Ex. extra 2

Seja $f : \mathbb R \to \mathbb R$ localmente integrável e periódica de período $2 \pi$. Seja

(1)

\begin{align} s_N(x) = \frac{a_0}{2} + \sum_{n=1}^N (a_n \cos(nx) + b_n \sin(nx)), \quad x \in \mathbb R, \quad N \in \mathbb N, \end{align}

a soma parcial de ordem $N$ da série de Fourier de $f$. Numa tentativa de relacionar $s_N$ com $f$, mostre que, para cada $x \in \mathbb R$ e $N \in \mathbb N$,

(2)

\begin{eqnarray} s_N(x) & = & \frac{1}{\pi} \int_{-\pi}^\pi f(t)\, \Big[ \frac{1}{2} + \sum_{n=1}^N (\cos(nt) \cos(nx) + \sin(nt) \sin(nx)) \Big] \, dt \\ \\ & = & \frac{1}{\pi} \int_{-\pi}^\pi f(t)\, \Big[ \frac{1}{2} + \sum_{n=1}^N \cos(n(t-x)) \Big] \, dt \\ \\ & = & \frac{1}{\pi} \int_{-\pi-x}^{\pi-x} f(x+u)\, \Big[ \frac{1}{2} + \sum_{n=1}^N \cos(nu) \Big] \, du \\ \\ & = & \frac{1}{\pi} \int_{-\pi}^{\pi} \frac{f(x+u)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du \\ \\ & = & \frac{1}{\pi} \int_{0}^{\pi} \frac{f(x+u)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du \; + \; \frac{1}{\pi} \int_{0}^{\pi} \frac{f(x-u)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du. \end{eqnarray}

Depois mostre que, para cada $x \in \mathbb R$ e $N \in \mathbb N$ e no caso de existirem e serem finitos os limites laterais $f(x+0)$ e $f(x-0)$,

(3)

\begin{eqnarray} 0 & \leq & \Big| s_N(x) - \frac{f(x+0)+f(x-0)}{2} \Big| \\ \\ & = & \Big| \frac{1}{\pi} \int_{0}^{\pi} \frac{f(x+u)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du \; - \; \frac{f(x+0)}{2} \; + \; \frac{1}{\pi} \int_{0}^{\pi} \frac{f(x-u)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du \; - \; \frac{f(x-0)}{2} \Big| \\ \\ & = & \Big| \frac{1}{\pi} \int_{0}^{\pi} \frac{f(x+u)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du \; - \; \frac{1}{\pi} \int_{0}^{\pi} \frac{f(x+0)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du \\ \\ & & +\;\; \frac{1}{\pi} \int_{0}^{\pi} \frac{f(x-u)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du \; - \; \frac{1}{\pi} \int_{0}^{\pi} \frac{f(x-0)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du \Big| \\ \\ & \leq & \Big| \frac{1}{\pi} \int_{0}^{\pi} \frac{f(x+u)-f(x+0)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du \Big| \\ \\ & & + \;\; \Big| \frac{1}{\pi} \int_{0}^{\pi} \frac{f(x-u)-f(x-0)}{2 \sin \frac{u}{2}} \sin \big[\big(N+\frac{1}{2}\big)u\big] \, du \Big|. \end{eqnarray}