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Wikipedia gives Parseval's theorem as follows. Parseval's Theorem
,
is the Fourier transform of x(t).
And Plancheral's theorem is given as
,
where is the fourier analog.
Both look similar to me. What exactly is the difference between the two?
According to the links you shared, and using Wikipedia's notation, Parseval assumes $A(x),B(x)$ are two square integrable (with respect to the Lebesgue measure), complex-valued functions on $\mathbb {R}$ of period $ 2\pi $ with Fourier series
$$A(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$$ and $$B(x)=\sum_{n=-\infty}^\infty b_ne^{inx}$$ Then $$\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{2\pi} \int_{-\pi}^\pi A(x)\overline{B(x)} \, \mathrm{d}x$$ Plancherel provides a continuous version of Parsevals theorem (in a sense that $a_n$ and $b_n$ are now the $f(x)$ and $g(x)$, respectively and $A(x),B(x)$ are replaced by $ \widehat{f}(\xi), \widehat{g}(\xi)$)
