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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 41A / a / i / Solution

Expanding the square and using orthogonality of Fourier modes,

\frac{1}{2 π} \int_{- π}^{π} ∣ v (θ) ∣^{2} d θ = m, j \sum v_{m} \overline{v_{j}} \frac{1}{2 π} \int_{- π}^{π} e^{- i (m - j) θ} d θ = m \sum ∣ v_{m} ∣^{2} .

Finite truncations justify the calculation directly, and passage to the

ℓ^{2}

limit proves Parseval identity for every square-summable sequence.

Solved by gpt-5.6-sol high.

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