# Lecture 19: Tempered Distributions
###### tags: `224a`
Distributions are objects which are more general than functions. There are at least three motivations for studying them in the context of mathematical physics:
1. We would like objects which model "point" sources and "roughl" forcing functions.
2. We would like to be able to differentiate whatever these objects are, so we can put them in differential equations.
3. We would like to be able to take the Fourier transform of them.
We will begin by studying tempered distributions, which are a particular subclass of all distributions.
We adopt the perspective that a function $f(x)$ takes a number $x$ as an input and ouputs another number. We can think of $x$ as "probing" or "testing" the function. The key idea is that being able to test each point is unforgiving and leads to "pathological" functions, and we can have a much better behaved theory if we *restrict the class of tests*.
A tempered distribution takes a *Schwartz function* as input and outputs a number. Formally, it is a continuous linear functional on the space of Schwartz functions.
$\renewcommand{\S}{\mathcal{S}}$
The key facts allowing us to extend continuous operators on $\S(R)$ to $\S'(R)$ are:
1. The map $g\mapsto T_g$ where $$T_g\phi:=\int g(x)\phi(x)dx$$ is multiplication is easy to see that this map is continuous and 1-1. This yields an embedding of $\S(R)$ into $\S'(R)$
3. The embedding is *dense*, though we will not prove this (the interested reader can see the appendix of RS V.3). This implies that every continuous operator $A:\S(R)\rightarrow \S(R)$ has a *unique* extension $\hat{A}$ to $\S'(R)$.
4. The extension $\hat{A}$ must satisfy (in order to be an extension) $$(\hat{A}T_g)(\phi)=T_{Ag}(\phi)=\int (Ag)(x)\phi(x)dx \quad \forall g\in \S(R).$$
This relation can then be used to derive the *definition* for any $T\in \S'(R)$, as well as a rule for computing the extension.
We worked this out for the cases $A=$ multiplication by a slowly growing function or differentiation. The formal development almost exactly follows Reed and Simon V.3, with a couple of extra examples thrown in, so I will not rewrite everything and refer the reader to that section. Ignore any mentions of Frechet spaces and Topology if those things make you uncomfortable.

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
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