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,

More formally, the result holds as stated provided ${\displaystyle f}$ is a square-integrable function or, more generally, in Lp space ${\displaystyle L^{2}[-\pi ,\pi ].}$ 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 ${\displaystyle f\in L^{2}(\mathbb {R} ),}$

Another similar identity is a one which gives the integral of the fourth power of the function ${\displaystyle f\in L^{4}[-\pi ,\pi ]}$ in terms of its Fourier coefficients given ${\displaystyle f}$ has a finite-length discrete Fourier transform with ${\displaystyle M}$ number of coefficients ${\displaystyle c\in \mathbb {C} }$.