Parseval's identity | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-06

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Parseval%27s_identity


00:03:45 Generalization of the Pythagorean theorem
00:06:39 See also



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SUMMARY
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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 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,

  
    
      
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    {\displaystyle \Vert f\Vert _{L_{p}^{2}(-\pi ,\pi )}^{2}=\int _{-\pi }^{\pi }|f(x)|^{2}\,dx=2\pi \sum _{n=-\infty }^{\infty }|c_{n}|^{2}}
  where the Fourier coefficients cn of ƒ are given by

  
    
      
        
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    {\displaystyle c_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)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),

  
    
      
        
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    {\displaystyle \int _{-\infty }^{\infty }|{\hat {f}}(\xi )|^{2}\,d\xi =\int _{-\infty }^{\infty }|f(x)|^{2}\,dx.}