PeterWeylteoremet

The Peter-Weyl theorem is a fundamental result in harmonic analysis and representation theory of compact groups. It describes how the matrix coefficients of finite-dimensional irreducible unitary representations of a compact group G form a Fourier-type basis for functions on G and for the space L2(G).

Let G be a compact topological group with Haar measure dg. Denote by {π} the set of equivalence classes of finite-dimensional irreducible unitary representations of G, and for each such π with dimension dπ choose an orthonormal basis of its representation space Vπ. Define the matrix coefficients fijπ(g) = ⟨ei, π(g) ej⟩ for i, j = 1,…, dπ. The theorem asserts two principal facts. First, the linear span of all matrix coefficients fijπ is dense in C(G) with respect to the uniform norm, so every continuous function on G can be uniformly approximated by finite linear combinations of these coefficients. Second, in L2(G) (with respect to the Haar measure), the family {√dπ fijπ} over all π and all indices i, j forms an orthonormal basis. Consequently, the regular representation of G on L2(G) decomposes as a direct sum of the irreducible representations, with each π occurring with multiplicity equal to its dimension: L2(G) ≃ ⊕π (Hπ ⊗ Hπ*).

The theorem also implies orthogonality relations for matrix coefficients (Schur orthogonality) and provides a Fourier-type expansion for functions on G. Special cases include ordinary Fourier analysis on finite groups and harmonic analysis on compact Lie groups like tori and SU(2).