\part{Advanced}

\chapter{$L^p$ analysis on Fourier series}

The fundamentals part of this book develops the theory of Fourier analysis with reference only to fundamental real analysis and basic complex analysis, however the introduction of more advanced tools such as Lebesgue integration, $L^p$ spaces and ideas from functional analysis allows for a much more precise study of Fourier series and integrals. This is because the values of the integrals in the Fourier coefficients remain the same when a $\lambda$-null set of mappings of $f$ are modified; measure theory offers the tools to formally address this fact and work considering the possibilities of these modified mappings.

On a less important note, the domain of periodic functions on $[-\frac{T}{2}, \frac{T}{2}]$ can be considered as the circle group when adding domain elements; Fourier series essentially exist on the circle group.

In this chapter, we will employ the Lebesgue integral rather than the Riemann integral so that we can study an alternatives to the fundamental theorem of Fourier series, notably the fact that the Fourier series of $L^2$ functions converges pointwise almost everywhere.

This is a slightly weaker type of convergence being considered, however it applies to a larger class of functions as well as being a Hilbert space, meaning that we can apply functional analysis to study Fourier series!




\begin{theorem}[Carleson's theorem]
Let $f$ be a function in $L^2$, then the Fourier series of $f$ converges to $f$ pointwise almost everywhere.
\end{theorem}


\chapter{$L^p$ analysis on Fourier integral}

Functional analysis have since become a major tool in the study of Fourier analysis, since (Lebesgue) integrable functions form their own space (called a Banach space) on which the Fourier transform is a functor.

In the same vein as the previous chapter, we apply functional analysis and measure theory to commence a deeper study on the Fourier integral, and use them to generalize the Fourier integral for other spaces.

$\mathcal{F}\{f\} \in L^\infty$
$\| \widehat{f} \|_{\infty} \leq \| f \|_{1} $
\begin{theorem}[Plancherel theorem]
\[ f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R}) \implies \| \mathcal{F}\{f\} \|_2 = \sqrt{2\pi} \| f \|_2 \]
\end{theorem}



\chapter{Fourier integral on $\mathbb{R}^n$}

One can consider analogous Fourier integral representations for functions on $\mathbb{R}^n$.


\begin{definition}[$\mathbb{R}^n$ Fourier transform]
	\[f(\mathbf{x}) = \int_{\mathbb{R}} e^{-i \mathbf{x} \cdot \xi} \mathbf{dx}\]
\end{definition}


One can consider Schwartz spaces for $\mathbb{R}^n$-functions, and indeed the multivariate Fourier transform is a bijection on this space too.

$ \widehat{f(\mathbf{R}\mathbf{x})}(\xi) = \widehat{f}(\mathbf{R}\xi$

Fourier transform of radial functions are radial


\begin{theorem}[Fundamental theorem of Fourier integral  on $\mathbb{R}^n$]
\[f(\mathbf{x})= \int_{\mathbb{R}^n} \widehat{f}(\xi) e^{i\mathbf{x} \cdot \xi} d\xi\]
\end{theorem}

Plancherel theorem generalizes


\begin{lemma}[Riemann-Lebesgue lemma for $\mathbb{R}^n$]
Let $f$ be a $L^1(\mathbb{R}^n)$ function, then the following holds (i.e its Fourier transform's tails tend to 0).
\[ \lim_{\xi \to \pm\infty} \mathcal{F}\{f\}(\xi) = 0 \]
\[ f \in L^1 (\mathbb{R}^n) \implies \lim_{\xi \to \pm\infty} \mathcal{F}\{f\}(\xi) = 0 \]
\end{lemma}






\chapter{Fourier transform and complex analysis}

Various techniques from complex analysis give greater insight into the Fourier transform.

\section{Paley-Wiener theorem}