# Abstract Peter-Weyl theory for semicomplete orthonormal sets

## TL;DR

This paper extends harmonic analysis on compact groups by developing a Peter-Weyl theory for semicomplete orthonormal sets, introducing the prime-Parseval subspace, and exploring its properties and implications.

## Contribution

It constructs a theory for semicomplete orthonormal sets on compact groups, including explicit construction and analysis of the prime-Parseval subspace, broadening the classical Peter-Weyl framework.

## Key findings

- Existence of semicomplete orthonormal sets proved via explicit construction.
- The prime-Parseval subspace is dense in L^2 and serves as the Fourier transform domain.
- The approach provides new insights into harmonic analysis on compact groups.

## Abstract

The central concept in the harmonic analysis of a compact group is the completeness of Peter-Weyl orthonormal basis as constructed from the matrix coefficients of a maximal set of irreducible unitary representations of the group, leading ultimately to the direct sum decomposition of its $L^{2}-$ space. A Peter-Weyl theory for a semicomplete orthonormal set is also possible and is here developed in this paper for compact groups. Existence of semicomplete orthonormal sets on a compact group is proved by an explicit construction of the standard Riemann-Lebesgue semicomplete orthonormal set. This approach gives an insight into the role played by the $L^{2}-$ space of a compact group, which is discovered to be just an example (indeed the largest example for every semicomplete orthonormal set) of what is called a prime-Parseval subspace, which we proved to be dense in the usual $L^{2}-$ space, serves as the natural domain of the Fourier transform and breaks up into a direct-sum decomposition. This paper essentially gives the harmonic analysis of the prime-Parseval subsapce of a compact group corresponding to any semicomplete orthonormal set, with an introduction to what is expected for all connected semisimple Lie groups through the notion of a $K-$semicomplete orthonormal set.