Fourier Analysis on Compact Lie Groups: Peter–Weyl, SU(2), and Spherical Harmonics

This expository article introduces Fourier analysis on compact Lie groups through the Peter–Weyl theorem. Starting from classical Fourier series on the circle, it explains how matrix coefficients of irreducible unitary representations provide a generalization of complex exponentials. After reviewing smooth manifolds, Lie groups, Lie algebras, and finite-dimensional representations, the article introduces Haar measure and proves the density form of the Peter–Weyl theorem using convolution and compact operators. Schur orthogonality and complete reducibility then yield the corresponding orthonormal decomposition of L²(G). Applications include Fourier expansions on S¹ and SU(2), the construction of faithful finite-dimensional representations of compact Lie groups, and spherical harmonics through the identification S² ≅ SO(3)/SO(2). Particular attention is given to normalization conventions and convergence in L². The text is intended for readers familiar with linear algebra, differential calculus, integration, and basic Hilbert space theory.