Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 27H / c / Solution

Regard the same triangular functions as cutoffs in the frequency variable. They satisfySince $\hat{f}\ge 0$, monotone convergence givesFubini's theorem and evenness of ${\theta }_n$ givePart (b) shows that ${\hat{\theta }}_n\ge 0$. Moreover, Fourier inversion at zero yieldsConsequentlyPassing to the limit provesThus one may take the universal constant $\alpha =2\pi$.

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