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--- simpsons [2021/06/24 11:49] – tahzibi
+++ simpsons [Unknown date] (current) – removed - external edit (Unknown date) 127.0.0.1
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-Simpsons e Fourier:
-{{youtube>3c2xgVn6KCc?small}}
-Teorema de Parseval:
-{{youtube>9kjiIYJPU0o?small}}
-Sejam $f, g : [-\pi, \pi] \rightarrow \mathbb{R}$ funções Rieman integráveis e $f(x) \sim \sum_{-\infty}^{\infty} c_n e^{inx}, g(x) \sim \sum_{-\infty}^{\infty} \gamma_n e^{inx} $ respectivas séries de Fourier. Então:
-  - $\lim_{N \rightarrow \infty} \frac{1}{2\pi} \int_{-\pi}^{\pi} |f(x)- S_N(f, x)|^2 dx = 0.$
-  - $\frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \overline{g}(x) dx = \sum_{-\infty}^{\infty} c_n \overline{\gamma}_n$
-  - $\frac{1}{2\pi} \int_{-\pi}^{\pi} |f(x)|^2  dx = \sum_{-\infty}^{\infty} |c_n|^2.$
-Observe que a terceira afirmação é consequeência imediata da segunda, assumindo $f=g.$