# Lecture 16: Fourier Transform
###### tags: `224a`
$\newcommand{\F}{\mathcal{F}}$
$\newcommand{\R}{\mathbb{R}}$
So far in the course, we studied differential operators on bounded intervals. We began with the Laplacian $-d^2/dx^2$ on $[a,b]$, which we "solved" using Fourier series, and then generalized this to Schrodinger operators of type $L=-d^2/dx^2+q(x)$ on $[a,b]$. The generalization involved developing the theory of compact operators, and then observing that the Green's function $(z-L)^{-1}$ (which is really just the resolvent at some point not in $\sigma(L)$) is compact, so $L$ inherits the (complete) set of eigenvectors of the compact case.
In the next three lectures we will generalize this to differential operators on $\R$. The analogue of Fourier series is the Fourier transform, which will give us a spectral representation of the momentum and Laplacian operators on $\R$. We will study Schrodinger operators on $\R$ by observing that $(z-L)^{-1}$ is *bounded* for a judicious choice of $z$.
In this lecture we develop the basic theory of Schwartz spaces and the Fourier transform, covering the following topics. The treatment is very similar to Dimock 1.1.4 so I will not (re)write notes.
Schwartz Space
Fourier Transform
Multiplication and Differentiation
Inversion Theorem, Plancharel
Extension to a unitary operator on L^2
The punch line of this lecture was a spectral representation for the momentum operator $A=-id/dx$ on the Schwartz space $S(\R)$:
$$ \F A = M_x \F\quad on \quad S(\R)$$
where $\F$ is the $L^2$ Fourier transform. In particular, this means that
$$ \F A \F^{-1} = M_x \quad on \quad S(\R)\quad (*).$$
where $M_x$ is multiplication by $x$.
For the purposes of spectral theory, however $S(\R)$ is not a convenient subspace of $L^2$ to work with; the right one is the "maximal domain"
$$D(M_x):=\{f\in L^2(\R): M_xf\in L^2(\R)\}.$$
In order to make sense of the identiy $(*)$ on this extended domain, however, we need to extend $A$ to $\F^{-1}D(M_x)$, and this is done by *defining* $\hat{A}:\F^{-1}D(M_x)\rightarrow L^2(\R)$ by $\hat{A} f = \F^{-1} M_x \F f$. The domain of $\hat{A}$ thus obtained is called a *Sobolev space* and the resulting notion of derivative is called a *weak derivative*.
In the next lecture we will define the spectral theory of such operators and show how to do this for any self-adjoint unbounded operator (which we will also define).

The official schedule is posted at https://www.jointmathematicsmeetings.org/meetings/national/jmm2023/2270_program_ss94.html#title The Zoom link for virtual attendance/virtual talks is: https://berkeley.zoom.us/j/94982941411?pwd=RWRXeVdoZUNVTmdqK1FIMksrZFllZz09 Friday, January 6 (101 Hynes Convention Center) 1-2pm: Matt Colbrook, The foundations of infinite-dimensional spectral computations (Zoom) 2-3pm: JosuĂ© Tonelli-Cueto, Condition-based Low-Degree Approximation of Real Polynomial Systems

1/3/2023Lecture 1: Propositional Logic A proposition is a declarative sentence that is either true or false but not both. A proposition has a truth value which is either $T$ or $F$. A letter used to denote a proposition is called a propositional variable. If $p$ is a proposition, the negation of $p$, denoted $\lnot p$, is the proposition "it is not the case that $p$". The truth value of $\lnot p$ is the opposite of that of $p$. If $p$ and $q$ are propositions, the conjunction of $p$ and $q$ (denoted $p\land q$) is the proposition "p and q." Its truth value is $T$ when both $p$ and $q$ are $T$, and it is $F$ otherwise. If $p$ and $q$ are propositions,he disjunction of $p$ and $q$ is the proposition "p or q". Its truth value is $T$ when at least one of $p$ or $q$ is $T$, and $F$ if both $p$ and $q$ are $F$. If $p$ and $q$ are propositions, the conditional $p\rightarrow q$ is the proposition "if $p$ then $q$". Its truth values are given by the following truth table (which lists its truth values in terms of those of $p$ and $q$): [see Table 5 in Rosen] The converse of $p\rightarrow q$ is $q\rightarrow p$. A compound proposition consists of logical operations (the four listed above) applied to propositions or propositional variables, possibly with parentheses. The truth value of a compound proposition can be mechanically determined given the truth values of its constituents.

12/12/2022Lecture 14 Basic Terminology, Handshaking Theorem A graph is a pair $G=(V,E)$ where $V$ is a finite set of vertices and $E$ is a finite multiset of $2-$element subsets of $V$, called edges. If $E$ has repeated elements $G$ is called a multigraph, otherwise it is called a simple graph. We will not consider directed graphs or graphs with loops (edges from a vertex to itself). Two vertices $x,y\in V$ are adjacent if ${x,y}\in E$. An edge $e={x,y}$ is said to be incident with $x$ and $y$. The degree of a vertex $x$ is the number of edges incident with it. Theorem. In any graph $G=(V,E)$, $$ \sum_{v\in V} deg(v) = 2|E|.$$ Proof. Let $I={(e,v):e\in E,v\in V,\textrm{$e$ is incident with $v$ in $G$}}$ be the set of edge-vertex incidences in $G$. We will count the number of incidences in two ways: (i) Observe that every edge participates in exactly two incidences, and no incidence participates in more than one edge. Thus, the total number of incidences is $|I|=2|E|$.

12/12/2022Prove that every simple graph with $n$ vertices and $k$ edges has at least $n − k$ connected components (hint: induction, or contradiction). Let $p\ge 3$ be a prime. Consider the graph $G=(V,E)$ with $V={0,1,2,\ldots, (p-1)}$ and $$E={{x,y}:x-y\equiv 2(mod p)\lor x-y\equiv -2 (mod p)}$$ Show that $G$ is connected. Does $G$ have an Euler circuit? Prove your answer. Let $n\ge 1$ be an integer and let $k\le n/2$. Consider the graph $G=(V,E)$ with $V={S\subseteq {1,2,\ldots,n}: |S|=k}$ and $$E = {{S,T}:|(S-T)\cup (T-S)|=1}.$$ What are the degrees of the vertices in $G$? Is $G$ connected? For which values of $n$ and $k$ does $G$ have an Eulerian circuit? For which values of $n$ and $k$ is $G$ $2-$colorable? Prove your answers.

11/8/2022
Published on ** HackMD**

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