Parseval's identity

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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,

where the Fourier coefficients of are given by

More formally, the result holds as stated provided is a square-integrable function or, more generally, in Lp space 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

Generalization of the Pythagorean theorem[edit]

The identity is related to the Pythagorean theorem in the more general setting of a separable Hilbert space as follows. Suppose that is a Hilbert space with inner product Let be an orthonormal basis of ; i.e., the linear span of the is dense in and the are mutually orthonormal:

Then Parseval's identity asserts that for every

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 be the Hilbert space and setting for

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

The assumption that is total is necessary for the validity of the identity. If 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.

See also[edit]

References[edit]

  • "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.
  • Siktar, Joshua (2019), Recasting the Proof of Parseval's Identity, Turkish Journal of Inequalities, arXiv:1907.08331. [1]