Exercise 8.17

Question

Exercise 17: 
    Assume $f$ is bounded and monotonic on $[-\pi,\pi)$, with Fourier coefficients $c_n$ as given by (62).

(a) Use Exercise 17 of Chapter 6 to prove that $\{nc_n\}$ is a bounded sequence.

(b) Combine (a) with Exercise 16 and with Exercise 14(e) of Chapter 3, to conclude that $$\lim_{N\rightarrow\infty}s_N(f;x)=\frac{f(x+)+f(x-)}{2}$$ for every $x$.

(c) Assume only that $f\in\mathscr R$ on $[-\pi,\pi]$ and that $f$ is monotonic in some segment $(\alpha,\beta)\subset[-\pi,\pi]$. Prove that the conclusion of (b) holds for every
    $x\in(\alpha,\beta)$.

analambanomenos · Accepted Answer

(a) Using Exercise 6.17, with $G(x)=(i/n)e^{-inx}$,
    \begin{align*}
        nc_n &= \frac{n}{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}\,dx \
        &= \frac{i}{2\pi}\big(f(\pi)e^{-in\pi}-f(-\pi)e^{in\pi}\big)-\frac{i}{2\pi}\int_{-\pi}^\pi e^{-inx}\,df \
        |nc_n| &\le \frac{1}{2\pi}\big(f(\pi)-f(-\pi)\big)+\frac{1}{2\pi}\int_{-\pi}^\pi 1\,df \
        &= \frac{1}{\pi}\big(f(\pi)-f(-\pi)\big)
    \end{align*}

(b) Since $|nc_n|\le M=\big(f(\pi)-f(-\pi)\big)/\pi$, we can apply Exercise 3.14(e) to get $$\lim_{Nightarrow\infty}s_N(f;x)=\lim_{Nightarrow\infty}\sigma_N(f;x)=\frac{f(x+)+f(x-)}{2}.$$
    The last equality is from Exercise 16.

(c) Letting $$	ilde{f}(x)=
    \begin{cases}
        f(\alpha) & -\pi\le x\le\alpha \
        f(x) & \alpha\le x\le\beta \
        f(\beta) & \beta\le x\le\pi
    \end{cases}$$
    then $	ilde{f}$ is monotonically increasing on $[-\pi,\pi]$, so by part (b) we have $$\lim_{Nightarrow\infty}s_N(	ilde{f};x)=\frac{	ilde{f}(x+)+	ilde{f}(x-)}{2}$$ for all $-\pi\le x\le\pi$.
    Hence by the localization theorem, since $f(x)=	ilde{f}(x)$ for $\alpha<x<\beta$, we have $$\lim_{Nightarrow\infty}s_N(f;x)=\lim_{Nightarrow\infty}s_N(	ilde{f};x)=\frac{	ilde{f}(x+)+
    	ilde{f}(x-)}{2}=\frac{f(x+)+f(x-)}{2}$$ for $\alpha<x<\beta$.

Exercise 8.17

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Exercise 8.17

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