the Peter-Weyl theorem
/ PAY-ter VILE /
Fourier analysis says any reasonable periodic function is a sum of pure waves e^(i n theta). The Peter-Weyl theorem is the vast generalization: for any compact group, the 'pure waves' are the matrix entries of its irreducible representations, and every square-integrable function on the group is built from them. It is harmonic analysis liberated from the circle, the bridge from group representations to function theory on the group.
Let G be a compact (Hausdorff, topological) group with its normalized Haar measure. The Peter-Weyl theorem has three intertwined conclusions. (1) Density: the matrix coefficients of the finite-dimensional irreducible unitary representations span a dense subspace of the continuous functions C(G) (uniform norm) and of L^2(G). (2) Decomposition of L^2: as a representation of G x G acting by left and right translation, L^2(G) decomposes as a Hilbert-space direct sum over the irreducibles pi (each appearing with multiplicity equal to its dimension): L^2(G) ~ (direct sum over pi) of (V_pi tensor V_pi^*), where the V_pi are the finite-dimensional irreducible representation spaces. (3) Complete reducibility / finite-dimensionality: every irreducible unitary representation of a compact group is finite dimensional, and every unitary representation decomposes into irreducibles. The recipe to use it: integrate a function against a matrix coefficient (a 'generalized Fourier coefficient') to extract its component in each irreducible, exactly as one extracts Fourier coefficients on the circle.
Peter-Weyl is the foundation of representation theory of compact groups, of character theory (characters chi_pi = trace of pi form an orthonormal basis of the class functions), and of the spectral theory of the Laplacian on a compact symmetric space. For the circle G = U(1) it reduces exactly to classical Fourier series, the irreducibles being the characters theta |-> e^(i n theta). Honesty: the theorem genuinely needs COMPACTNESS. For noncompact groups (like the Lorentz group or SL(2, R)) irreducible unitary representations are typically infinite dimensional, there is no finite-multiplicity discrete decomposition of all of L^2, and one needs the far harder Plancherel theory with continuous spectra. Compactness, via the averaging supplied by Haar measure, is what forces finite dimensionality and clean decomposition.
For G = U(1) ~ S^1, the irreducible unitary representations are the 1-dimensional characters chi_n(theta) = e^(i n theta), n in Z. Peter-Weyl says these span a dense subspace of L^2(S^1) and give the orthogonal decomposition L^2(S^1) = (direct sum over n in Z) of C e^(i n theta) — which is precisely the classical theory of Fourier series.
On U(1), Peter-Weyl IS the theory of Fourier series; the characters e^(in theta) are the irreducibles.
Peter-Weyl needs compactness. For noncompact groups irreducible unitary representations are generally infinite dimensional and L^2 has continuous spectrum; the clean finite-multiplicity decomposition fails.