Suppose that ${\displaystyle A(x)}$  and ${\displaystyle B(x)}$  are two complex-valued functions on ${\displaystyle \mathbb {R} }$  of period ${\displaystyle 2\pi }$  that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series

${\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{inx}}$ 

and

${\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{inx}}$ 

respectively. Then

where ${\displaystyle i}$  is the imaginary unit and horizontal bars indicate complex conjugation. Substituting ${\displaystyle A(x)}$  and ${\displaystyle {\overline {B(x)}}}$ :

${\displaystyle {\begin{aligned}\sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(\sum _{n=-\infty }^{\infty }a_{n}e^{inx}\right)\left(\sum _{n=-\infty }^{\infty }{\overline {b_{n}}}e^{-inx}\right)\,\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}e^{i1x}+a_{2}e^{i2x}+\cdots \right)\left({\overline {b_{1}}}e^{-i1x}+{\overline {b_{2}}}e^{-i2x}+\cdots \right)\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}e^{i1x}{\overline {b_{1}}}e^{-i1x}+a_{1}e^{i1x}{\overline {b_{2}}}e^{-i2x}+a_{2}e^{i2x}{\overline {b_{1}}}e^{-i1x}+a_{2}e^{i2x}{\overline {b_{2}}}e^{-i2x}+\cdots \right)\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}{\overline {b_{1}}}+a_{1}{\overline {b_{2}}}e^{-ix}+a_{2}{\overline {b_{1}}}e^{ix}+a_{2}{\overline {b_{2}}}+\cdots \right)\mathrm {d} x\end{aligned}}}$ 

As is the case with the middle terms in this example, many terms will integrate to ${\displaystyle 0}$  over a full period of length ${\displaystyle 2\pi }$  (see harmonics):

${\displaystyle {\begin{aligned}\sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}&={\frac {1}{2\pi }}\left[a_{1}{\overline {b_{1}}}x+ia_{1}{\overline {b_{2}}}e^{-ix}-ia_{2}{\overline {b_{1}}}e^{ix}+a_{2}{\overline {b_{2}}}x+\cdots \right]_{-\pi }^{+\pi }\\[6pt]&={\frac {1}{2\pi }}\left(2\pi a_{1}{\overline {b_{1}}}+0+0+2\pi a_{2}{\overline {b_{2}}}+\cdots \right)\\[6pt]&=a_{1}{\overline {b_{1}}}+a_{2}{\overline {b_{2}}}+\cdots \\[6pt]\end{aligned}}}$ 

More generally, if ${\displaystyle A(x)}$  and ${\displaystyle B(x)}$  are instead two complex-valued functions on ${\displaystyle \mathbb {R} }$  of period ${\displaystyle P}$  that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series

${\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{2\pi ni\left({\frac {x}{P}}\right)}}$ 

and

${\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{2\pi ni\left({\frac {x}{P}}\right)}}$ 

respectively. Then

Even more generally, given an abelian locally compact 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 ${\displaystyle \mathbb {R} }$ , G^ is also ${\displaystyle \mathbb {R} }$  and the unitary transform is the Fourier transform on the real line. When G is the cyclic group Z_n, again it is self-dual and the Pontryagin–Fourier transform is what is called discrete Fourier transform in applied contexts.

Parseval's theorem can also be expressed as follows: Suppose ${\displaystyle f(x)}$  is a square-integrable function over ${\displaystyle [-\pi ,\pi ]}$  (i.e., ${\displaystyle f(x)}$  and ${\displaystyle f^{2}(x)}$  are integrable on that interval), with the Fourier series

${\displaystyle f(x)\simeq {\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }(a_{n}\cos(nx)+b_{n}\sin(nx)).}$ 

Then^[4][5][6]

${\displaystyle {\frac {1}{\pi }}\int _{-\pi }^{\pi }f^{2}(x)\,\mathrm {d} x={\frac {a_{0}^{2}}{2}}+\sum _{n=1}^{\infty }\left(a_{n}^{2}+b_{n}^{2}\right).}$ 

In electrical engineering, Parseval's theorem is often written as:

${\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,\mathrm {d} t={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|X(\omega )|^{2}\,\mathrm {d} \omega =\int _{-\infty }^{\infty }|X(2\pi f)|^{2}\,\mathrm {d} f}$ 

where ${\displaystyle X(\omega )={\mathcal {F}}_{\omega }\{x(t)\}}$  represents the continuous Fourier transform (in normalized, unitary form) of ${\displaystyle x(t)}$ , and ${\displaystyle \omega =2\pi f}$  is frequency in radians per second.

The interpretation of this form of the theorem is that the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.

For discrete time signals, the theorem becomes:

${\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|X_{2\pi }({\phi })|^{2}\mathrm {d} \phi }$ 

where ${\displaystyle X_{2\pi }}$  is the discrete-time Fourier transform (DTFT) of ${\displaystyle x}$  and ${\displaystyle \phi }$  represents the angular frequency (in radians per sample) of ${\displaystyle x}$ .

Alternatively, for the discrete Fourier transform (DFT), the relation becomes:

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

where ${\displaystyle X[k]}$  is the DFT of ${\displaystyle x[n]}$ , both of length ${\displaystyle N}$ .

We show the DFT case below. For the other cases, the proof is similar. By using the definition of inverse DFT of ${\displaystyle X[k]}$ , we can derive

${\displaystyle {\begin{aligned}{\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}&={\frac {1}{N}}\sum _{k=0}^{N-1}X[k]\cdot X^{*}[k]={\frac {1}{N}}\sum _{k=0}^{N-1}\left[\sum _{n=0}^{N-1}x[n]\,\exp \left(-j{\frac {2\pi }{N}}k\,n\right)\right]\,X^{*}[k]\\[5mu]&={\frac {1}{N}}\sum _{n=0}^{N-1}x[n]\left[\sum _{k=0}^{N-1}X^{*}[k]\,\exp \left(-j{\frac {2\pi }{N}}k\,n\right)\right]={\frac {1}{N}}\sum _{n=0}^{N-1}x[n](N\cdot x^{*}[n])\\[5mu]&=\sum _{n=0}^{N-1}|x[n]|^{2},\end{aligned}}}$ 

where ${\displaystyle *}$  represents complex conjugate.

Parseval's theorem is closely related to other mathematical results involving unitary transformations:

• Parseval's identity
• Plancherel's theorem
• Wiener–Khinchin theorem
• Bessel's inequality

• Parseval's Theorem on Mathworld