= Solution
For the <method of lines> in part (a), the <Fourier symbol> of the spatial operator is
$$
\lambda_h(\theta)=h^{-2}\left(-\frac52+\frac83\cos\theta-\frac16\cos2\theta\right)=-\frac4{h^2}\sin^2\frac\theta2\left(1+\frac13\sin^2\frac\theta2\right)\le0.
$$
Each <Fourier mode> therefore evolves by $\widehat U(\theta,t)=e^{t\lambda_h(\theta)}\widehat U(\theta,0)$, whose multiplier has <modulus> at most one. The <discrete Parseval identity> gives
$$
\boxed{\|U(t)\|_{\ell_h^2}\le\|U(0)\|_{\ell_h^2}},\qquad\|U\|_{\ell_h^2}^2=h\sum_{m\in\mathbb Z}|U_m|^2.
$$
Thus the <fourth-order centered second derivative> gives an \b[unconditionally stable semidiscrete evolution in the <L2 norm>]. No time integration method has yet been chosen, so no time-step restriction belongs to this conclusion.
An arbitrary <L2 function> does not have well-defined point samples. For the stated initial data one may use <cell-average projection>, $U_m(0)=h^{-1}\int_{x_m-h/2}^{x_m+h/2}u_0(x)\,dx$. The <Cauchy-Schwarz inequality> gives $h\sum_m|U_m(0)|^2\le\|u_0\|_{L^2}^2$, providing a bounded initialization for the stability estimate.
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