\part{Fundamentals}

Fourier analysis is often ones introduction to the field of harmonic analysis. Abstract harmonic analysis studies expansion of functions by means of different kernels rather than the trigonometric functions. Generalizing ideas from Fourier analysis to utilize different functions may bring some new symmetries to light, and may perhaps offer expansions for functions on more abstract spaces!

This book is therefore about taking the ideas of Fourier analysis, generalizing them and applying them to different kernels (possibly existing on different spaces).

This book requires a background in functional analysis.

\chapter{Generalized Fourier transform}




\chapter{Hankel transform}


\[ \mathcal{F}\{f\}(\boldsymbol{\xi}) = \int_{\mathbb{R}^n} f(\mathbf{x}) e^{-i \langle \mathbf{x} , \boldsymbol{\xi} \rangle } d\mathbf{x} \]

\section{Hankel transform}


For radial functions,there exists a different set of orthogonal functions that better suits radial symmetry.


\begin{definition}[Hankel transform]
Integral transform that extends the notion of Fourier transform for radial functions
\[ \mathrm{H}_n\{f\}(\rho ) = \int_{0}^{\infty} f(r) J_{n}(\rho r)r dr \]
\begin{itemize}
	\item $f \in L^1$ is a Lebesgue intergable radial function
	\item $\mathrm{H}_n\{f\} $ is the radial Hankel transform of order $n$
	\item $J_n$ is the Bessel function of the first kind of order $n$
\end{itemize}

