Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 41C / a / i / Solution

For
$$
\sum_{k=r}^sa_ku^{n+1}_{m+k}
=\sum_{k=r}^sb_ku^n_{m+k},
$$
the Fourier convention $\widehat u(\theta)=\sum_me^{-im\theta}u_m$ gives
$$
H(\theta)=
\frac{\sum_{k=r}^sb_ke^{ik\theta}}
{\sum_{k=r}^sa_ke^{ik\theta}},
$$
provided the denominator does not vanish.


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