Parseval's identity

From Wikipedia, the free encyclopedia

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 the Pythagorean theorem for inner-product spaces.

Informally, the identity asserts that the sum of the squares of the Fourier coefficients of a function is equal to the integral of the square of the function,

$\sum _{{n=-\infty }}^{\infty }|c_{n}|^{2}={\frac {1}{2\pi }}\int _{{-\pi }}^{\pi }|f(x)|^{2}\,dx,$

where the Fourier coefficients c_n of ƒ are given by

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

More formally, the result holds as stated provided ƒ is square-integrable or, more generally, in L²[−π,π]. 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 ƒ ∈ L²(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 〈•,•〉. Let (e_n) 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:

$\langle e_{m},e_{n}\rangle ={\begin{cases}1&{\mbox{if}}\ m=n\\0&{\mbox{if}}\ m\not =n.\end{cases}}$

Then Parseval's identity asserts that for every x ∈ H,

$\sum _{n}|\langle x,e_{n}\rangle |^{2}=\|x\|^{2}.$

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 L²[−π,π], and setting e_n = e^−inx for n ∈ 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; i.e., 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 ≥, yielding Bessel's inequality. This general form of Parseval's identity can be proved using the Riesz–Fischer theorem.

References

Hazewinkel, Michiel, ed. (2001), "Parseval equality", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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 .

v t e Functional analysis

Local Notions on Sets	Absolutely convex Absorbing Balanced Bounded Cones (Linear, Convex cone) Convex Radial Star-shaped Symmetric

Types of TVSs	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (Tame) Hilbert LF Locally convex Mackey Montel Nuclear Normed (Norm) Reflexive Riesz Smith Stereotype Topological tensor product (of Hilbert spaces) Webbed

Spaces of Maps	Dual space Dual topology Operator topologies Strong Topology (Polar, Operator) Topology of uniform convergence Weak topology (operator) Ultrastrong topology Ultraweak topology

Types of Linear operators	Adjoint Bilinear (form) Bounded (Unbounded) Closed Continuous (Discontinuous) Compact Fredholm Hilbert–Schmidt Functionals (Positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Operations on Sets	Algebraic interior (core) Interior Minkowski addition Polar

Banach Algebras	C-algebras Spectrum (of a C-algebra, Spectral radius) Spectral theory

Important Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (Hyperplane separation) Kakutani fixed-point Lomonosov's Invariant Subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Derivatives (Fréchet derivative, Gâteaux derivative, Functional, Holomorphy) Differentiation in Fréchet spaces Integrals (Bochner, Dunford, Gelfand–Pettis, Regulated, Weak) Functional calculus Inverse function theorem (Nash–Moser theorem) Vector measure

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.

Parseval's identity

Generalization of the Pythagorean theorem

See also

References