\part{Fundamentals}


\chapter{Fourier series}


Fourier analysis is the fundamental (perhaps described as most elementary) field of harmonic analysis, studying functions between Euclidean spaces and their trigonometric series representations.

Abstract harmonic analysis generalizes the ideas and tools of Fourier analysis by considering more complex spaces that may require a different type of sum (or integral) decomposition to best suit the space's symmetry.

Trigonometric series were used notably by Euler, Lagrange, and Gauss in various applications, however Fourier and Bessel were the first mathematicians to think about them as a way of representing an arbitrary function; it was his boldness that brought this field to life!



Taylor series are based on a premise that a Taylor series should have the same derivatives as the function it models. Fourier series work by means of a different train of thought, we're projecting a function onto a space spanned by infinite 'sinusoids'.



\section{Fourier series}

\begin{definition}[Trigonometric series]
%\[ a_0 + \sum^{N}_{n=1} a_n \sin (nx)\]
%\[ \sum^{N}_{n=0} b_n \cos (nx)\]
	\[ a_0 +\sum^{\infty}_{n=1} [ a_n \cos (n \frac{2\pi}{P}x)  + b_n \sin (n\frac{2\pi}{P}x) ] \]
\[ \sum^{\infty}_{n=-\infty} c_n e^{in\frac{2\pi}{P}x}\]
\end{definition}


Due to convenience, we will study Fourier analysis by means of the its complex form.

We also have the following formulae to convert the coefficients between the 2 representations.

\begin{proposition}
\[c_0 = a_0\]
\[c_n = \frac{a_n - ib_n}{2}\]
\[c_{-n} = \frac{a_n + ib_n}{2}\]
\end{proposition}





We are familiar with the Taylor series decomposition for analytic functions by matching derivatives at some reference point, can we also model some rich class of functions by means of a trigonometric series? This was the exact motivation of Fourier and Bessel.

If we restrict our attention to $L^1 ([0,2\pi])$ functions (complex functions integrable on $[0,2\pi]$), we can devise a general method by using the \emph{orthogonality of trigonometric functions on $[0,2\pi]$}, in other words, we can abuse the following identities.
\[\int_{[0,2\pi]} e^{inx}e^{imx} dx = 0\]


Let's say we have some function $f : [0,2\pi] \to \mathbb{C}$ and we're hoping to model a trigonometric series converging to $f$ on $[0,2\pi]$. 
\[ \int_{[0,2\pi] } f(x)e^{inx}dx  \]
\[ = \int_{[0,2\pi] } [\sum^{\infty}_{n=-\infty} \widehat{f}(n) e^{inx}]e^{imx}dx  \]
\[ =  \sum^{\infty}_{n=-\infty} \widehat{f}(n) \int_{[0,2\pi] }  e^{inx}e^{imx}dx  \]
\[ =  \widehat{f}(n) \int_{[0,2\pi] }  e^{imx}e^{imx}dx + \widehat{f}(-n) \int_{[0,2\pi] }  e^{-imx}e^{imx}dx  \]
\[ =  2 \pi \widehat{f}(-n)  \]

If we rearrange for $\widehat{f}(-n)$ and then flip $-n \to n$, we have a method to calculate the coefficients we want so that the trigonometic series equals $f$ (well, at least for sufficiently nice functions)!
\[ \widehat{f}(n)=  \frac{1}{2 \pi} \int_{[0,2\pi]}  f(x)e^{-inx}dx \]

There's one slightly subtle but mathematically crucial caveat to this method; the 3rd equality swaps the infinite sum and the integral. This cannot be done in general, so we must keep in mind that this trigonometric series converges to $f$ iff this sum-integral swap is permitted (we will disuss this in further detail later).

We expanded on the interval $[-\pi,\pi]$, though one can generalize this to any interval $[-\frac{T}{2},\frac{T}{2}]$ by considering some $2\pi / T$
So far our theory had been about developing Fourier series on $[-\pi,\pi]$, however this can be generalized to $[-\frac{T}{2},\frac{T}{2}]$.


\[ \widehat{f}(n)=  \frac{\langle f ,e^{-inx} \rangle}{\langle  e^{-inx} , e^{-inx} \rangle} \]



\begin{definition}[Fourier series]
%Infinite series on a periodic space $ L^1( (-\pi , \pi] ) $ relating to a periodic function $f$.
For a real function $f$ defined on $[-\frac{T}{2},\frac{T}{2}]$, the \emph{Fourier series of $f$ on $[-\frac{T}{2},\frac{T}{2}]$} (discrete spectrum) is the following related trigonometric series (if it exists).
\[ f \sim \sum^{\infty}_{n = -\infty} \widehat{f}(n) e^{in \omega x} \]
\begin{itemize}
	\item $\omega = \frac{2\pi}{T}$ is the fundamental frequency
	\item $ \widehat{f}(n) =  \frac{1}{T}\int^{\frac{T}{2}}_{-\frac{T}{2}} f(n) e^{- i n \omega x } dx $ are the \emph{Fourier coefficients}
\end{itemize}
\end{definition}

We remind the reader that a Fourier series of a function does not necessarily equal the function, even if this is the main motivation for its definition (just like Taylor series). Indeed, it is quite complicated to determine when and in what sense (pointwise or uniformly, everywhere or almost everywhere etc.) a Fourier series converges to its function. We will come back and answer this question later.

The Fourier coefficient function is worthy of much study, and it obeys many elementary, yet interesting properties.

\begin{proposition}
\begin{itemize}
	\item $f \in L^2 \implies \| f \|_{2}^{2}  = \sum^{\infty}_{n = -\infty} | \widehat{f}(n) |^2 $
	\item $ \widehat{af+bg}(n) = a\widehat{f}(n) + b \widehat{g}(n) $
	\item $ \widehat{f(x-a)}(n) = e^{-in \omega a}\widehat{f}(n) $
	\item $ \widehat{f'}(n)= in\omega \widehat{f}(n) $
	\item $ \widehat{f^{*}}(n)= [\widehat{f}(-n)]^{*} $
\end{itemize}
\end{proposition}


The fourier coefficients together are called the (discrete) spectrum of a function. Their modulus and argument are called the amplitude and phase spectrum respectively.


\section{Forced series development}

Furthermore, one can also consider a Fourier sine series or Fourier cosine series on $[0,2\pi]$ (or generally $[0,T]$), that only expands the function in terms of one type of sinusoid (sine or cosine)

\section{Bessel's inequality}


\begin{lemma}[Bessel's inequality (Fourier coefficients)]
Let $f$ be a piecewise continuous periodic function whose Fourier series exists, then the following holds.
	\[ \sum^{\infty}_{n=-\infty} |\hat{f}(n)|^2 \leq \frac{1}{T} \int^{\frac{T}{2}}_{-\frac{T}{2}} |f(t)|^2 dt\]
\end{lemma}

Considering partial sums of the Fourier series as $s_n (t)$, the Bessel inequality hinges off the fact that $\int^{\frac{T}{2}}_{-\frac{T}{2}} (f(t)-s_n(t))(f(t)-s_n(t))^{*} dt$ must be positive and real since it is the integral of the square of a modulus $zz^{*}=|z|^2$

\section{Riemann-Lebesgue lemma}

To prove the fundamental theorem of Fourier series, we will eventually need to employ the \emph{Riemann-Lebesgue lemma}, which states that Fourier coefficients tend to 0 for larger frequencies.

To make mathematical proofs less complicated, chunks of the proof that amount to a result in and of itself will be set aside as its own lemma so that t can be applied to the proof and any other proofs.

\begin{lemma}[Riemann-Lebesgue lemma (Fourier coefficients)]
Let $f$ be a piecewise continuous periodic function whose Fourier series exists, then the following holds.
\[ \lim_{n \to \pm\infty} \hat{f}(n) = 0 \]
\end{lemma}

The intuition is that  because continuous functions have close image elements associated for close domain elements, the more oscilliatory trigonometric functions bounce at a higher frequency and are therefore required less and less in the series.

This is one of the lemmas we will require to propound the fundamental theorem of Fourier series; a theorem that assures that for any piecewise smooth periodic function $f$, the Fourier series converge to $\frac{f(t+)+f(t-)}{2}$ pointwise! Our analysis in this theorem will be based on analyzing the partial sums of a Fourier series, and this lemma will be the final touch to prove the theorem.


\section{Fundamental theorem of Fourier series}


This theorem allows us to understand when and how Fourier series of a function converge to the function itself.

\begin{theorem}[Fundamental theorem of Fourier series]
Let $f$ be a piecewise smooth periodic function, then the fourier series of $f$ converges pointwise to the following.
\[\sum^{\infty}_{n=-\infty} \hat{f}(n)e^{i n \omega x} = \frac{f(x+) + f(x-)}{2}\]
\end{theorem}

This theorem doesn't give a necessary condition; it is a possibility that some non-piecewise smooth functions converges to its Fourier series, and there are examples of this (such as the Weierstrass function)



\begin{proposition}
\[  \widehat{f * g}(n) = \hat{f}(n)\hat{g}(n) \]
\[  \widehat{fg}(n) = \sum_{m \in \mathbb{Z}} \hat{f}(m)\hat{g}(n-m) \]
\end{proposition}



Furthermore, this theorem operates within the realm of elementary real analsis, playing with a Riemann integral intil our result is derived. It is indeed possible to find a wider class of functions that converges in some weaker sense; this however requires working with the Lebesgue integral and spaces of functions called $L^p$ spaces, covered in the advanced section of this book.


\section{Parseval's theorem}


\begin{definition}[Parseval's theorem]
Let $f,g$ be piecewise smooth periodic functions, then the following holds.
	\[\frac{1}{T} \int^{\frac{T}{2}}_{-\frac{T}{2}} f(x)g^{*}(x)dx  \sum^{\infty}_{n=-\infty} \hat{f}(n)\hat{g}(n)^{*}\]
\end{definition}


Parseval's theorem can. be used to derive many interesting series. Though the fundamental theorem of Fourier series can do this too, Parseval's theorem gives more of an edge to create such series.

\section{Convolution kernels}

Taylor's theorem considers partial Taylor series as apart of its result; it is a good start to study partial Fourier series too. We know that since it is a finite sum that it indeed converges, so things are much nicer.

Note that this section requires familiarity with the convolution operator, discussed in the advanced section of Real Analysis. The idea of this section is to show a way of using convolutions with a function to form their partial Fourier series. This streamlines some of our previous observations that we required in the fundamental theorem of Fourier series.

\subsection{Dirichlet kernel}

We can actually represent partial Fourier series by means of a special integral transform on the related function $f$; partial Fourier series have an integral representation!

For an informal derivation, let's operate under the ansatz that such a integral transform exists, then we have the following.

\[\sum^{N}_{n=-N} \widehat{f}(n) e^{inx} = \int^{2\pi}_{0} f(x)g(x,t) dt \]

Since we know how Fourier series coefficients are calculated, we replace the Fourier coefficient symbols with what they calculate to.

\[\sum^{N}_{n=-N} \widehat{f}(n) e^{inx} = \int^{2\pi}_{0} f(x)g(x,t) dt \]

Applying the linearity of the integral (which is possible in general since we are considering finite sums), it becomes clear what kind of kernel our integral transform must take!


Our kernel is $\frac{\sum^{N}_{n=-N} e^{-inx}}{2\pi}$. FOr the sake of a clean notation, we can consider partial Fourier series as the convolution of $f$ and $\frac{\sum^{N}_{n=-N} e^{inx}}{2\pi}$.

Let's make some definitions.


\begin{definition}[Dirichlet kernel]
\[  D_n (x) = \sum^{n}_{k=-n} e^{ikx} = \frac{ \sin ( \frac{n+1}{2} x )}{ \sin (  \frac{x}{2} ) }\]
\end{definition}


\begin{proposition}
\[ (D_n * f)(x) = 2\pi \sum^{n}_{k=-n} \widehat{f}(n) e^{i n x } \]
$ \int^{\pi}_{-\pi} D_n(x) dx = 2\pi $
$  D_n(2\pi k) = 2n+1 $
$  D_n \text{ is even } $
$ (D_n * f)(x) = 2\pi \sum^{n}_{k=-n} \widehat{f}(n) e^{i n x }$
$ \exists f'(x) \implies \lim_{n \to \infty} (D_n * f)(x) = f(x) $
$ \exists f'(x) \implies \lim_{n \to \infty} (D_n * f)(x) = \frac{1}{2}[f(x^{+})+f(x^{-})] $
$ \exists f'(x) \text{p.w} \implies \lim_{n \to \infty} (D_n * f)(x) = f(x) $
\end{proposition}


The Dirichlet kernel can be used to make the proof of the fundamental theorem of Fourier series much more mechanical since we now have prepackaged results that could be applied to hasten the proof, rather than toying around with integrals.

\subsection{Fejér kernel}


The Cesàro summation of the first $n$ partial Fourier series can also be represented as an integral transform. If one were to reverse engineer under this assumption like we did to derive the Dirichlet kernel, we would end up discovering the \emph{Fejér kernel}.




\section{Poisson summation formula}




















