity](/theorems/434), $\|U^n\|_{\ell_2} = \|\hat{U}^n\|_*$ for each $n$, so stability in $\ell_2$ is equivalent to boundedness of $\|\hat{U}^n\|_*$. [guided] The discrete Fourier transform diagonalises constant-coefficient difference schemes: each Fourier mode $e^{im\theta}$ evolves independently, multiplied by the amplification factor $H(\theta)$ at each time step. After $n$ steps, the mode $\theta$ has been multiplied by $[H(\theta)]^n$. Substituting $U_m^n = \hat{U}^n(\theta) e^{im\theta}$ into $\sum_k b_k U_{m+k}^{n+1} = \sum_k c_k U_{m+k}^n$: \begin{align*} \hat{U}^{n+1}(\theta) \underbrace{\sum_k b_k e^{ik\theta}}_{=: B(\theta)} = \hat{U}^n(\theta) \underbrace{\sum_k c_k e^{ik\theta}}_{=: C(\theta)}, \end{align*} so $\hat{U}^{n+1}(\theta) = H(\theta) \hat{U}^n(\theta)$ where $H(\theta) = C(\theta)/B(\theta)$. Iterating: $\hat{U}^n(\theta) = [H(\theta)]^n \hat{U}^0(\theta)$. The bridge between the spatial domain and the Fourier domain is [Parseval's Identity](/theorems/434): $\|U^n\|_{\ell_2} = \|\hat{U}^n\|_*$. This is an isometry — the $\ell_2$ norm is preserved exactly under the Fourier transform. Therefore, bounding $\|U^n\|_{\ell_2}$ is equivalent to bounding $\|\hat{U}^n\|_*$, and the latter involves only the pointwise amplification $|H(\theta)|^n$. [/guided] [/step] [step:Prove the forward direction: $|H(\theta)| \leq 1$ implies stability] Assume $|H(\theta)| \leq 1$ for all $\theta \in [-\pi, \pi]$. Then for every $\theta$: \begin{align*} |\hat{U}^n(\theta)| = |H(\theta)|^n |\hat{U}^0(\theta)| \leq 1^n \cdot |\hat{U}^0(\theta)| = |\hat{U}^0(\theta)|. \end{align*} Squaring, multiplying by $\frac{1}{2\pi}$, and integrating over $[-\pi, \pi]$: \begin{align*} \|\hat{U}^n\|_*^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |H(\theta)|^{2n} |\hat{U}^0(\theta)|^2 \, d\mathcal{L}^1(\theta) \leq \frac{1}{2\pi} \int_{-\pi}^{\pi} |\hat{U}^0(\theta)|^2 \, d\mathcal{L}^1(\theta) = \|\hat{U}^0\|_*^2. \end{align*} By [Parseval's Identity](/theorems/434), $\|U^n\|_{\ell_2} = \|\hat{U}^n\|_* \leq \|\hat{U}^0\|_* = \|U^0\|_{\ell_2}$. The scheme is stable with constant $c = 1$. [/step] [step:Prove the converse: instability when $|H(\theta_0)| > 1$ for some $\theta_0$] Suppose there exists $\theta_0 \in [-\pi, \pi]$ with $|H(\theta_0)| > 1$. Write $|H(\theta_0)| = 1 + 2\varepsilon$ for some $\varepsilon > 0$. Since $H$ is continuous (as a ratio of trigonometric polynomials), there exists an interval $[\theta_1, \theta_2] \ni \theta_0$ with $\theta_2 - \theta_1 > 0$ such that $|H(\theta)| \geq 1 + \varepsilon$ for all $\theta \in [\theta_1, \theta_2]$. Choose the initial data in Fourier space as $\hat{U}^0(\theta) = \mathbb{1}_{[\theta_1, \theta_2]}(\theta)$. After $n$ time steps: \begin{align*} \|\hat{U}^n\|_*^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |H(\theta)|^{2n} |\hat{U}^0(\theta)|^2 \, d\mathcal{L}^1(\theta) \geq \frac{1}{2\pi} \int_{\theta_1}^{\theta_2} |H(\theta)|^{2n} \, d\mathcal{L}^1(\theta). \end{align*} Since $|H(\theta)| \geq 1 + \varepsilon$ on $[\theta_1, \theta_2]$: \begin{align*} \|\hat{U}^n\|_*^2 \geq \frac{1}{2\pi}(1 + \varepsilon)^{2n}(\theta_2 - \theta_1) \to \infty \quad \text{as } n \to \infty. \end{align*} By [Parseval's Identity](/theorems/434), $\|U^n\|_{\ell_2} = \|\hat{U}^n\|_* \to \infty$, so the scheme is unstable. [guided] The converse direction constructs explicit initial data that exposes the instability. The idea is to concentrate all the initial energy in the Fourier modes where $|H(\theta)| > 1$, so that every time step amplifies the solution. Since $H$ is a ratio of trigonometric polynomials $C(\theta)/B(\theta)$, it is a continuous function of $\theta$. If $|H(\theta_0)| = 1 + 2\varepsilon > 1$, continuity guarantees an interval $[\theta_1, \theta_2]$ around $\theta_0$ where $|H(\theta)| \geq 1 + \varepsilon$. Choosing $\hat{U}^0 = \mathbb{1}_{[\theta_1, \theta_2]}$ places all the initial energy in this amplified region. The initial $\ell_2$ norm is finite: $\|\hat{U}^0\|_*^2 = \frac{\theta_2 - \theta_1}{2\pi} > 0$. After $n$ steps: \begin{align*} \|\hat{U}^n\|_*^2 = \frac{1}{2\pi}\int_{\theta_1}^{\theta_2} |H(\theta)|^{2n} \, d\mathcal{L}^1(\theta) \geq \frac{(\theta_2 - \theta_1)}{2\pi}(1 + \varepsilon)^{2n}. \end{align*} Since $(1 + \varepsilon)^{2n} \to \infty$ exponentially, the $\ell_2$ norm of the numerical solution grows without bound. By [Parseval's Identity](/theorems/434), $\|U^n\|_{\ell_2} \to \infty$, so the scheme is unstable. Note that the initial data $\hat{U}^0 = \mathbb{1}_{[\theta_1, \theta_2]}$ corresponds (via inverse Fourier transform) to a legitimate $\ell_2$ sequence. The construction shows that instability is not an artifact of pathological initial data — it occurs for a smooth, compactly supported Fourier profile. [/guided] [/step]