Plancherel

Plancherel refers to a family of results in harmonic analysis named after a French mathematician whose work helped establish how Fourier analysis interacts with L^2 spaces. The most familiar result is Plancherel’s theorem, which provides a precise link between a function and its Fourier transform.

In its standard form on Euclidean space, Plancherel’s theorem states that, with a suitable normalization of the Fourier transform, the Fourier transform maps square-integrable functions to square-integrable functions and preserves the L^2 norm. Concretely, the integral of the square magnitude of a function equals the integral of the square magnitude of its transform, up to a constant determined by the chosen convention. This makes the Fourier transform an isometry from L^2 to L^2 and yields Parseval’s identity as a special case for Fourier series.

Beyond the real line and Euclidean spaces, Plancherel theory extends to locally compact groups, forming part of non-commutative harmonic analysis. In this general setting one introduces a Plancherel measure on the unitary dual of the group and expresses L^2(G) as a direct integral of Hilbert spaces corresponding to irreducible representations. The Fourier transform then becomes a field of operators, and the Plancherel formula equates L^2-norms with a corresponding integral over representations.

Applications of Plancherel theory appear in signal processing, quantum mechanics, and representation theory, where it provides a rigorous foundation for switching between time/space and frequency descriptions and for decomposing functions according to the spectral content of more general symmetry groups.