Topics in Hilbert Spaces, Spectral Theory, and Harmonic Analysis

TL;DR

This paper provides an overview of Hilbert spaces, spectral theory, and harmonic analysis, including foundational concepts, spectral theorem derivations, and applications to finite groups and $L^2$ spaces.

Contribution

It offers a comprehensive exposition of spectral theory in Hilbert spaces with new derivations and applications to harmonic analysis on groups and function spaces.

Findings

Derived spectral theorem for finite and infinite-dimensional Hilbert spaces.

Applied spectral theory to finite abelian groups.

Extended applications to $L^2$ spaces on the circle.

Abstract

General expository paper concerning topics in Hilbert spaces, spectral theory, and harmonic analysis. The preliminary section includes basic Banach algebra and Hilbert space theory with a digression on Riesz bases. The second and third sections are focused on deriving the spectral theorem for Hilbert spaces in finite and arbitrary dimension, respectively. The fourth and fifth sections present applications of the general theory: first to finite abelian groups, then to the space $L^2(S^1,\mu)$.