Planchereltype

Planchereltype refers to objects in harmonic analysis that admit a Plancherel-type formula. In its standard form, such a formula provides a Fourier transform that is a unitary isomorphism between L^2-spaces on a locally compact group G and a direct integral of irreducible unitary representations, with respect to a Plancherel measure μ on the unitary dual Ĝ. Concretely, for suitable f in L^1∩L^2(G), f̂(π) = ∫G f(x)π(x) dx, and the Plancherel identity ∥f∥2^2 = ∫Ĝ ∥f̂(π)∥HS^2 dμ(π) holds. Objects with this property are often required to be locally compact, unimodular, and of type I so that the unitary dual can be well behaved.

Examples include locally compact abelian groups, where the classical Plancherel theorem applies; compact groups, where the Peter-Weyl theorem yields a discrete Plancherel measure; and more generally certain nonabelian groups such as nilpotent Lie groups (e.g., the Heisenberg group). For semisimple Lie groups one has a Harish-Chandra Plancherel formula, a refinement of the Plancherel theory for noncompact cases. The term Planchereltype is used in literature to describe spaces or representations that enjoy such a Fourier-analytic decomposition, even when the full general theory is complex or not yet established.

Applications of Planchereltype decompositions include harmonic analysis on groups and homogeneous spaces, representation theory, and signal processing on non-Euclidean domains. Limitations include the need for structural hypotheses (type I, amenability in some settings) and the potential absence of a simple, explicit Plancherel measure in general settings.