Ex. extra 1

Seja $f : [-\pi,\pi] \to \mathbb R$ integrável e seja, para cada $N \in \mathbb N$,

(1)

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

a soma parcial de ordem $N$ da série de Fourier de $f$. Mostre, sucessivamente, que

(2)

\begin{align} \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) s_N(x) \, dx = \frac{1}{2}a_0^2 + \sum_{n=1}^N (a_n^2 + b_n^2), \end{align}

(3)

\begin{align} \frac{1}{\pi} \int_{-\pi}^{\pi} [s_N(x)]^2 \, dx = \frac{1}{2}a_0^2 + \sum_{n=1}^N (a_n^2 + b_n^2), \end{align}

(4)

\begin{align} \frac{1}{\pi} \int_{-\pi}^{\pi} [f(x) - s_N(x)]^2 \, dx = \frac{1}{\pi} \int_{-\pi}^{\pi} [f(x)]^2 \, dx - \Big( \frac{1}{2}a_0^2 + \sum_{n=1}^N (a_n^2 + b_n^2) \Big) \end{align}

e, finalmente, a desigualdade de Bessel:

(5)

\begin{align} \frac{1}{2}a_0^2 + \sum_{n=1}^\infty (a_n^2 + b_n^2) \leq \frac{1}{\pi} \int_{-\pi}^{\pi} [f(x)]^2 \, dx. \end{align}