Parseval's identity

Parseval's identity equates a function's squared norm with the sum or integral of the squared magnitudes of its Fourier coefficients or transform. It expresses preservation of energy under Fourier representation.

Fourier series: A representation of a periodic function as a sum of sine and cosine waves, or complex exponentials. For periodic functions, the identity equates the integral of squared magnitude with the sum of squared Fourier coefficients.
Fourier coefficient: A scalar measuring a function's component along a particular frequency in a Fourier expansion. The identity adds the squared magnitudes of these coefficients to recover total energy.
Signal energy: The total squared magnitude of a signal, summed or integrated over its domain. Fourier-domain energy can be computed from coefficients instead of directly from the signal.
Marc-Antoine Parseval: A French mathematician whose name is attached to an equality for sums of squared Fourier coefficients. The identity is named for Parseval, who presented an early form in work on trigonometric series.
Fourier transform: An integral transform that decomposes a function into its frequency components. Its continuous-frequency version relates a function's energy to the energy of its transform.
L2 space: A space of measurable functions whose squared magnitudes have finite integral, identified up to equality almost everywhere. The identity is naturally stated for functions with finite squared norm.
Power spectral density: A function describing how the power of a signal is distributed across frequencies. Parseval-type relations connect integrated spectral power to time-domain mean-square power.
Joseph Fourier: A French mathematician and physicist whose work established the use of trigonometric series to study heat flow. Fourier series supplied the setting in which the identity's coefficient-energy relation developed.
Inner product space: A vector space equipped with an inner product, which defines lengths and angles. The identity follows from preservation of inner products under a complete orthonormal expansion.
Hilbert space: A complete inner product space in which Cauchy sequences converge to elements of the space. Completeness supports expansions in infinite orthonormal systems and their norm identities.

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