# 帕塞瓦尔定理

## 帕塞瓦尔定理的陈述

$A(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$

$B(x)=\sum_{n=-\infty}^\infty b_ne^{inx}.$

$\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{2\pi} \int_{-\pi}^\pi A(x)\overline{B(x)} \, dx,$

More generally, given an abelian topological group G with Pontryagin dual G^, Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces L2(G) and L2(G^) (with integration being against the appropriately scaled Haar measures on the two groups.) When G is the unit circle T, G^ is the integers and this is the case discussed above. When G is the real line R, G^ is also R and the unitary transform is the Fourier transform on the real line. When G is the cyclic group Zn, again it is self-dual and the Pontryagin–Fourier transform is what is called discrete-time Fourier transform in applied contexts.

## 物理学和工程学上使用的记号

$\int_{-\infty}^\infty | x(t) |^2 \, dt = \int_{-\infty}^\infty | X(f) |^2 \, df$

$\sum_{n=-\infty}^\infty | x[n] |^2 = \frac{1}{2\pi} \int_{-\pi}^\pi | X(e^{i\phi}) |^2 d\phi$

$\sum_{n=0}^{N-1} | x[n] |^2 = \frac{1}{N} \sum_{k=0}^{N-1} | X[k] |^2$