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Parseval's identity - Wikipedia, the free encyclopedia

Parseval's identity

From Wikipedia, the free encyclopedia

In functional analysis, a branch of mathematics, Parseval's identity, also known as Parseval's equality, is the Pythagorean theorem for inner-product spaces. Let \left(H,\langle\cdot,\cdot\rangle\right) be a complete inner-product space (i.e., a Hilbert space) and B\subset H be an orthonormal basis in H (meaning that, for all x\in B and y\in B, \langle x,y\rangle=1 if x = y and \langle x,y\rangle=0 if x\ne y) which is total in it (meaning that \overline{span(B)}=H))). Then,

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

The totality assumption on B is crucial for the validity of the identity.If B is not total, then the equality in Parseval's identity must be replaced by \le, thus yielding Bessel's inequality. The origin of the name is in Parseval's theorem for Fourier series, which is a special case.

Parseval's identity can be proved using the Riesz-Fischer theorem.

[edit] See also

[edit] References

  • Johnson, Lee W.; and R. Dean Riess (1982). Numerical Analysis, 2nd ed., Reading, Mass.: Addison-Wesley. ISBN 0-201-10392-3. 

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