Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 41A / a / Solution

For a two-periodic Fourier series sampled at $2N$ equally spaced points, discrete averaging retains exactly the modes divisible by $2N$:
$$
I_N(h)-I(h)=\sum_{j\ne0}\widehat h_{2Nj}.
$$
If $|\widehat h_n|\leq Mc^{|n|}$ with $0<c<1$, then
$$
|I_N-I|\leq\frac{2Mc^{2N}}{1-c^{2N}},
$$
which is exponentially small.

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 Incoming links: Periodic trapezoidal Fourier aliasing