Chapter 2

Let \(f \in E\) and $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ denote the Fourier series of \(f\). Prove that there exist \(\left\\{A_{n}\right\\}_{n=0}^{\infty}\) and \(\left\\{\alpha_{n}\right\\}_{n=0}^{\infty}\), where \(-\frac{\pi}{2}<\alpha_{n} \leq \frac{\pi}{2}\), such that $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right]=A_{0}+\sum_{n=1}^{\infty} A_{n} \cos \left(n x-\alpha_{n}\right) . $$ In a similar way, prove that there exist \(\left\\{B_{n}\right\\}_{n=0}^{\infty}\) and \(\left\\{\beta_{n}\right\\}_{n=1}^{\infty}\), where \(-\frac{\pi}{2}<\) \(\beta_{n} \leq \frac{\pi}{2}\), such that $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right]=B_{0}+\sum_{n=1}^{\infty} B_{n} \sin \left(n x+\beta_{n}\right) . $$

Let \(f \in E[-\pi, \pi]\) and $$ f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ be the Fourier series of \(f\). Assume that \(f\) is \(\frac{\pi}{m}\)-periodic, for some \(m \in \mathbb{N}\). Prove that \(a_{n}=b_{n}=0\) for every \(n\) which is not divisible by \(2 m\).

For each natural number \(m\), let \(D_{m}(t)=\frac{1}{2}+\sum_{n=1}^{m} \cos n t\). (a) Calculate \(\int_{-\pi}^{\pi} D_{m}(t) \sin 100 t d t\). (b) Determine \(\frac{1}{\pi} \int_{-\pi}^{\pi}\left[D_{m}(t)\right]^{2} d t\), for \(m=100\). (c) Let \(g(t)= \begin{cases}\frac{\sin \frac{1}{2} t}{t}, & t \neq 0, \\\ \frac{1}{2}, & t=0 .\end{cases}\) Calculate \(\lim _{m \rightarrow \infty} \frac{1}{\pi} \int_{-\pi}^{\pi} D_{m}(t) g(t) d t\).

Assume that \(f\) is continuous on \([-\pi, \pi]\) and \(f(-\pi)=f(\pi)\). Prove that \(f\) can be uniformly approximated by trigonometric polynomials. That is, given \(\epsilon>0\) there exists a trigonometric polynomial \(T\) of some degree such that $$ |f(x)-T(x)|<\epsilon $$ for all \(x \in[-\pi, \pi]\).

Let \(f\) be \(a \pi\)-periodic function for which $$ f(x)= \begin{cases}\sin 2 x, & 0 \leq x \leq \frac{\pi}{2} \\ 0, & \frac{\pi}{2} \leq x \leq \pi\end{cases} $$ (a) Prove that for all \(x \in \mathbb{R}\) $$ f(x)=\frac{1}{\pi}+\frac{1}{2} \sin 2 x-\frac{2}{\pi} \sum_{n=1}^{\infty} \frac{\cos 4 n x}{(2 n-1)(2 n+1)} $$ and then prove that $$ \sum_{n=1}^{\infty} \frac{1}{(2 n-1)^{2}(2 n+1)^{2}}=\frac{\pi^{2}-8}{16} $$ (b) Determine the value of the sum $$ \frac{\sin 4 x}{1 \cdot 2 \cdot 3}+\frac{\sin 8 x}{3 \cdot 4 \cdot 5}+\frac{\sin 12 x}{5 \cdot 6 \cdot 7}+\cdots, \quad 0 \leq x \leq \pi $$

Let $$ f(x)= \begin{cases}2+\frac{2 x}{\pi}, & -\pi

Prove that if \(g\) is a piecewise continuous \(2 \pi\)-periodic function on \(\mathbb{R}\), then for every real \(a\) $$ \int_{-\pi+a}^{\pi+a} g(t) d t=\int_{-\pi}^{\pi} g(t) d t . $$

(a) Determine the Fourier series of the function $$ f(x)= \begin{cases}0, & -\pi

We define the function $$ f(x)= \begin{cases}\sin 2 x, & -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \\ 0, & \text { otherwise }\end{cases} $$ on the interval \([-\pi, \pi]\). (a) Determine the Fourier series of \(f\) on \([-\pi, \pi]\). (b) Determine the Fourier series of \(f^{\prime}\) on \([-\pi, \pi]\). (c) To what values does the Fourier series of \(f^{\prime}\) converge at the points \(x=\pm \frac{\pi}{2} ?\) (d) Calculate the sums $$ \sum_{k=1}^{\infty} \frac{1}{(2 k-3)^{2}(2 k+1)^{2}}, \quad \sum_{k=1}^{\infty} \frac{(2 k-1)(-1)^{k}}{(2 k-3)(2 k+1)} $$

Let \(f\) be a \(2 \pi\)-periodic piecewise continuous function and $$ f(x) \sim \sum_{n=1}^{\infty}\left[a_{n} \cos n x+b_{n} \sin n x\right] $$ its Fourier series on \([-\pi, \pi]\). Set $$ g(x)=\int_{-\pi}^{x}[f(t)+f(\pi-t)] d t $$ and let $$ g(x) \sim \frac{A_{0}}{2}+\sum_{n=1}^{\infty}\left[A_{n} \cos n x+B_{n} \sin n x\right] $$ be the Fourier series of \(g\) on \([-\pi, \pi]\). Express the \(A_{n}\) and \(B_{n}\) in terms of \(a_{n}\) and \(b_{n}\).