Paper 1, Section II, A

(a) A function $f(t)$ is periodic with period $2 \pi$ and has continuous derivatives up to and including the $k$ th derivative. Show by integrating by parts that the Fourier coefficients of $f(t)$

$\begin{aligned} &a_{n}=\frac{1}{\pi} \int_{0}^{2 \pi} f(t) \cos n t d t \\ &b_{n}=\frac{1}{\pi} \int_{0}^{2 \pi} f(t) \sin n t d t \end{aligned}$

decay at least as fast as $1 / n^{k}$ as $n \rightarrow \infty$

(b) Calculate the Fourier series of $f(t)=|\sin t|$ on $[0,2 \pi]$.

(c) Comment on the decay rate of your Fourier series.

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