\part{Advanced}


\chapter{Function sets and spaces}


If not for anything else, this chapter can serve as an introduction of a rather convenient notation.

\section{Sets of continuous and differentiable functions}
$C(\Omega)$ is the set of continuous functions on $\Omega$
$C^n(\Omega)$ is the set of $n$-differentiable functions on $\Omega$
$C^{\infty}(\Omega)$ is the set of smooth functions on $\Omega$
$C^{\omega}(\Omega)$ is the set of analytic functions on $\Omega$

$\Omega$ is an open set

These notations are used also in complex and vector analysis, although instead of $\Omega$ being open intervals, they must be open sets of $\mathbb{C}$ and $\mathbb{R}^n$ with their standard topologies.

Those familiar with linear algebra and recall the properties pertaining to continuous, differentiable, smooth, and analytic functions may realize that theese actually form a linear space! This is studied further in Functional Analysis.

\begin{proposition}
$C(\Omega)$ is a linear space.
$C^n(\Omega)$ is a linear space.
$C^{\infty}(\Omega)$ is a linear space.
$C^{\omega}(\Omega)$ is a linear space.
\end{proposition}

\subsection{Schwartz space}


It happens to have a very important place within the field of Fourier analysis; a subfield of harmonic analysis. Much of this advanced part of the book will be introducing preliminary or preparatory notions for the study of Fourier analysis.
\[\mathcal{S}(\mathbb{R}) = \{ f \in C^{\infty}(\mathbb{R}) : \forall n,m \in \mathbb{N} [ \exists M \in \mathbb{R}_{+}  [ \forall x \in \mathbb{R} [ | x^m f^{(n)}(x)| \leq M ] ] ]  \}\]

\[f \in \mathcal{S}(\mathbb{R})  \implies \mathcal{F}\{f\} \in \mathcal{S}(\mathbb{R}) \]
\[f \in \mathcal{S}(\mathbb{R})  \implies \forall n,m \in \mathbb{N} [ |f^{(n)}(x)| \leq M|x^{-m}| ] \]
\[f \in \mathcal{S}(\mathbb{R})  \implies \forall n \in \mathbb{N} [ \lim_{x \to \pm \infty} f^{(n)}(x)=0 ] \]
\[e^{-x^2} \in \mathcal{S}(\mathbb{R}) \]

\[f \in \mathcal{S}(\mathbb{R}) \implies \lim_{x \to \pm \infty} f(x) = 0\]

\[f \in \mathcal{S}(\mathbb{R}) \land P \text{ is a polynomial } \implies Pf \in \mathcal{S}(\mathbb{R})\]




\section{Sets of Riemann integrable functions}

$\mathcal{R}(\Omega)$ is the set of Riemann absolutely integrable functions on $\Omega$
$\mathcal{R}^n(\Omega)$ is the set of Riemann $n$-power integrable functions on $\Omega$

It is true that these sets of functions form linear spaces too, however as it turns out, there is a cleaner, more powerful alternative to the Riemann integral; the Lebesgue integral.

Using the Lebesgue definition of an integral allows for the following:
\item Allows more functions to have a well defined integral
\item Allows the integral to exist over more complex domains
\item Facilitates proofs regarding integrals (specifically when limits and integral signs may be swapped)
\item Can be generalized to consider integration for more abstract function spaces
\item Generates a generally nicer function space than the Riemann integral

The Lebesgue integral is covered my book Measure Theory.




\chapter{Useful theorems}

\section{Lagrange inversion theorem}























\chapter{Integral transforms}

This chapter aims to introduce concepts about integral transforms that find use in many areas of mathematics. Integral transforms often arise in functional analysis and harmonic analysis, however their study (and introduction) by methods of real analysis often goes a long way.


\section{Integral transforms}


\begin{definition}[Integral transform]
An integral transform is a transform (self-map) of a function space of the following form
\[\mathcal{T} \{f\}(t) = \int_{\Omega} I(x,t)f(x)dx\]
\begin{itemize}
\item $\mathcal{T}$ is the integral transform operator
\item $f : X \to Y$ is the input function
\item $I : X\times T \to Y$ is the \emph{integral transform kernel}
\end{itemize}

In functional analysis these are central objects of study, treated as mappings from a function space to $\mathbb{R}$ or $\mathbb{C}$ (this is called a functional).

Integral transforms are a generic object in mathematical analysis that appear in many contexts with many different use cases, for instance, the Laplace transform is used as a tool to transform differnetial equations into algebraic equations, however in probability theory it is used to calculate the generating function of the raw moments. It is useful to study integral transforms so that it is easier to see where their use may be beneficial as well as to calculate them with relative easily.

\section{Convolutions}


There exists a specific form of integral transforms that frequently arises in many places within mathematical analysis and its applications, taking the following form.
\[\int_{\Omega} f(x)g(x-t)dx\]

Note that by fixing the function $g$ and integral domain $\Omega$ as desired, one has essentially defined an integral transform. Though such an expression can be used in defining transforms, treating such integrals of this type as functions in themselves and using other integral transforms on them can display interesting results. Due to its prevalence, mathematicians created an operator to succintly represent and formally study such integral transforms; the \emph{convolution operator}.

%We will also be studying ways in which the convolution operator interacts with Fourier series. It is an operator for defining a certain type of integral transform, hence why it is discussed in this chapter.

\begin{definition}[Convolution operator]
Operator on $ \mathcal{R}(\mathbb{R}) $ functions such that the integral of $f$ is evaluated with $g$ as a weight function that is translated by $t$
\[ (f * g)(t) = \int_{-\infty}^{\infty} f(\tau )g(t -\tau) d \tau \]
\end{definition}

One can see that if both $f,g$ are absolutely integral on $\mathbb{R}$ then the convolution exists, hence this operator is well defined.

\begin{proposition}
\[  (f*g) = (g*f) \]
\[  \int^{\infty}_{-\infty} |(f*g)(t)|dt = \int^{\infty}_{-\infty} |f(t)|dt \int^{\infty}_{-\infty}|g(u)|du \]
\end{proposition}


As we will see, this operator interacts well with many integral transforms, and is considered as a major object of study in the field of functional analysis.




\section{Laplace transform}

A specific integral transform that serves as a powerful method for transforming differential equations into algebraic equations and real numbered equations.

Because the Laplace transform maps derivatives of a function to expressions in terms of the Laplace transform of the original function.



\begin{definition}[Heaviside step function]
The characteristic function of real numbers greater than or equal to 0, \(\chi_{\mathbb{R}_{+}}\), it serves useful for the Laplace inversion of function with a factor $e^{-sa}$
\[ H(t) = \begin{cases} 1 & t \geq 0 \\ 0 & t < 0 \end{cases} \]
\end{definition}


\begin{definition}[Laplace transform]
Integral transform on \((0,\infty)\) with kernel $e^{-st}$
\[ \mathcal{L} \{ f \} (s) = \int^{\infty}_{0} f(t) e^{-st} dt \]
\begin{itemize}
\item $f : [0,\infty) \to \mathbb{R}$ is a real valued function dominated by an exponential function
\item $\mathcal{L}\{f\} : S \to \mathbb{C} $ is the Laplace transform of $f$
\end{itemize}
\end{definition}

				\[\mathcal{L}\{f\} \in L^\infty([0,\infty])\]
				\[ \mathcal{L} \{ af+bg \} = a \mathcal{L}\{f\} + b \mathcal{L}\{g\} \]
				\[ \mathcal{L} \{ f \} \text{ is analytic on } \text{dom}( \mathcal{L}\{ f \})  \]
				\[ \mathcal{L} \{ t^n f \}(s) = (-1)^n \frac{d^n}{ds^n}\mathcal{L}\{f\}(s) \]
				\[ \mathcal{L} \{ e^{-at} f \}(s) = \mathcal{L}\{f\}(s+a) \]
				\[ \sum^{\infty}_{n=0} \frac{a_n}{n!}t^n \text{ is absolutely convergent } \implies \mathcal{L} \{ \sum^{\infty}_{n=0} \frac{a_n}{n!}t^n \}(s) = \sum^{\infty}_{n=0} \frac{a_n}{s^{n+1}} \]
				\[ \mathcal{L} \{ f(at) \}(s) = \frac{1}{a}\mathcal{L}\{f\}(\frac{s}{a}) \]
				\[\exists \mathcal{L}\{f'\}, \mathcal{L}\{f\} \implies \mathcal{L} \{ f' \}(s) = -f(0) + s \mathcal{L}\{f\}(s) \]
				\[\mathcal{L} \{ f^{(n)} \}(s) = s^{n} \mathcal{L}\{f\}(s) - \sum^{n-1}_{i=0} s^{n-i-1}f^{(i)}(0) \]
				\[ \mathcal{L} \{ f \} = 0 \implies f = 0 \text{ almost everywhere} \]
				\[ \mathcal{L} \{ f \} = \mathcal{L}\{g\} \implies f = g \text{ almost everywhere} \]
				\[  \mathcal{L}\{H(t-a) f(t-a) \} (s) = e^{-sa} \mathcal{L}\{f\}(s)\]


Here we list the Laplace transforms associated with common functions

\begin{itemize}
			\item $\mathcal{L}\{t^n\}(s) = \frac{\Gamma (n +1 )}{s^{n+1}} $
			\item $\mathcal{L}\{e^{-at}\}(s) =  \frac{1}{s+a} $
			\item $\mathcal{L}\{\ln(at)\}(s) =  \frac{\ln(a)-\gamma-\ln(s)}{s} $
			\item $\mathcal{L}\{\sin (at)\}(s) = \frac{a}{s^2 + a^2} $
			\item $\mathcal{L}\{\cos (at)\}(s) = \frac{s}{s^2 + a^2} $
			\item $\mathcal{L}\{J_0 (t)\}(s) = \frac{1}{\sqrt{1 +s^2}} $
\end{itemize}






\section{Hilbert transforms}


Later on, it was discovered to arise naturally within the study of signal processing (specifically in calculating the resulting function of a HPF).



\begin{definition}[Hilbert transform]
Integral transform on \((0,\infty)\) with kernel $e^{-st}$
	\[ \mathcal{Hi} \{ f \} (t) = \frac{1}{\pi}\mathbf{PV} \int^{\infty}_{-\infty} \frac{f(\tau)}{t-\tau} d\tau = \frac{(f * \frac{1}{t})(t)}{\pi}\]
\begin{itemize}
\item $f : \mathbb{R} \to \mathbb{R}$ is a real valued function dominated by an exponential function
\item $\mathcal{Hi}\{f\} : S \to \mathbb{C} $ is the Laplace transform of $f$
\end{itemize}
\end{definition}

Note how one can rewrite the integral transform by means of convolutions.
From the perspective of Functional analysis, the Hilbert transform is particulatly nice as it is a \emph{bounded linear operator}, however the meaning and proof of this property is best introduced within the scope of a functional analysis book, since this is best proven with access to the notion of an $L^p$ space.




\begin{proposition}
	\[\mathcal{Hi}\{\mathcal{Hi}\{f\}  \} = -f\]
\end{proposition}






\begin{proposition}
	\[\mathcal{Hi}\{  f^{(n)}(t)\} = \mathcal{Hi}\{ f\}^{(n)}\]
\end{proposition}



\begin{proposition}
	\[\int_{\mathbb{R}} f(t)\mathcal{Hi}\{ f \}(t)dt = 0\]
\end{proposition}


\subsection{Relation with Fourier transform}
Possibly inspired by the prevalence of both Fourier and Hilbert transforms in signal processing, this transform has been observed to obey some rather nice properties when composed with the Fourier transform.


\[\mathcal{Hi}\{\delta\}(t) = \frac{1}{\pi t}\]

\[\mathcal{F}\{\mathcal{Hi}\{f\}\}(\xi) = -i \mathrm{sgn}(\xi) \mathcal{F}\{f\}(\xi)\]


\[ |\mathcal{F}\{\mathcal{Hi}\{f\}\}(\xi)|^2 = | \mathcal{F}\{f\}(\xi) |^2\]

\subsection{Orthogonality of Hilbert transform}

\[ \int f(x)\mathcal{Hi}\{f\}^{*}(x) dx = 0 \]























\chapter{Distributions}


Yet another theory that arose from the field of harmonic analysis.

We have seen in our study of Fourier transforms that the Fourier transform does not exist for some functions (for instance, the identity function $f(x)=x$). However if we take the limit of the Fourier transform of the rectangular function as its corners approach infinity, we can essentially emulate something like the Fourier transform of the identity function.


One can prove that $\lim_{a \to \infty} \frac{1}{2\pi} \int^{\infty}_{-\infty} \frac{2\sin(a\xi)}{\xi} f(\xi) d\xi = f(0)$, where $\frac{2\sin(a\xi)}{\xi}$ happens to be the Fourier transform (continuous spectrum) of $\mathrm{rect}_{2a}$. Swapping limits and integrals here is out of the question because if we were to even try it, our limit wouldn't even exist anymore $\lim_{a \to \infty} \frac{2\sin(a\xi)}{\xi} f(\xi) $ is undefined.

So we have an 'integral transform', except the kernel is the limit of a function that must be taken outside of the integral.

This is really whay distributions are about; defining our idealized 'function' by how it would act if it were the kernel of an integral transform, rather than defining it by its scalar to scalar mappings. The example that I have introduced is actually the \emph{Dirac distribution}, often denoted as $\delta$.


\section{Schwartz distributions}

Integral transforms may only allow certain classes of functions to operate on; the same goes for distributions, although we will start out by looking at distributions on the Schwartz space due to its prevalence in literature. We define the notion and notation of a 'Schwartz distribution'.

\begin{definition}[Schwartz Distribution]
A \emph{Schwartz distribution} is a linear map $T : \mathcal{S} \to \mathbb{C}$.
We use the 
\[ \langle T , f+g \rangle = \langle T,f \rangle + \langle T,g \rangle \]
\[ \langle T , cf \rangle =  c \langle T,f \rangle \]
\end{definition}

Note that the same notation is used for inner products. Since integral transforms are linear (since they are integrals)

Note that despite the fact that distributions have functions as domains rather than scalars, we may give them a 'decorative' scalar domain notation like for normal functions for the sake of representing translations, reflections, scaling, and possible other transformations

\subsection{Delta distribution}

\begin{definition}[Delta distribution]
The \emph{delta distribution} is the Schwartz distribution $\delta(x)$ defined in the following manner.
\[\langle \delta , \phi \rangle = \lim_{a \to \infty} \frac{1}{2\pi} \int^{\infty}_{-\infty} \frac{2\sin(a\xi)}{\xi} \phi(\xi) d\xi = \phi(0) \]
\end{definition}

\[\langle \delta(x-c) , \phi \rangle = \phi(c) \]

\subsection{Functions representable as Schwartz distributions}

We have seen the Delta distribution as a Schwartz distribution that cannot be represented as a function, however there are distributions that are also functions; we can translate some functions into Schwartz distributions. This isn't possible for every real function, but a a pretty rich class of them work!
In the same vein as talking about distributions as defining functions by their behaviour as integral kernels, we can define Schwartz functions as diistributions in the following manner.

\[\langle T_f , \phi \rangle = \int^{\infty}_{-\infty} f(x)\phi(x)dx \]

When is such a distribution well-defined (i.e when does this integral converge)?
Since $\phi$ is a Schwartz function, we can prove that $f$ being Riemann absolutely integrable is enough for the integral to converge.

Although the identity function isn't Riemann absolutely integrbale on $\mathbb{R}$, it still manages to form a welldefined distribution due to the properties of Schwartz functions.

\section{Derivatives of distributions}

There is a notion of a derivative for distributions too.
If we consider the distribution of derivatives of classical functions, the product rule gives us the following.
\[\langle T_{f'} , \phi \rangle = -\langle T_{f} , \phi' \rangle \]

We will define the derivative of distributions to agree with this classical notion.

\begin{definition}[Derivative of a distribution]
\[\langle T' , \phi \rangle = -\langle T , \phi' \rangle \]
\end{definition}

Since we can differentiate any Schwartz function, the right hand side will always exist and hence distributions always have a derivative; something that doesn't hold for classical functions!


\[\langle T^{(n) , \phi \rangle = (-1)^n \langle T , \phi^{(n)} \rangle \]

\subsection{Jump formula}

Due to linearity, the following notations are well defined.
\[\langle cT , \phi \rangle = c\langle T , \phi \rangle \]
\[\langle T+S , \phi \rangle = \langle T , \phi \rangle + \langle S , \phi \rangle \]

Here is the jump formula.
\[T_{f}' = T_{f'} + [f(a+) - f(a-)] \delta (t-a) \]

for any continuous function, we have the following.
\[\langle fT , \phi \rangle = \langle T , f\phi \rangle\]

\[\langle T(at) , \phi(t) \rangle = \langle T(t) , \frac{\phi(\frac{t}{a})}{a} \rangle\]



\subsection{Convolution of distributions}

\begin{definition}[Convolution of distributions]
Given that $\langle T(t) , \phi(t+\tau) \rangle $ belongs to $\mathcal{S}(\mathbb{R})$, the convolution of distributions $S * T $ is defined as the following.
\[\langle S * T , \phi(t) \rangle = \langle S(t) , \langle T(t) , \phi(t+\tau) \rangle \rangle\]
\end{definition}