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.

Plancheral theorem wiki link

Both look similar to me. What exactly is the difference between the two?


1 Answer 1


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)$)


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