Generalisations of the theorem of Peter and Weyl - Stevenson, Elizabeth (Department of Mathematics and Applied Mathematics, 2025)
Introducing some classical concepts in the representation theory of compact groups, we study the historical background of the Fourier Theorem and how this result was used to solve the heat equation. Then we sketch the main steps of the proof of the Fourier Theorem, explaining how it can be interpreted as a special case of a more general argument, which is used in the proof of the Peter-Weyl Theorem. A comparative analysis of different arguments, which have been used to prove the Fourier Theorem and the Peter-Weyl Theorem in the literature, is illustrated in the present thesis. Therefore, we note that some arguments apply to compact connected abelian groups while others can't be applied. Inspecting the details of the proofs, we observe that the use of the Stone-Weierstrass Theorem plays a fundamental role, so we discuss the relevance of the Stone-Weiestrass Theorem in the proof of Peter-Weyl Theorem.