= Solution
{c}
Transforming the spatial shifts and putting $\xi=k\Delta x$ gives the real <amplification factor>
$$
\widehat U^{m+1}=G(\xi)\widehat U^m,\qquad
G(\xi)=\frac{1-\mu(1-\theta)\sin^2\xi}{1+4\mu\theta\sin^2(\xi/2)}.
$$
For $0\leq\theta\leq1$ the denominator is at least one, so the implicit step is a bounded <Fourier multiplier> on $\ell^2$. Set $v=\sin^2(\xi/2)\in[0,1]$ and use $\sin^2\xi=4v(1-v)$. The numerator is no greater than the denominator, so $G\leq1$. The other inequality is
$$
G\geq-1\iff
4\mu v\bigl[(1-2\theta)-(1-\theta)v\bigr]\leq2.
$$
If $\theta\geq1/2$, the bracket is nonpositive and this always holds. If $0\leq\theta<1/2$, the concave quadratic has its maximum at $v=(1-2\theta)/[2(1-\theta)]$, with value $(1-2\theta)^2/[4(1-\theta)]$. Therefore $|G|\leq1$ precisely under the stated condition
$$
\boxed{\mu(1-2\theta)^2\leq2(1-\theta)\quad(0\leq\theta<1/2),}
$$
and without a mesh-ratio restriction when $1/2\leq\theta\leq1$. The <Parseval identity> now gives
$$
\|U^{m+1}\|_{\ell^2}^2=\frac1{2\pi}\int|G|^2|\widehat U^m|^2dk
\leq\|U^m\|_{\ell^2}^2.
$$
Iteration proves stability with constant one. It is \b[conditional for $\theta<1/2$ and unconditional for $\theta\geq1/2$]. The sampled initial data have finite support, hence belong to $\ell^2$. This is the <Fourier stability threshold for a mixed-width heat scheme>.
\Image[/past-exam-of-the-numerical-solution-of-differential-equations-course-of-the-university-of-oxford/2010-heat-stability.png]
{title=The exact stable mesh-ratio region for the mixed-width heat scheme}
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