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. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which can have an uncountable infinity of basis vectors).

Informally, 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,

\Vert f\Vert _{L^{2}(-\pi ,\pi )}^{2}=\int _{-\pi }^{\pi }|f(x)|^{2}\,dx=2\pi \sum _{n=-\infty }^{\infty }|c_{n}|^{2}

where the Fourier coefficients $c_{n}$ of $f$ are given by

c_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx.

More formally, the result holds as stated provided $f$ is a square-integrable function or, more generally, in Lp space $L^{2}[-\pi ,\pi ].$ 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 $f\in L^{2}(\mathbb {R} ),$

\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 $H$ is a Hilbert space with inner product $\langle \,\cdot \,,\,\cdot \,\rangle .$ Let $\left(e_{n}\right)$ be an orthonormal basis of $H$ ; i.e., the linear span of the $e_{n}$ is dense in $H,$ and the $e_{n}$ are mutually orthonormal:

\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 $x\in H,$

\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 $H$ be the Hilbert space $L^{2}[-\pi ,\pi ],$ and setting $e_{n}={\frac {e^{-inx}}{\sqrt {2\pi }}}$ for $n\in \mathbb {Z} .$

More generally, Parseval's identity holds in any inner product space, not just separable Hilbert spaces. Thus suppose that $H$ is an inner-product space. Let $B$ be an orthonormal basis of $H$ ; that is, an orthonormal set which is total in the sense that the linear span of $B$ is dense in $H.$ Then

\|x\|^{2}=\langle x,x\rangle =\sum _{v\in B}\left|\langle x,v\rangle \right|^{2}.

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

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.

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Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

Parseval's identity

Generalization of the Pythagorean theorem

See also

References