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. The identity asserts the equality of the energy of a periodic signal (given as the integral of the squared amplitude of the signal) and the energy of its frequency domain representation (given as the sum of squares of the amplitudes). Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which can have an uncountable infinity of basis vectors). 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,
‖ f ‖ L 2 ( − π , π ) 2 = 1 2 π ∫ − π π | f ( x ) | 2 d x = ∑ n = − ∞ ∞ | f ^ ( n ) | 2 , {\displaystyle \Vert f\Vert _{L^{2}(-\pi ,\pi )}^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|f(x)|^{2}\,dx=\sum _{n=-\infty }^{\infty }|{\hat {f}}(n)|^{2},}
where the Fourier coefficients f ^ ( n ) {\displaystyle {\hat {f}}(n)} of f {\displaystyle f} are given by
f ^ ( n ) = 1 2 π ∫ − π π f ( x ) e − i n x d x . {\displaystyle {\hat {f}}(n)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx.}
The result holds as stated, provided f {\displaystyle f} is a square-integrable function or, more generally, in Lp space L 2 [ − π , π ] . {\displaystyle 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 ∈ L 2 ( R ) , {\displaystyle f\in L^{2}(\mathbb {R} ),}
∫ − ∞ ∞ | f ^ ( ξ ) | 2 d ξ = ∫ − ∞ ∞ | f ( x ) | 2 d x . {\displaystyle \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 {\displaystyle H} is a Hilbert space with inner product ⟨ ⋅ , ⋅ ⟩ . {\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle .} Let ( e n ) {\displaystyle \left(e_{n}\right)} be an orthonormal basis of H {\displaystyle H} ; i.e., the linear span of the e n {\displaystyle e_{n}} is dense in H , {\displaystyle H,} and the e n {\displaystyle e_{n}} are mutually orthonormal:
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