Parseval's identity

In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. The identity asserts the equality of the energy of a periodic signal (given as the integral of the squared amplitude of the signal) and the energy of its frequency domain representation (given as the sum of squares of the amplitudes). Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which can have an uncountable infinity of basis vectors).

The identity asserts that the sum of squares of the Fourier coefficients of a function is equal to the integral of the square of the function, $${\displaystyle \Vert f\Vert _{L^{2}(-\pi ,\pi )}^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|f(x)|^{2}\,dx=\sum _{n=-\infty }^{\infty }|{\hat {f}}(n)|^{2},}$$ where the Fourier coefficients ${\displaystyle {\hat {f}}(n)}$ of ${\displaystyle f}$ are given by $${\displaystyle {\hat {f}}(n)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx.}$$

The result holds as stated, provided ${\displaystyle f}$ is a square-integrable function or, more generally, in L^p space ${\displaystyle L^{2}[-\pi ,\pi ].}$ A similar result is the Plancherel theorem, which asserts that the integral of the square of the Fourier transform of a function is equal to the integral of the square of the function itself. In one-dimension, for ${\displaystyle f\in L^{2}(\mathbb {R} ),}$$${\displaystyle \int _{-\infty }^{\infty }|{\hat {f}}(\xi )|^{2}\,d\xi =\int _{-\infty }^{\infty }|f(x)|^{2}\,dx.}$$

Generalization of the Pythagorean theorem

The identity is related to the Pythagorean theorem in the more general setting of a separable Hilbert space as follows. Suppose that ${\displaystyle H}$ is a Hilbert space with inner product ${\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle .}$ Let ${\displaystyle \left(e_{n}\right)}$ be an orthonormal basis of ${\displaystyle H}$; i.e., the linear span of the ${\displaystyle e_{n}}$ is dense in ${\displaystyle H,}$ and the ${\displaystyle e_{n}}$ are mutually orthonormal:

${\displaystyle \langle e_{m},e_{n}\rangle ={\begin{cases}1&{\mbox{if}}~m=n\\0&{\mbox{if}}~m\neq n.\end{cases}}}$

Then Parseval's identity asserts that for every ${\displaystyle x\in H,}$$${\displaystyle \sum _{n}\left|\left\langle x,e_{n}\right\rangle \right|^{2}=\|x\|^{2}.}$$

This is directly analogous to the Pythagorean theorem, which asserts that the sum of the squares of the components of a vector in an orthonormal basis is equal to the squared length of the vector. One can recover the Fourier series version of Parseval's identity by letting ${\displaystyle H}$ be the Hilbert space ${\displaystyle L^{2}[-\pi ,\pi ],}$ and setting ${\displaystyle e_{n}=e^{inx}}$ for ${\displaystyle n\in \mathbb {Z} .}$

More generally, Parseval's identity holds in any inner product space, not just separable Hilbert spaces. Thus suppose that ${\displaystyle H}$ is an inner-product space. Let ${\displaystyle B}$ be an orthonormal basis of ${\displaystyle H}$; that is, an orthonormal set which is total in the sense that the linear span of ${\displaystyle B}$ is dense in ${\displaystyle H.}$ Then $${\displaystyle \|x\|^{2}=\langle x,x\rangle =\sum _{v\in B}\left|\langle x,v\rangle \right|^{2}.}$$

The assumption that ${\displaystyle B}$ is total is necessary for the validity of the identity. If ${\displaystyle B}$ is not total, then the equality in Parseval's identity must be replaced by ${\displaystyle \,\geq ,}$ yielding Bessel's inequality. This general form of Parseval's identity can be proved using the Riesz–Fischer theorem.

The statement also holds for arbitrary Hilbert spaces, not necessarily separable. When the Hilbert space is not separable any orthonormal basis is uncountable and we need to generalize the concept of a series to an unconditional sum as follows: let ${\displaystyle \{e_{r}\}_{r\in \Gamma }}$ an orthonormal basis of a Hilbert space (where ${\displaystyle \Gamma }$ have arbitrary cardinality), then we says that ${\textstyle \sum _{r\in \Gamma }a_{r}e_{r}}$ converges unconditionally if for every ${\displaystyle \epsilon >0}$ there exists a finite subset ${\displaystyle A\subset \Gamma }$ such that $${\displaystyle \left\|\sum _{r\in B}a_{r}e_{r}-\sum _{r\in C}a_{r}e_{r}\right\|<\epsilon }$$ for any pair of finite subsets ${\displaystyle B,C\subset \Gamma }$ that contains ${\displaystyle A}$ (that is, such that ${\displaystyle A\subset B\cap C}$). Note that in this case we are using a net to define the unconditional sum.

See also

• Parseval's theorem  – Theorem in mathematics

References

• "Parseval equality", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• Johnson, Lee W.; Riess, R. Dean (1982), Numerical Analysis (2nd ed.), Reading, Mass.: Addison-Wesley, ISBN   0-201-10392-3 .
• Titchmarsh, E (1939), The Theory of Functions (2nd ed.), Oxford University Press.
• Zygmund, Antoni (1968), Trigonometric Series (2nd ed.), Cambridge University Press (published 1988), ISBN   978-0-521-35885-9 .