# Non-Standard Analysis, Multiplication of Schwartz Distributions, and Delta-
Like Solution of Hopf’s Equation

###### Abstract

Non-Standard Analysis, Multiplication of Schwartz Distributions, and Delta-
Like Solution of Hopf’s Equation
Guy Berger

We construct an algebra of generalized functions
${}^{*}\mathcal{E}(\mathbb{R}^{d})$. We also construct an embedding of the
space of Schwartz distributions $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ into
${}^{*}\mathcal{E}(\mathbb{R}^{d})$ and thus present a solution of the problem
of multiplication of Schwartz distributions which improves J.F. Colombeau’s
solution. As an application we prove the existence of a weak delta-like
solution in ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ of the Hopf equation. This
solution does not have a counterpart in the classical theory of partial
differential equations. Our result improves a similar result by M. Radyna
obtained in the framework of perturbation theory.

A Thesis Presented to the Faculty of

California Polytechnic State University

San Luis Obispo

In Partial Fulfillment of the Requirements for the Degree

Master of Science in Mathematics

by

Guy Berger

September 2005

AUTHORIZATION FOR REPRODUCTION OF MASTER’S THESIS

I grant permission for the reproduction of this thesis in its entirety or any
of its parts, without further authorization from me.

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APPROVAL PAGE

TITLE: Non-Standard Analysis, Multiplication of Schwartz Distributions, and
Delta-Like Solution of Hopf’s Equation

AUTHOR: Guy Berger

DATE SUBMITTED: September 16, 2005

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Acknowledgements

I would like to thank the faculty and staff of the Cal Poly Mathematics
Department, who have made my experience over the past two years a positive
one.

I am especially grateful to my supervisor, Dr. Todor D. Todorov, for his help,
guidance, and support throughout the past year.

Key words and phrases: Schwartz distributions, multiplication of Schwartz
distributions, Colombeau’s algebra of generalized functions, non-standard
analysis, saturation principle, conservation law, Hopf equation, weak
solution, shock wave.
AMS Subject Classification: 26E35, 30G06, 46F10, 46F30, 46S10, 46S20, 35D05,
35L67, 35L65.

###### Contents

  1. 1 Introduction
  2. 2 Ordered Fields
  3. 3 Filters and Ultrafilters
  4. 4 Ultrafilter on $\mathcal{D}(\mathbb{R}^{d})$
  5. 5 Non-Standard Numbers
  6. 6 Internal Sets
  7. 7 Saturation Principle in ${}^{*}\mathbb{C}$
  8. 8 Non-Standard Smooth Functions
  9. 9 Internal Sets and Saturation Principle in ${}^{*}\mathcal{E}(\mathbb{R}^{d})$
  10. 10 Weak Equality
  11. 11 Schwartz Distributions
  12. 12 Embedding of Schwartz Distributions in ${}^{*}\mathcal{E}(\mathbb{R}^{d})$
  13. 13 Conservation Laws in ${}^{*}\mathcal{E}(\Omega)$ and the Hopf Equation
  14. 14 Generalized Delta-like Solution of the Hopf Equation

##  1 Introduction

In what follows
$\mathcal{E}(\mathbb{R}^{d})=\mathcal{C}^{\infty}(\mathbb{R}^{d})$ denotes the
class of $\mathcal{C}^{\infty}$-functions on $\mathbb{R}^{d}$. Also
$\mathcal{D}(\mathbb{R}^{d})=\mathcal{C}_{0}^{\infty}(\mathbb{R}^{d})$ denotes
the class of test functions on $\mathbb{R}^{d}$ and
$\mathcal{D}^{\prime}(\mathbb{R}^{d})$ stands for the space of Schwartz
distributions (Schwartz generalized functions) on $\mathbb{R}^{d}$ (H.
Bremermann [1]).

The algebra of generalized functions ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ is
a particular non-standard extension of the class
$\mathcal{E}(\mathbb{R}^{d})$. The field of the scalars ${}^{*}\mathbb{C}$ of
the algebra ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ is a particular non-standard
extension of the field of complex numbers $\mathbb{C}$ and the field of the
real scalars ${}^{*}\mathbb{R}$ is a non-standard extension of $\mathbb{R}$.
That means that both ${}^{*}\mathbb{C}$ and ${}^{*}\mathbb{R}$ are non-
Archimedean fields containing non-zero infinitesimals, i.e. generalized
numbers $h$ such that $0<|h|<1/n$ for all $n\in\mathbb{N}$. Since the
involvement of non-Archimedean fields in applied mathematics is somewhat
unusual, we start with a summary of the relevant definitions and results in
the theory of ordered fields and non-Archimedean fields (Section 2).

In Sections 3-4 we present the basic facts of the theory of free filters and
ultrafilters (C. C. Chang and H. J. Keisler [2]). We construct a particular
ultrafilter on the space of test functions $\mathcal{D}(\mathbb{R}^{d})$ which
is important for the embedding of Schwartz distributions in the algebra
${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$.

In Sections 5-6 we present the construction of the fields of the complex and
real non-standard numbers ${}^{*}\mathbb{C}$ and ${}^{*}\mathbb{R}$. In
Section 7 we prove the Saturation Principle in ${}^{*}\mathbb{C}$ which plays
a role in non-standard analysis similar to the role of the completeness of
$\mathbb{R}$ and $\mathbb{C}$ in usual (standard) analysis. These sections
might be viewed as an introduction to non-standard analysis (A. Robinson
[12]). We should note that our exposition of non-standard analysis does not
require any background in mathematical logic or model theory.

The construction of the algebra ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ is
presented in Section 8; in short, ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ is a
differential associative commutative algebra of generalized functions similar
to (but much larger than) the class
$\mathcal{E}(\mathbb{R}^{d})=\mathcal{C}^{\infty}(\mathbb{R}^{d})$. In Section
9 we state the Saturation Principle for ${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$,
playing the role of the completeness property.

In Section 12 we construct the chain of embeddings
$\mathcal{E}(\mathbb{R}^{d})\subset\mathcal{D}^{\prime}(\mathbb{R}^{d})\subset{{}^{*}\mathcal{E}(\mathbb{R}^{d})}$.
These embeddings presents a solution of the problem of multiplication of
Schwartz distributions similar to but different from Colombeau’s solution of
the same problem (J.F. Colombeau [3]). The problem of multiplication of
Schwartz distributions has an interesting and dramatic history. Soon after the
distribution theory was invented by L. Schwartz, he proved that the space of
distributions $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ cannot be supplied with
an associative and commutative product that reproduces the usual product in
the spaces $\mathcal{C}^{k}(\mathbb{R}^{d}),k=0,1,2,\dots$. This negative
result, known as Schwartz Impossibilities Result (L. Schwartz [14]), was the
reason this problem was considered for a long time as unsolvable. In the late
1980’s Jean F. Colombeau offered a solution of the problem of multiplication
of distributions by constructing an algebra of generalized functions
$\mathcal{G}(\mathbb{R}^{d})$ with the chain of algebraic embeddings
$\mathcal{E}(\mathbb{R}^{d})\subset\mathcal{D}^{\prime}(\mathbb{R}^{d})\subset\mathcal{G}(\mathbb{R}^{d})$
thus avoiding Schwartz Impossibilities Result (since $k=\infty$). One
(slightly disturbing) feature of Colombeau’s solution is that the set of
scalars $\overline{\mathbb{C}}$ of the algebra $\mathcal{G}(\mathbb{R}^{d})$
is a ring with zero divisors, not a field as any set of scalars should be. In
this respect our solution of the problem of multiplication of Schwartz
distributions presents an important improvement of Colombeau’s theory: the set
of scalars ${}^{*}\mathbb{C}$ of the algebra
${}^{*}\mathcal{E}(\mathbb{R}^{d})$ is an algebraically complete
$c^{+}$-saturated field (Section 7). As a consequence, the set of the real
scalars ${}^{*}\mathbb{R}$ is a real closed Cantor complete field. We should
notice that the fact that ${}^{*}\mathcal{E}(\mathbb{R}^{d})$ is a
differential algebra (not merely a linear space) is important for our goals in
applied mathematics, in particular, for studying generalized solutions of non-
linear partial differential equations such as shock-wave and delta-like
solutions. Notice that these are the solutions after the formation of the
shock in many conservation law type equations.

In Section 14 we prove the existence of a weak delta-like solution of the Hopf
equation $u_{t}(x,t)+u(x,t)u_{x}(x,t)=0$ in the framework of
${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$. This solution has counterparts neither
in the spaces of classical functions such as
$\mathcal{C}^{k}(\mathbb{R}^{d}),\;k=1,2,\dots,\infty$, nor in the spaces of
Schwartz distributions such as $\mathcal{D}^{\prime}(\mathbb{R}^{d})$. Our
result improves a similar result by M. Radyna [11] obtained in the spirit of
perturbation theory.

##  2 Ordered Fields

We will begin by defining ordered fields and giving examples of some well
known and some lesser known orderings on the (non-archimedean) fields of
rational functions and Laurent series.

###### Definition 2.1

Let $\mathbb{K}$ be a field (ring). $\mathbb{K}$ is called orderable if there
exists a nonempty subset $\mathbb{K}_{+}\subset\mathbb{K}$ such that

  * (1)

$0\notin\mathbb{K}_{+}$

  * (2)

$x,y\in\mathbb{K}_{+}\Longrightarrow x+y,xy\in\mathbb{K}_{+}$

  * (3)

For every non-zero $x\in\mathbb{K}$, either $x\in\mathbb{K}_{+}$ or
$-x\in\mathbb{K}_{+}$

$\mathbb{K}_{+}$ generates an order relation $<_{\mathbb{K}}$ on $\mathbb{K}$
as follows: $x<_{\mathbb{K}}y$ i f f  $y-x\in\mathbb{K}_{+}.$

$(\mathbb{K},<_{\mathbb{K}})$ is called a _totally ordered field_ or simply an
_ordered field_.

We will write $<$ instead of $<_{\mathbb{K}}$ when it is clear from context
which field’s order relation we are referring to.

In addition to $(\mathbb{R},<)$ (where $<$ is the usual order on $\mathbb{R}$)
there are many (more interesting) examples of ordered fields. But first let us
make a short detour:

###### Example 2.1

$\mathbb{C}$, the set of complex numbers, is not orderable.

_Proof_ Suppose there exists a subset $\mathbb{C}_{+}$ satisfying Definition
2.1. Then consider:
Case 1 Suppose $i\in\mathbb{C}_{+}$. Then $i\cdot i=-1\in\mathbb{C}_{+}$,
implying $(-1)\cdot(-1)=1\in\mathbb{C}_{+}$. This is impossible since
$-1+1=0\notin\mathbb{C}_{+}$.
Case 2 Suppose $i\notin\mathbb{C}_{+}$. Then $-i\in\mathbb{C}_{+}$, implying
$(-i)\cdot(-i)=-1\in\mathbb{C}_{+}$, leading to the same contradiction as in
Case 1. $\blacktriangle$

The previous example can be generalized as follows:

###### Theorem 2.1

A field $\mathbb{K}$ is orderable  i f f it is formally real. This means that
for every $n\in\mathbb{N}$ and every $x_{k}\in\mathbb{K}$

$\sum_{k=1}^{n}x_{k}^{2}=0\quad implies\quad x_{k}=0\quad for\quad all\quad
k.$

For details on the subject of formally real fields and the proof of this
theorem, see (Van Der Waerden [18], Chapter 11).

###### Definition 2.2

Let $\mathbb{K}$ and $\mathbb{L}$ be ordered fields. If
$\varphi:\;\mathbb{K}\longrightarrow\mathbb{L}$ is a field homomorphism that
preserves order, i.e. $x<_{\mathbb{K}}y$ implies
$\varphi(x)<_{\mathbb{L}}\varphi(y)$, then $\varphi$ is said to be an ordered
field homomorphism.
Ordered field isomorphisms and ordered field embeddings are defined similarly.

###### Remark 2.1

There exists an ordered field embedding from $\mathbb{Q}$ into any ordered
field $\mathbb{K}$. We call it the canonical embedding of $\mathbb{Q}$ into
$\mathbb{K}$ and it is defined by: $\sigma(0)=0$, $\sigma(n)=n\cdot 1$ and
$\sigma(-n)=-\sigma(n)$ for $n\in\mathbb{N}$, and
$\sigma(p/q)=\sigma(p)/\sigma(q)$ for $p,\,q\in\mathbb{Z},\,q\neq 0$.

From now on, if $x\in\mathbb{N},\mathbb{Z},$ or $\mathbb{Q}$, we will refer to
$x$ and $\sigma(x)\in\mathbb{K}$ interchangeably.

###### Example 2.2

Let $(\mathbb{R},<)$ be the field of real numbers with the usual order, and
let $\mathbb{R}(x)$ be the set of rational functions in the variable $x$ with
coefficients in $\mathbb{R}$. Note that we may think of $\mathbb{R}$ as a
subfield of $\mathbb{R}(x)$, as represented by the constant functions. Then
define $\mathbb{R}(x)_{+}=\\{R(x):R(x)\in\mathbb{R}(x)$ and there exists
$x_{0}\in\mathbb{R}$ such that $R(x)>0$ whenever $x>x_{0}\\}.$
The ordered field generated by $\mathbb{R}(x)_{+}$, which we will refer to
simply as $\mathbb{R}(x)$, has some surprising properties. Namely:

  * (i)

$\mathbb{R}(x)$ contains infinitely large elements like $f(x)=x$. This means
that $f(x)>n$ for all $n\in\mathbb{N}$ (let $x_{0}=n$).

  * (ii)

$\mathbb{R}(x)$ contains positive infinitesimals like $g(x)=\frac{1}{x}$. This
means that $0<g(x)<\frac{1}{n}$ for all $n\in\mathbb{N}$ (let $x_{0}=n$).

###### Remark 2.2

Let $\mathbb{L}$ be an ordered integral domain (an ordered ring without zero
divisors) and $\mathbb{K}$ be the field of fractions of $\mathbb{L}$. Define

$\mathbb{K}_{+}=\\{\frac{x}{y}:x,y\in\mathbb{L}_{+}\;or\;-x,-y\in\mathbb{L}_{+}\\}$

The order generated by $\mathbb{K}_{+}$ is the only one which extends the
order in $\mathbb{L}$. It is said to be the order inherited from $\mathbb{L}$.

###### Example 2.3

With Remark (2.2) in mind, we may revisit Example (2.2).
If $\mathbb{R}[x]$ is the ring of polynomials over $\mathbb{R}$, we may define
$\mathbb{R}[x]_{+}=\\{P(x)\in\mathbb{R}[x]$: $lead(P)>0\\}$, where $lead(P)$
is the leading coefficient of $P(x)$.
The order generated on $\mathbb{R}[x]$ by $\mathbb{R}[x]_{+}$ can be extended
to $\mathbb{R}(x)$ since $\mathbb{R}(x)$ is the field of fractions of
$\mathbb{R}[x]$. That is, we may redefine
$\mathbb{R}(x)_{+}=\\{\frac{P(x)}{Q(x)}:P(x),Q(x)\in\mathbb{R}[x]\;and\;lead(P),lead(Q)>0\;or\;lead(P),lead(Q)<0\\}$
This definition is equivalent to that given previously and the orders
generated by the two are in fact one and the same. Now we may plainly see that
$f(x)=x$ is indeed an infinitely large element since $lead(x-n)=1>0$ for all
$n\in\mathbb{N}$.
Similarly, $g(x)=\frac{1}{x}$ is a positive infinitesimal because
$\frac{1}{n}-\frac{1}{x}=\frac{x-n}{nx}$ and $lead(x-n),lead(nx)>0$ for all
$n\in\mathbb{N}$.

###### Example 2.4

The set

$\mathbb{R}(x^{\mathbb{Z}})=\\{\sum_{n=m}^{\infty}\frac{a_{n}}{x^{n}}:\quad
a_{n}\in\mathbb{R}\quad,\quad m\in\mathbb{Z},\quad and\quad a_{m}\neq 0\\}$

of Laurent series with coefficients in $\mathbb{R}$ is a field under normal
polynomial addition and multiplication.
We may define an order on $\mathbb{R}(x^{\mathbb{Z}})$ by

$\mathbb{R}_{+}(x^{\mathbb{Z}})=\\{\sum_{n=m}^{\infty}\frac{a_{n}}{x^{n}}\in\mathbb{R}(x^{\mathbb{Z}}):\quad
a_{m}>0\\}$

Here, an element such as

$\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}+\ldots$

is infinitesimal.

###### Example 2.5

The field of (formal) Laurent Series may also be defined as follows:

$\mathbb{R}(x^{\mathbb{Z}})=\\{\sum_{n=m}^{\infty}a_{n}{x^{n}}:\quad
a_{n}\in\mathbb{R},\quad m\in\mathbb{Z},\quad and\quad a_{m}\neq 0\\}$

If we now let

$\mathbb{R}_{+}(x^{\mathbb{Z}})=\\{\sum_{n=m}^{\infty}a_{n}{x^{n}}\in\mathbb{R}(x^{\mathbb{Z}}):\quad
a_{m}>0\\}$

then even a series that is divergent for all x, such as

$x+2x^{2}+6x^{3}+\ldots+n!x^{n}+\ldots$

is an infinitesimal in this field.

##  3 Filters and Ultrafilters

In this section we define and give examples of filters on an arbitrary
infinite set. Having done so, we will prove the existence of an ultrafilter
using the axiom of choice.

###### Definition 3.1

Let $I$ be an infinite set and let
$\mathcal{F}\subset\mathcal{P}(I),\;\mathcal{F}\neq\varnothing$. If
$\mathcal{F}$ satisfies:

  * (F1)

If $A\in\mathcal{F}$ and $A\subset B\subset I$, then $B\in\mathcal{F}$.

  * (F2)

$A,\,B\in\mathcal{F}$ implies $A\cap B\in\mathcal{F}$

  * (F3)

$\varnothing\notin\mathcal{F}$

then $\mathcal{F}$ is a filter on $I$. If it is also true that

  * (F4)

$\bigcap_{A\in\mathcal{F}}A=\varnothing$

then $\mathcal{F}$ is called a free filter on $I$. A filter $\mathcal{F}$ is
called countably incomplete if:

  * (F5)

There exists a sequence of decreasing sets $I=I_{0}\supset I_{1}\supset
I_{2}\supset\ldots$ in $\mathcal{F}$ such that
$\bigcap_{n=0}^{\infty}I_{n}=\varnothing$.

If $\mathcal{F}$ is a filter on $I$, it follows immediately from the
definition that:

  * (i)

$I\in\mathcal{F}$

  * (ii)

$\mathcal{F}$ is closed under finite intersections.

  * (iii)

If $A\in\mathcal{F}$ then $I\setminus A\notin\mathcal{F}$

  * (iv)

If $\mathcal{F}$ is countably incomplete, then $\mathcal{F}$ is free.

###### Definition 3.2

A filter $\mathcal{U}$ on a set $I$ is called an ultrafilter if for every
filter $\mathcal{F}$ on $I$, $\mathcal{U}$ is not a proper subset of
$\mathcal{F}$. That is, there is no filter $\mathcal{F}$ on $I$ that properly
contains $\mathcal{U}$.

###### Theorem 3.1

Let $\mathcal{U}$ be a filter on $I$. Then $\mathcal{U}$ is an ultrafilter on
$I$ i f f for every $A\subset I$ either $A\in\mathcal{U}$ or $I\setminus
A\in\mathcal{U}$.

_Proof_ Suppose $\mathcal{U}$ is an ultrafilter on $I$ and $A,I\setminus
A\notin\mathcal{U}$. Let $\widehat{\mathcal{U}}=\\{X\;:\;A\cup
X\in\mathcal{U}\\}$. It is not hard to check that $\widehat{\mathcal{U}}$ is a
filter that properly contains $\mathcal{U}$, since $I\setminus
A\in\widehat{\mathcal{U}}$. Thus $\mathcal{U}$ cannot be an ultrafilter. To
prove the other direction, suppose, to the contrary, that $\mathcal{U}$ is not
an ultrafilter. Then there exists a filter $\mathcal{V}$ which is a proper
extension of $\mathcal{U}$. Let $A\in\mathcal{V}\setminus\mathcal{U}$. Then,
since $\mathcal{U}\subset\mathcal{V}$, we have that $A\cap X\neq\varnothing$
for all $X\in\mathcal{U}$. But $I\setminus A\in\mathcal{U}$ by assumption, so
$A\cap\left(I\setminus A\right)\neq\varnothing$, a contradiction.
$\blacktriangle$.

###### Example 3.1

Let $I=\mathbb{N}$, and fix $n\in\mathbb{N}$. Then
$\mathcal{U}=\\{X\;:\;X\subset\mathbb{N},\;n\in X\\}$ is an ultrafilter on
$\mathbb{N}$. However, $\mathcal{U}$ is clearly not free.

###### Example 3.2

The filter $\mathcal{F}_{r}(\mathbb{N})$ consisting of all cofinite sets of
natural numbers is called the Fréchet filter on $\mathbb{N}$. It is free and
countably incomplete since
$\bigcap_{n=1}^{\infty}\left(\mathbb{N}\setminus\\{n\\}\right)=\varnothing$
However, since neither the set of even numbers nor the set of odd numbers is
in $\mathcal{F}_{r}(\mathbb{N})$, by Theorem (3.1), it is not an ultrafilter.

###### Theorem 3.2

Let $I$ be an infinite set and $\mathcal{F}$ a free filter on $I$. Then there
exists a free ultrafilter $\mathcal{U}$ on $I$ such that
$\mathcal{F}\subseteq\mathcal{U}$.

_Proof_ Let $\widehat{\mathcal{F}}$ be the set of all filters on $I$ that
contain $\mathcal{F}$. $\widehat{\mathcal{F}}\neq\varnothing$ since
$\mathcal{F}\in\widehat{\mathcal{F}}$. Let $\widehat{\mathcal{F}}$ be ordered
by set inclusion, and consider a linearly ordered subset
$\mathcal{M}\subseteq\widehat{\mathcal{F}}$. Define
$\widehat{\mathcal{M}}=\bigcup_{M\in\mathcal{M}}M$. Note that if
$A\in\widehat{\mathcal{M}}$ then $A\in M$ for some $M\in\mathcal{M}$. Thus, if
$A\subset B\subset I$, it follows that $B\in M$, implying
$B\in\widehat{\mathcal{M}}$. Also, if $A,B\in\widehat{\mathcal{M}}$, then we
must have that $A\in M_{1}$ and $B\in M_{2}$ for some
$M_{1},M_{2}\in\mathcal{M}$. Since $\mathcal{M}$ is linearly ordered, we may
assume without loss of generality that $M_{1}\subset M_{2}$. Thus $A,B\in
M_{2}$. Hence $A\cap B\in M_{2}$, implying $A\cap B\in\widehat{\mathcal{M}}$.
Finally, we have that $\varnothing\notin\widehat{\mathcal{M}}$ because
otherwise $\varnothing$ would be an element of some filter $M\in\mathcal{M}$,
which is impossible. We have just shown that $\widehat{\mathcal{M}}$ is itself
a filter. But since the choice of $\mathcal{M}$ was arbitrary, we can conclude
that every linearly ordered subset of $\widehat{\mathcal{F}}$ has an upper
bound in $\widehat{\mathcal{F}}$. Thus, by Zorn’s Lemma,
$\widehat{\mathcal{F}}$ has a maximal element, $\mathcal{U}$, which is an
ultrafilter on $I$ containing $\mathcal{F}$. Also, since
$\mathcal{F}\subset\mathcal{U}$ and $\mathcal{F}$ is free, we have that
$\bigcap_{A\in\mathcal{U}}A\subseteq\bigcap_{A\in\mathcal{F}}A=\varnothing$.
Therefore $\mathcal{U}$ is free. $\blacktriangle$.

##  4 Ultrafilter on $\mathcal{D}(\mathbb{R}^{d})$

Here we define an ultrafilter on $\mathcal{D}(\mathbb{R}^{d})$, the set of
test functions, in order to construct ordered, non-archimedean fields of non-
standard real and complex numbers, ${}^{*}\mathbb{R}$ and ${}^{*}\mathbb{C}$,
respectively.

###### Definition 4.1

Let $\mathcal{D}(\mathbb{R}^{d})$ be the set of test functions on
$\mathbb{R}^{d}$. That is,
$\mathcal{D}(\mathbb{R}^{d})=\mathcal{C}^{\infty}_{0}(\mathbb{R}^{d})$. For
every $n\in\mathbb{N}$, define the basic set $\mathcal{B}_{n}$ by

$\displaystyle\mathcal{B}_{n}=\\{$
$\displaystyle\varphi\in\mathcal{D}(\mathbb{R}^{d})\,:\ $
$\displaystyle\varphi\text{ is real-valued and symmetric},$
$\displaystyle\varphi(x)=0\text{ for all }x\in\mathbb{R}^{d},\|x\|\geq 1/n,$
$\displaystyle\int\varphi=1$ $\displaystyle\int x^{\alpha}\varphi=0\text{ for
all }\alpha\in\mathbb{N}^{d}_{0},1\leq|\alpha|\leq n,$ $\displaystyle
1\leq\int|\varphi|<1+\frac{1}{n}\\}$

and $\mathcal{B}_{0}=\mathcal{D}(\mathbb{R}^{d})$.

###### Theorem 4.1

  * (i)

$\mathcal{B}_{n}\neq\varnothing$ for all n.

  * (ii)

$\mathcal{B}_{0}\supset\mathcal{B}_{1}\supset\mathcal{B}_{2}\supset\ldots$

  * (iii)

$\bigcap_{n}\mathcal{B}_{n}=\varnothing$

_Proof_ For the proof of (i), see (Oberguggenberger and Todorov [10]). (ii)
follows from Definition (4.1). For (iii), suppose there were a function
$\varphi$ such that $\varphi\in\bigcap_{n}\mathcal{B}_{n}$ for all $n$. Then
consider $\widehat{\varphi}(\xi)=\int\varphi(x)e^{i\xi x}dx$, the Fourier
transform of $\varphi$. Since $\varphi\in\mathcal{D}(\mathbb{R}^{d})$,
$\widehat{\varphi}$ is entire (_Bremermann_ [1] Lemma 8.11, p.85). Therefore,
we can write
$\widehat{\varphi}(\xi)=\sum_{\alpha\in\mathbb{N}_{0}^{d}}\frac{(\partial^{\alpha}\widehat{\varphi})(0)}{\alpha!}\xi^{\alpha}$.
But $0=i^{|\alpha|}\int x^{\alpha}\varphi(x)dx=\left.i^{|\alpha|}\int
x^{\alpha}\varphi(x)e^{i\xi
x}dx\right|_{\xi=0}=(\partial^{\alpha}\widehat{\varphi})(0)$ for all
$\alpha\neq 0$. It follows that $\widehat{\varphi}$ is constant. However, by
the same lemma as before, we also have that
$\lim_{|\xi|\to\infty}\widehat{\varphi}(\xi)=0$. Thus $\widehat{\varphi}=0$,
implying $\varphi(x)=0$. This contradicts the property that $\int\varphi=1$.
Hence $\bigcap_{n}\mathcal{B}_{n}=\varnothing$. $\blacktriangle$.

###### Definition 4.2

Define the basic filter $\mathcal{F}_{\mathcal{B}}$ on
$\mathcal{D}(\mathbb{R}^{d})$ by

$\mathcal{F}_{\mathcal{B}}=\\{\Phi\subseteq\mathcal{D}(\mathbb{R}^{d})\;:\;\mathcal{B}_{n}\subseteq\Phi\text{
for some }n\in\mathbb{N}\\}.$

Since each $\mathcal{B}_{n}$ is itself an element of
$\mathcal{F}_{\mathcal{B}}$, it follows from Theorem (4.1) that
$\mathcal{F}_{\mathcal{B}}$ is countably incomplete, and therefore free. Thus,
by Theorem (3.2), there exists an ultrafilter $\mathcal{U}$ on
$\mathcal{D}(\mathbb{R}^{d})$ containing $\mathcal{F}_{\mathcal{B}}$. We shall
keep $\mathcal{U}$ fixed in what follows.

##  5 Non-Standard Numbers

We will now use the ultrafilter defined in the previous section to construct
fields of non-standard real and complex numbers.

###### Definition 5.1

Let $\mathcal{U}$ be as before, and let
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$ be the ring of functionals from
$\mathcal{D}(\mathbb{R}^{d})$ to $\mathbb{C}$ supplied with pointwise addition
and multiplication. We shall denote these functionals as “families”
$(A_{\varphi})$ and treat the domain $\mathcal{D}(\mathbb{R}^{d})$ as an
“index set”.
We may define the operations of absolute value, real part mapping, imaginary
part mapping, and complex conjugation on the elements of
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$ by:

$\displaystyle|(A_{\varphi})|=(|A_{\varphi}|)$
$\displaystyle\Re(A_{\varphi})=(\Re A_{\varphi})$
$\displaystyle\Im(A_{\varphi})=(\Im A_{\varphi})$
$\displaystyle\overline{(A_{\varphi})}=(\overline{A_{\varphi}})$

Also, we may define an embedding of $\mathbb{C}$ into
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$ by $c\rightarrow(C_{\varphi})$
where $C_{\varphi}=c$ for all $\varphi\in\mathcal{D}(\mathbb{R}^{d})$.
Define an equivalence relation $\sim_{\mathcal{U}}$ on
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$ by

$(A_{\varphi})\sim_{\mathcal{U}}(B_{\varphi})\;\;if\;\;\\{\varphi\in\mathcal{D}(\mathbb{R}^{d}):A_{\varphi}=B_{\varphi}\\}\in\mathcal{U}$

Finally, let
${}^{*}\mathbb{C}=\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}/\sim_{\mathcal{U}}$.
That is, ${}^{*}\mathbb{C}$ consists of equivalence classes of functionals in
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$. We may write
$\langle(A_{\varphi})\rangle$ to represent these classes, but to simplify
notation we will denote by $\langle A_{\varphi}\rangle\in$ ${}^{*}\mathbb{C}$
the non-standard number (equivalence class of functionals) with representative
$(A_{\varphi})$. ${}^{*}\mathbb{C}$ is called a field of complex non-standard
numbers.
${}^{*}\mathbb{C}$ inherits the operations and embedding mentioned above from
$\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$. With the embedding in mind, we
shall treat elements of $\mathbb{C}$ as their images in ${}^{*}\mathbb{C}$.
A non-standard number $\langle A_{\varphi}\rangle$ is called real if

$\\{\varphi\in\mathcal{D}(\mathbb{R}^{d}):A_{\varphi}\in\mathbb{R}\\}\in\mathcal{U}$

We denote the set of all real non-standard numbers by ${}^{*}\mathbb{R}$ and
supply it with an order relation as follows:

$\langle A_{\varphi}\rangle>_{{}^{*}\mathbb{R}}0\;\text{ if
}\;\\{\varphi\in\mathcal{D}(\mathbb{R}^{d}):A_{\varphi}>0\\}\in\mathcal{U}$

###### Theorem 5.1

  * (i)

Every number $\gamma\in\ ^{*}\mathbb{C}$ can be uniquely represented in the
form $\gamma=\alpha+\beta i$ where $\alpha,\beta\in$ ${}^{*}\mathbb{R}$ and
$\alpha=\Re\gamma$, $\beta=\Im\gamma$, and
$|\gamma|=\sqrt{\alpha^{2}+\beta^{2}}$.

  * (ii)

${}^{*}\mathbb{C}$ is an algebraically closed non-Archimedean field of
characteristic zero. $\mathbb{C}$ is a subfield of ${}^{*}\mathbb{C}$.

  * (iii)

${}^{*}\mathbb{R}$ is a totally ordered non-Archimedean real closed field.
Moreover, $\alpha>0$ in ${}^{*}\mathbb{R}$ i f f $\alpha=\beta^{2}$ for some
$\beta\in$ ${}^{*}\mathbb{R}$, $\beta\neq 0$. $\mathbb{R}$ is an ordered
subfield of ${}^{*}\mathbb{R}$.

_Proof_ (i) Let $\gamma\in\,^{*}\mathbb{C}$. Then $\gamma=\langle
C_{\varphi}\rangle$ for some
$(C_{\varphi})\in\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$. But for each
$\varphi$, $C_{\varphi}=A_{\varphi}+B_{\varphi}i$, where $A_{\varphi}=\Re
C_{\varphi}$ and $B_{\varphi}=\Im C_{\varphi}$. Thus $\langle
C_{\varphi}\rangle=\langle A_{\varphi}\rangle+\langle B_{\varphi}\rangle i$.
To prove uniqueness, suppose that $\langle C_{\varphi}\rangle=\langle
D_{\varphi}\rangle+\langle E_{\varphi}\rangle i$ also. Then

$\\{\varphi:\Re C_{\varphi}=D_{\varphi}\\}\cap\\{\varphi:\Re
C_{\varphi}=A_{\varphi}\\}=\\{\varphi:A_{\varphi}=D_{\varphi}\\}\in\mathcal{U}$

because $\mathcal{U}$ is closed under intersections. Therefore $\langle
A_{\varphi}\rangle=\langle D_{\varphi}\rangle$.
The same argument can be applied to show that $\langle
B_{\varphi}\rangle=\langle E_{\varphi}\rangle$.
The proof for $|\gamma|$ is similar.

(ii) It is not hard to check that $\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$
really is a ring, and that $\sim_{\mathcal{U}}$ really is an equivalence
relation. It follows that ${}^{*}\mathbb{C}$ is a (commutative) ring. To prove
that ${}^{*}\mathbb{C}$ is a field, we must show that each non-zero element
has a multiplicative inverse. For any non-zero $\gamma\in\,^{*}\mathbb{C}$, we
may choose a representative $(C_{\varphi})$ such that $C_{\varphi}\neq 0$ for
all $\varphi$. Let $D_{\varphi}=1/C_{\varphi}$ and $\delta=\langle
D_{\varphi}\rangle$. Then $\delta\gamma=\langle 1\rangle$.
Let

$P(x)=\sum_{k=0}^{n}\alpha_{k}x^{k},\;\alpha_{k}\in\,^{*}\mathbb{C}\text{ for
all k }$

be a polynomial in ${}^{*}\mathbb{C}[x]$. Define

$P_{\varphi}(x)=\sum_{k=0}^{n}A_{k,\varphi}x^{k}$

where $\alpha_{k}=\langle A_{k,\varphi}\rangle$ for each k. Since each
$P_{\varphi}(x)$ is a polynomial over $\mathbb{C}$, there exists a number
$C_{\varphi}\in\mathbb{C}$ such that $P_{\varphi}(C_{\varphi})=0$. If we let
$\gamma=\langle C_{\varphi}\rangle$, it follows that $P(\gamma)=0$ in
${}^{*}\mathbb{C}$.
That $\mathbb{C}$ is a subfield of ${}^{*}\mathbb{C}$ is clear from the
embedding.

(iii) The trichotomy of the order relation on ${}^{*}\mathbb{R}$ follows from
the trichotomy of the order relation on $\mathbb{R}$. For suppose
$\mathcal{A}=\\{\varphi:A_{\varphi}<B_{\varphi}\\}$,
$\mathcal{B}=\\{\varphi:A_{\varphi}=B_{\varphi}\\}$, and
$\mathcal{C}=\\{\varphi:A_{\varphi}>B_{\varphi}\\}$, for some non-standard
real numbers $\langle A_{\varphi}\rangle$, $\langle B_{\varphi}\rangle$. Note
that $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are mutually disjoint.
Therefore, at most one of $\mathcal{A}$, $\mathcal{B}$, or $\mathcal{C}$ can
be in $\mathcal{U}$. Also,
$\mathcal{A}\cup\mathcal{B}\cup\mathcal{C}=\mathcal{D}(\mathbb{R}^{d})\in\mathcal{U}$.
We can use this to prove that one of $\mathcal{A}$, $\mathcal{B}$, or
$\mathcal{C}$ must be in $\mathcal{U}$. For suppose that none of
$\mathcal{A}$, $\mathcal{B}$, or $\mathcal{C}$ is in $\mathcal{U}$. Then by
Theorem (3.1), $\mathcal{B}\cup\mathcal{C}\in\mathcal{U}$ and
$\mathcal{A}\cup\mathcal{C}\in\mathcal{U}$. Taking the intersection of these
two sets, we would have $\mathcal{C}\in\mathcal{U}$, a contradiction.
$\blacktriangle$

###### Definition 5.2

Define the sets of infinitesimal, finite, and infinitely large numbers as
follows:

$\mathcal{I}(^{*}\mathbb{C})=\\{x\in\,^{*}\mathbb{C}:|x|<1/n\text{ for all
}n\in\mathbb{N}\\}$
$\mathcal{F}(^{*}\mathbb{C})=\\{x\in\,^{*}\mathbb{C}:|x|<n\text{ for some
}n\in\mathbb{N}\\}$
$\mathcal{L}(^{*}\mathbb{C})=\\{x\in\,^{*}\mathbb{C}:|x|>n\text{ for all
}n\in\mathbb{N}\\}$

It is not hard to prove that $\mathcal{F}(^{*}\mathbb{C})$ is a subring of
${}^{*}\mathbb{C}$ and $\mathcal{I}(^{*}\mathbb{C})$ is a maximal ideal in
$\mathcal{F}(^{*}\mathbb{C})$.

###### Example 5.1

Define $(R_{\varphi})\in\mathbb{C}^{\mathcal{D}(\mathbb{R}^{d})}$ by

$R_{\varphi}=\sup\\{\|x\|:x\in{\rm{supp}}\varphi\\}$

where ${\rm{supp}}\varphi=\overline{\\{x\in\mathbb{R}^{d}:\varphi(x)\neq
0\\}}$ is the support of $\varphi$. The non-standard number $\rho=\langle
R_{\varphi}\rangle$ is a (positive) infinitesimal. For let
$\mathcal{A}=\\{\varphi:0<R_{\varphi}<1/n\\}$. Then for any
$\varphi\in\mathcal{B}_{n+1}$, we have $\varphi\in\mathcal{A}$, by the
definition of $\mathcal{B}_{n+1}$. Thus $\mathcal{B}_{n+1}\subset\mathcal{A}$,
implying $\mathcal{A}\in\mathcal{U}$. $\rho$ is called the canonical
infinitesimal in ${}^{*}\mathbb{C}$.

###### Definition 5.3

Define the standard part mapping $st:$
${}^{*}\mathbb{R}\rightarrow\mathbb{R}\cup\\{\pm\infty\\}$ by

$st(x)=\begin{cases}\sup\\{r\in\mathbb{R}:r<x\\}&\text{if
$x\in\mathcal{F}(^{*}\mathbb{R})$}\\\ \infty&\text{if
$x\in\mathcal{L}(^{*}\mathbb{R}_{+})$}\\\ -\infty&\text{if
$x\in\mathcal{L}(^{*}\mathbb{R}_{-})$}\end{cases}$

We may extend this definition to ${}^{*}\mathbb{C}$ by
$st(x+yi)=st(x)+st(y)i$.

###### Theorem 5.2

If $x\in\mathcal{F}(^{*}\mathbb{C})$ then $x$ has a unique asymptotic
expansion: $x=r+dx$ where $r\in\mathbb{C}$ and
$dx\in\mathcal{I}(^{*}\mathbb{C})$. In fact, $r=st(x)$.

_Proof_ We will prove the case for $x\in\mathcal{F}(^{*}\mathbb{R})$. The
general result will follow. Let $x\in\mathcal{F}(^{*}\mathbb{R})$. First note
that $x-st(x)\in\mathcal{I}(^{*}\mathbb{R})$, for otherwise we would have
$|x-st(x)|>1/n$ for some n, implying either that $st(x)>x$ or that
$st(x)+1/2n<x$. In either case, this is a contradiction to Definition (5.3).
To prove uniqueness, suppose that $x=r+dx$ and $x=s+dy$ are two expansions of
$x$. Then we would have $r-s=dx-dy$, implying that
$r-s\in\mathcal{I}(^{*}\mathbb{R})$. But since $r-s\in\mathbb{R}$, $r-s=0$.
Hence $r=s$. Therefore $r+dx=r+dy$, implying $dx=dy$. $\blacktriangle$

##  6 Internal Sets

In non-standard analysis, internal sets play the role of the “good” sets, in a
similar way to the measurable sets in Lebesgue theory.
In what follows we will use the abbreviation a.e. to mean that the set of
functions for which some statement is true is in $\mathcal{U}$.

###### Definition 6.1

Let $\mathbb{A}\subseteq\mathbb{C}$. The non-standard extension of
$\mathbb{A}$ is

${}^{*}\mathbb{A}=\\{\langle
A_{\varphi}\rangle\in\,^{*}\mathbb{C}:A_{\varphi}\in\mathbb{A}\;a.e.\\}$

A set $\mathcal{A}$ of non-standard numbers is called internal standard if it
is the non-standard extension of some subset of $\mathbb{C}$. The set of all
internal standard sets is denoted by ${}^{\sigma}\mathcal{P}(\mathbb{C})$.

###### Example 6.1

The non-standard extensions of the intervals $(a,\,b)$, $[a,b]$, $(a,\infty)$,
etc. are

${}^{*}(a,b)=\\{x\in\ ^{*}\mathbb{R}:\,a<x<b\\}$ ${}^{*}[a,b]=\\{x\in\
^{*}\mathbb{R}:\,a\leq x\leq b\\}$ ${}^{*}(a,\infty)=\\{x\in\
^{*}\mathbb{R}:\,a<x\\},\;etc.$

###### Definition 6.2

Let
$(\mathbb{A}_{\varphi})\in\mathcal{P}(\mathbb{C})^{\mathcal{D}(\mathbb{R}^{d})}$
be a family of subsets of $\mathbb{C}$. We define the internal set generated
by $(\mathbb{A}_{\varphi})$ by

$\langle\mathbb{A}_{\varphi}\rangle=\\{\langle
A_{\varphi}\rangle\in\,^{*}\mathbb{C}:A_{\varphi}\in\mathbb{A}_{\varphi}\;a.e.\\}$

A set is called external if it is not internal.

###### Example 6.2

Let $\mathbb{A}_{\varphi}=(0,R_{\varphi})$, where $R_{\varphi}$ is as in
Example (5.1). Then the internal set $\langle\mathbb{A}_{\varphi}\rangle$
generated by $(\mathbb{A}_{\varphi})$ is the internal interval $(0,\rho)$. It
is important to note (and easy to check) that this coincides with the more
natural definition for $(0,\rho)$ given by

$(0,\rho)=\\{x\in\,^{*}\mathbb{R}:0<x<\rho\\}$

A set $S\subset\mathbb{R}^{d}$ is called relatively compact if its closure
$\overline{S}$ is compact in $\mathbb{R}^{d}$. Unless it is specified
otherwise, we shall call Lebesgue measurable sets of $\mathbb{R}^{d}$ simply
measurable sets.

###### Definition 6.3

An internal set $\large\langle\mathbb{A}_{\varphi}\large\rangle$ of
${}^{*}\mathbb{R}^{d}$ is called $*$-measurable ($*$-compact, $*$-relatively-
compact, $*$-closed, $*$-open, etc.) if

$\mathbb{A}_{\varphi}\text{\;is measurable (compact, relatively compact,
closed, etc.) in \;}\mathbb{R}^{d}\text{ for a.e. }\varphi$

Let $\rho\in{{}^{*}\mathbb{R}}$ denote a positive infinitesimal in
${}^{*}\mathbb{R}$ (for example, $\rho$ might be the positive infinitesimal
defined in (Example 5.1)). We shall keep $\rho$ fixed in what follows.

###### Definition 6.4

Let $\rho$ be a positive infinitesimal in ${}^{*}\mathbb{R}$. We define the
following (external) sets of non-standard numbers:

$\displaystyle\mathcal{M}_{\rho}({{}^{*}\mathbb{C}})=\\{x\in{{}^{*}\mathbb{C}}\mid\;|x|\leq\rho^{-n}\text{\>
for some\;}n\in\mathbb{N}\\}$
$\displaystyle\mathcal{N}_{\rho}({{}^{*}\mathbb{C}})=\\{x\in{{}^{*}\mathbb{C}}\mid\;|x|<\rho^{n}\text{\>
for all\;}n\in\mathbb{N}\\}$
$\displaystyle\mathcal{F}_{\rho}(^{*}\mathbb{C})=\\{x\in{{}^{*}\mathbb{C}}\mid\;|x|<1/\sqrt[n]{\rho}\text{\;
for all\; }n\in\mathbb{N}\\},$
$\displaystyle\mathcal{I}_{\rho}(^{*}\mathbb{C})=\\{x\in{{}^{*}\mathbb{C}}\mid\;|x|\leq\sqrt[n]{\rho}\text{\;
for some\; }n\in\mathbb{N}\\},$
$\displaystyle\mathcal{C}_{\rho}(^{*}\mathbb{C})=\\{x\in{{}^{*}\mathbb{C}}\mid\;\sqrt[n]{\rho}<|x|<1/\sqrt[n]{\rho}\text{\;
for all\; }n\in\mathbb{N}\\}.$

The numbers in $\mathcal{M}_{\rho}(^{*}\mathbb{C})$ and
$\mathcal{N}_{\rho}(^{*}\mathbb{C})$ are called $\rho$-moderate and
$\rho$-null non-standard numbers, respectively. Similarly, the numbers in
$\mathcal{F}_{\rho}(^{*}\mathbb{C})$, $\mathcal{I}_{\rho}(^{*}\mathbb{C})$ and
$\mathcal{C}_{\rho}(^{*}\mathbb{C})$ are called $\rho$-finite,
$\rho$-infinitesimal and $\rho$-constant, respectively.

##  7 Saturation Principle in ${}^{*}\mathbb{C}$

###### Theorem 7.1

Let $\\{\mathcal{A}_{n}\\}$ be a sequence of internal sets in
${}^{*}\mathbb{C}$ such that

$\bigcap_{n=0}^{m}\mathcal{A}_{n}\neq\varnothing$

for all $m\in\mathbb{N}$. (The sequence $\\{\mathcal{A}_{n}\\}$ satisfies the
finite intersection property.) Then

$\bigcap_{n=0}^{\infty}\mathcal{A}_{n}\neq\varnothing.$

_Proof_ Since each $\mathcal{A}_{n}$ is internal,

$\mathcal{A}_{n}=\langle\mathbb{A}_{n,\varphi}\rangle\text{ ,
$\mathbb{A}_{n,\varphi}\subseteq\mathbb{C}$}$

Also, since for each $m$ it is given that
$\bigcap_{n=0}^{m}\mathcal{A}_{n}\neq\varnothing$, this implies that for each
$m$ there exists a non-standard number $\langle C_{m,\varphi}\rangle\in$
${}^{*}\mathbb{C}$ such that

$\langle
C_{m,\varphi}\rangle\in\bigcap_{n=0}^{m}\langle\mathbb{A}_{n,\varphi}\rangle$

or, in other words,

$\langle C_{m,\varphi}\rangle\in\langle\mathbb{A}_{n,\varphi}\rangle\text{ for
}0\leq n\leq m$

This means that for a.e. $\varphi$ and $0\leq n\leq m$,

$C_{m,\varphi}\in\mathbb{A}_{n,\varphi}$

Remembering that $\mathcal{U}$ is closed under finite intersections, we see
that for a.e. $\varphi$,

$C_{m,\varphi}\in\bigcap_{n=0}^{m}\mathbb{A}_{n,\varphi}$

Hence for each $m$,

$\bigcap_{n=0}^{m}\mathbb{A}_{n,\varphi}\neq\varnothing\;a.e.$

We may assume without loss of generality that $\mathbb{A}_{0,\varphi}$ is non-
empty for all $\varphi$. (Else define
$\mathbb{A^{\prime}}_{0,\varphi}=\mathbb{A}_{0,\varphi}$ if
$\mathbb{A}_{0,\varphi}\neq\varnothing$ and
$\mathbb{A^{\prime}}_{0,\varphi}=\mathbb{C}$ otherwise. Then it will still be
true that $\mathcal{A}_{0}=\langle\mathbb{A^{\prime}}_{0,\varphi}\rangle$ .)
Next define a function
$\mu:\mathcal{D}(\mathbb{R}^{D})\longrightarrow\mathbb{N}\cup\\{\infty\\}$ by

$\mu(\varphi)=\max\\{m\in\mathbb{N}_{0}\cup\\{\infty\\}\,|\,\bigcap_{n=0}^{m}\mathbb{A}_{n,\varphi}\neq\varnothing\\}$

Notice that $\mu$ is defined for all $\varphi$ due to our assumption for
$\mathbb{A}_{0,\varphi}$.
Thus we have

$\bigcap_{n=0}^{\mu(\varphi)}\mathbb{A}_{n,\varphi}\neq\varnothing\text{ for
all }\varphi\in\mathcal{D}(\mathbb{R}^{d})$

Hence for every $\varphi\in\mathcal{D}(\mathbb{R}^{d})$ there exists (by Axiom
of Choice) $A_{\varphi}$ such that
$A_{\varphi}\in\bigcap_{n=0}^{\mu(\varphi)}\mathbb{A}_{n,\varphi}$.
We intend to show that

$\langle A_{\varphi}\rangle\in\bigcap_{n=0}^{\infty}\mathcal{A}_{n}$

or, equivalently, that for every $m$, $A_{\varphi}\in\mathbb{A}_{m,\varphi}$
for a.e. $\varphi$.
If $\varphi$ is such that
$\bigcap_{n=0}^{m}\mathbb{A}_{n,\varphi}\neq\varnothing$, this implies that
$0\leq m\leq\mu(\varphi)$. Thus $A_{\varphi}\in\mathbb{A}_{m,\varphi}$, by the
choice of $A_{\varphi}$. Therefore,

$\\{\varphi\,|\,\bigcap_{n=0}^{m}\mathbb{A}_{n,\varphi}\neq\varnothing\\}\subseteq\\{\varphi\,|\,A_{\varphi}\in\mathbb{A}_{m,\varphi}\\}$

But the set on the left is in $\mathcal{U}$, and so the set on the right is
also, as required. $\blacktriangle$

##  8 Non-Standard Smooth Functions

Having constructed the fields ${}^{*}\mathbb{R}$ and ${}^{*}\mathbb{C}$, the
natural next step is to look at functions on these fields. However, for our
purposes we will focus on a certain class of function contained in
${}^{*}\mathbb{C}^{{}^{*}\mathbb{R}}$. In what follows
$\mathcal{E}(\mathbb{R})$ is the set of $\mathcal{C}^{\infty}$-functions from
$\mathbb{R}$ into $\mathbb{C}$.

###### Definition 8.1

A function $f\in\,^{*}\mathbb{C}^{{}^{*}\mathbb{R}}$ is called internal smooth
if there exists a family
$(f_{\varphi})\in\mathcal{E}(\mathbb{R})^{\mathcal{D}(\mathbb{R}^{d})}$ such
that for every $x=\langle X_{\varphi}\rangle\in\,^{*}\mathbb{R}$

$f(x)=\langle f_{\varphi}(X_{\varphi})\rangle$

The set of all internal smooth functions will be denoted by
${}^{*}\mathcal{E}(\mathbb{R})$.

###### Remark 8.1

${}^{*}\mathcal{E}(\mathbb{R})$ may equivalently be defined as the set of
equivalence classes $\langle f_{\varphi}\rangle$ of families of functions in
$\mathcal{E}(\mathbb{R})^{\mathcal{D}(\mathbb{R}^{d})}$, where the equivalence
relation is as usual:

$(f_{\varphi})\sim_{\mathcal{U}}(g_{\varphi})\text{ if
}f_{\varphi}=g_{\varphi}\text{ for a.e. }\varphi$

It is not hard to prove that the value of an internal function does not depend
on the choice of representatives. If $\langle X_{\varphi}\rangle=\langle
Y_{\varphi}\rangle\in\,^{*}\mathbb{R}$ and $\langle f_{\varphi}\rangle=\langle
g_{\varphi}\rangle\in\,^{*}\mathcal{E}(\mathbb{R})$, then

$\\{\varphi\,|\,f_{\varphi}(X_{\varphi})=g_{\varphi}(X_{\varphi})\\}\cap\\{\varphi\,|\,g_{\varphi}(X_{\varphi})=g_{\varphi}(Y_{\varphi})\\}\subseteq\\{\varphi\,|\,f_{\varphi}(X_{\varphi})=g_{\varphi}(Y_{\varphi})\\}$

Since $\mathcal{U}$ is closed under intersections, $\langle
f_{\varphi}(X_{\varphi})\rangle=\langle g_{\varphi}(Y_{\varphi})\rangle$.

The operations of addition, multiplication, and partial differentiation in
${}^{*}\mathcal{E}(\mathbb{R}^{d})$ are inherited from
$\mathcal{E}(\mathbb{R}^{d})$. Also, $\mathcal{E}(\mathbb{R}^{d})$ is embedded
in ${}^{*}\mathcal{E}(\mathbb{R}^{d})$ by $f\longrightarrow\,^{*}f$ where
${}^{*}f=\langle f_{\varphi}\rangle$, $f_{\varphi}=f$ for all
$\varphi\in\mathcal{D}(\mathbb{R}^{d})$.
In what follows integrable means Lebesgue integrable.

###### Definition 8.2

Let
$\large\langle\mathbb{X}_{\varphi}\large\rangle\subseteq\,^{*}\mathbb{R}^{d}$
be a $*$-measurable internal set and let $\large\langle
f_{\varphi}\large\rangle\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}$ be an internal
function. We say that $\large\langle f_{\varphi}\large\rangle$ is
$*$-integrable over $\large\langle\mathbb{X}_{\varphi}\large\rangle$ if

$f_{\varphi}\text{\; is integrable over\; }\mathbb{X}_{\varphi}\text{ for a.e.
}\varphi$

If $\large\langle f_{\varphi}\large\rangle$ is $*$-integrable over
$\large\langle\mathbb{X}_{\varphi}\large\rangle$, we define the integral:

(1) $\int_{\large\langle\mathbb{X}_{\varphi}\large\rangle}\large\langle
f_{\varphi}\large\rangle(x)\,dx=\left<\int_{\mathbb{X}_{\varphi}}f_{\varphi}(x)\,dx\right>.$

We also say that the integral converges in ${}^{*}\mathbb{C}$ (since it is a
number in ${}^{*}\mathbb{C}$). Notice that as long as the integral converges
for a.e. $\varphi$, we include this object in the equivalence class, even if
the integral diverges for other $\varphi$.

##  9 Internal Sets and Saturation Principle in
${}^{*}\mathcal{E}(\mathbb{R}^{d})$

We define internal sets in ${}^{*}\mathcal{E}(\mathbb{R}^{d})$ similarly to
those of ${}^{*}\mathbb{C}$.

###### Definition 9.1

  * (i)

Let
$(\mathbb{F}_{\varphi})\in\mathcal{P}(\mathcal{E}(\mathbb{R}^{d}))^{\mathcal{D}(\mathbb{R}^{d})}$
be a family of subsets of $\mathcal{E}(\mathbb{R}^{d})$. We define the
internal set generated by $(\mathbb{F}_{\varphi})$ by

$\langle\mathbb{F}_{\varphi}\rangle=\\{\langle
f_{\varphi}\rangle\in\,^{*}\mathcal{E}(\mathbb{R}^{d}):f_{\varphi}\in\mathbb{F}_{\varphi}\;a.e.\\}$

A set is called external if it is not internal.

  * (ii)

An internal set $\mathcal{F}$ is called standard if there exists
$\mathbb{F}\subset\mathcal{E}(\mathbb{R}^{d})$ such that
$\mathcal{F}=\langle\mathbb{F}\rangle$. In this case we may also write
$\mathcal{F}=\,^{*}\mathbb{F}$.

###### Theorem 9.1

Let $\\{\mathcal{F}_{n}\\}$ be a sequence of internal sets in
${}^{*}\mathcal{E}(\mathbb{R}^{d})$ such that

$\bigcap_{n=0}^{m}\mathcal{F}_{n}\neq\varnothing$

for all $m\in\mathbb{N}$. (The sequence $\\{\mathcal{F}_{n}\\}$ satisfies the
finite intersection property.) Then

$\bigcap_{n=0}^{\infty}\mathcal{F}_{n}\neq\varnothing.$

_Proof_ The proof is almost identical to that of (Theorem 7.1).

###### Definition 9.2

We define the following (external) subsets of
${{}^{*}\mathcal{E}(\mathbb{R}^{d})}$:

$\displaystyle\mathcal{F}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{F}(^{*}\mathbb{C})\right]\\},$
$\displaystyle\mathcal{I}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{I}(^{*}\mathbb{C})\right]\\},$
$\displaystyle\mathcal{M}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\left\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{M}_{\rho}(^{*}\mathbb{C})\right]\right\\},$
$\displaystyle\mathcal{N}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\left\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{N}_{\rho}(^{*}\mathbb{C})\right]\right\\},$
$\displaystyle\mathcal{F}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\left\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{F}_{\rho}(^{*}\mathbb{C})\right]\right\\},$
$\displaystyle\mathcal{I}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\left\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{I}_{\rho}(^{*}\mathbb{C})\right]\right\\},$
$\displaystyle\mathcal{C}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))=\left\\{f\in{{}^{*}\mathcal{E}(\mathbb{R}^{d})}\mid(\forall\alpha\in\mathbb{N}_{0}^{d})(\forall
x\in\mathcal{F}(^{*}\mathbb{R}^{d}))\left[\partial^{\alpha}f(x)\in\mathcal{C}_{\rho}(^{*}\mathbb{C})\right]\right\\}.$

The functions in
$\mathcal{F}(^{*}\mathcal{E}(\mathbb{R}^{d})),\mathcal{I}(^{*}\mathcal{E}(\mathbb{R}^{d}))$,
$\mathcal{M}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))$,
$\mathcal{N}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d})$,
$\mathcal{F}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d})$,
$\mathcal{I}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d})$, and
$\mathcal{C}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d})$ are called  finite,
infinitesimal, $\rho$-moderate, $\rho$-null, $\rho$\- finite,
$\rho$-infinitesimal and $\rho$-constant functions, respectively. For more
details we refer to (Lightstone and Robinson [7]) and (Wolf and Todorov [17].

##  10 Weak Equality

###### Definition 10.1

Let $x,y\in\,^{*}\mathbb{C}$, $f,\,g\in\,^{*}\mathcal{E}(\mathbb{R}^{d})$

  * (i)

$x\approx y$ if $x-y\in\mathcal{I}(^{*}\mathbb{C})$

  * (ii)

$x\overset{\rho}{=}y$ if $x-y\in\mathcal{N}_{\rho}(^{*}\mathbb{C})$

  * (iii)

$f\approx g$ if $f-g\in\mathcal{I}(^{*}\mathcal{E}(\mathbb{R}^{d}))$

  * (iv)

$f\overset{\rho}{=}g$ if
$f-g\in\mathcal{N}_{\rho}(^{*}\mathcal{E}(\mathbb{R}^{d}))$

  * (v)

$f\cong g$ if $\int f(x)\tau(x)\,dx=\int g(x)\tau(x)\,dx$ for every

$\tau\in\mathcal{D}(\mathbb{R}^{d})$

  * (vi)

$f\overset{\rho}{\simeq}g$ if $\int f(x)\tau(x)\,dx\overset{\rho}{=}\int
g(x)\tau(x)\,dx$ for every

$\tau\in\mathcal{D}(\mathbb{R}^{d})$

  * (vii)

$f\approxeq g$ if $\int f(x)\tau(x)\,dx\approx\int g(x)\tau(x)\,dx$ for every

$\tau\in\mathcal{D}(\mathbb{R}^{d})$

It is not hard to prove that each of these weak equalities forms an
equivalence relation in its respective space. Many results in non-standard
analysis hold weakly in the sense of one of these weak equalities.

##  11 Schwartz Distributions

At this point, we must take a short detour to present some basic definitions
and results from the Schwartz theory.

###### Definition 11.1

A distribution is a mapping
$F:\mathcal{D}(\mathbb{R}^{d})\longrightarrow\mathbb{C}$ that satisfies the
following conditions:

  * (i)

Linearity:

$F[c_{1}\tau_{1}+c_{2}\tau_{2}]=c_{1}F[\tau_{1}]+c_{2}F[\tau_{2}]$ for all
$c_{1},c_{2}\in\mathbb{C}$ and

$\tau_{1},\tau_{2}\in\mathcal{D}(\mathbb{R}^{d})$.

  * (ii)

Continuity: Let $\\{\tau_{k}\\}$ be a sequence in
$\mathcal{D}(\mathbb{R}^{d})$. Suppose there exists $R$ such that
${\rm{supp}}\tau_{k}\subseteq\\{x:\;|x|<R\\}$ for all $k$. Also, suppose there
exists $\tau\in\mathcal{D}(\mathbb{R}^{d})$ such that
$\partial^{\alpha}\tau_{k}\longrightarrow\partial^{\alpha}\tau$ for all
$\alpha\in\mathbb{N}_{0}^{d}$ uniformly as $k\longrightarrow\infty$. Then
$F[\tau_{k}]\longrightarrow F[\tau]$.

We will denote by $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ the set of all such
distributions.

We supply $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ with the usual pointwise
addition and scalar multiplication. In addition, we define partial
differentiation by

$(\partial^{\alpha}F)[\tau]=(-1)^{|\alpha|}F[\partial^{\alpha}\tau]$

and multiplication by a smooth function $g\in\mathcal{E}(\mathbb{R}^{d})$ by

$(gF)[\tau]=F[g\tau]$

Both of these operations are well-defined since
$\partial^{\alpha}\tau,\,g\tau\in\mathcal{D}(\mathbb{R}^{d})$.
$\mathcal{L}_{loc}(\mathbb{R}^{d})$, the set of locally integrable functions,
is embedded in $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ by the mapping

$S(f)=\int f(t)\tau(t)dt$

It is not hard to show that this embedding preserves the operations mentioned
above.
Finally, we define the convolution of a distribution with a test function by

$(F\ast\tau)(x)=F[\tau(x-t)]$

###### Theorem 11.1

If $F\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$ and
$\tau\in\mathcal{D}(\mathbb{R}^{d})$ then
$(F\ast\tau)(x)\in\mathcal{E}(\mathbb{R}^{d})$ and
$\partial^{\alpha}(F\ast\tau)=F\ast\partial^{\alpha}\tau$.

Before proving this theorem, we will state (without proof) a result from
analysis. See (Rudin [13] p.148):

###### Lemma 11.1

Suppose

$\lim_{n\to\infty}f_{n}(x)=f(x)\quad(x\in E).$

Put

$M_{n}=\sup_{x\in E}|f_{n}(x)-f(x)|.$

Then $f_{n}\longrightarrow f$ uniformly on $E$ if and only if
$M_{n}\longrightarrow 0$ as $n\longrightarrow\infty$ .

_Proof of the theorem_ We will prove the theorem for the case d=1. The general
result will follow.
Let $f(x)=(F\ast\tau)(x)$. Fixing $x$, we wish to show that

$f(x+h)-f(x)\longrightarrow 0\text{ as }h\longrightarrow 0$

Note that

$\displaystyle f(x+h)-f(x)$ $\displaystyle=(F\ast\tau)(x+h)-(F\ast\tau)(x)$
$\displaystyle=F[\tau(x+h-t)]-F[\tau(x-t)]$
$\displaystyle=F[\tau(x+h-t)-\tau(x-t)]$

by the linearity of $F$. Let

$\psi(t)=\tau(x+h-t)-\tau(x-t)$

$\psi$ is itself a test function, and if we restrict $|h|<1$, then the support
of $\psi$ and all its derivatives is contained in $E=\\{y\,|\,|y|\leq
r+|x|+1\\}$, where $r$ is the radius of the support of $\tau$. It is clear
that any sequence $\\{\psi_{h_{n}}\\}$ where $h_{n}\longrightarrow 0$ as
$n\longrightarrow\infty$ converges pointwise to 0 for all $x$ and $t$ (by the
uniform continuity of $\tau$). Also, since one compact set, $E$, contains the
support of $\psi_{h_{n}}$ for all n, and since each $\psi_{h_{n}}$ is
continuous, $M_{n}=\sup_{t\in E}|\psi_{h_{n}}(t)|$ is achieved by
$\psi_{h_{n}}$ for each $n$. Thus $M_{n}\to 0$, implying that
$\\{\psi_{h_{n}}\\}\to 0$ uniformly, by the Lemma. Therefore, since $F$ is
continuous in the sense of (Definition 11.1),

$F[\psi_{h_{n}}]\longrightarrow F[0]=0$

Since $x$ was chosen arbitrarily, this proves that $f=F\ast\tau$ is
continuous.
To prove that $f^{\prime}$ exists and that
$f^{\prime}(x)=(F\ast\tau^{\prime})(x)$, we must show that

$\frac{f(x+h)-f(x)}{h}-(F\ast\tau^{\prime})(x)\longrightarrow 0\text{ as
}h\longrightarrow 0$

Note that

$\displaystyle\frac{f(x+h)-f(x)}{h}-(F\ast\tau^{\prime})(x)$
$\displaystyle=\frac{(F\ast\tau)(x+h)-(F\ast\tau)(x)}{h}-(F\ast\tau^{\prime})(x)$
$\displaystyle=F\left[\frac{\tau(x+h-t)-\tau(x-t)}{h}-\tau^{\prime}(x-t)\right]$

Now, if we let

$\chi(t)=\frac{\tau(x+h-t)-\tau(x-t)}{h}-\tau^{\prime}(x-t)$

we can use the same argument as before to show that

$F[\chi_{h_{n}}(t)]\longrightarrow F[0]=0\text{ as }h\longrightarrow 0$

This proves that $(F\ast\tau)^{\prime}=F\ast\tau^{\prime}$. Since
$\tau^{\prime}$ is itself a test function, the same proof works to show that
$(F\ast\tau)^{\prime\prime}=F\ast\tau^{\prime\prime}$ and so on. For functions
of several variables, the same argument can be applied in each variable to
show the general result. $\blacktriangle$

Before we can prove the embedding of the distributions in
${}^{*}\mathcal{E}(\mathbb{R}^{d})$, we need a result showing that
distributions can be “approximated” in a way by a certain sequence of test
functions.

###### Theorem 11.2

Let $\\{\delta_{n}\\}$ be a sequence in $\mathcal{D}(\mathbb{R}^{d})$ such
that $\delta_{n}\in\mathcal{B}_{n}$ for every $n$. Then for any distribution
$T\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$, $T\ast\delta_{n}\to T$ weakly. (A
sequence of distributions $\\{F_{k}\\}$ converges weakly to a distribution $F$
if $F_{k}[\tau]\to F[\tau]$ for all test functions $\tau$.)

Before proving this theorem, we need two lemmas:

###### Lemma 11.2

Let $\\{\delta_{n}\\}$ be as above and let $\tau$ be any test function. Then
there exists $R$ such that ${\rm{supp}}(\delta_{n}\ast\tau)\subset\\{x:|x|\leq
R\\}$. Also, $\partial^{\alpha}(\delta_{n}\ast\tau)\to\tau$ uniformly for
every $\alpha\in\mathbb{N}_{0}^{d}$.

_Proof_ For each $n$, ${\rm{supp}}\delta_{n}\subset\\{x:|x|\leq 1/n\\}$. In
particular, ${\rm{supp}}\delta_{n}\subset\\{x:|x|\leq 1\\}$. If we let
$R_{\tau}$ be the radius of the support of $\tau$ and set $R=R_{\tau}+1$, then
it is not hard to see that ${\rm{supp}}(\delta_{n}\ast\tau)\subset\\{x:|x|\leq
R\\}$.
As before, to show the uniform convergence it is enough to prove that

$\sup_{|x|\leq R}|(\delta_{n}\ast\tau)(x)-\tau(x)|\to 0$

Recalling that $\int\delta_{n}=1$, we see that

$\displaystyle\sup_{|x|\leq R}|(\delta_{n}\ast\tau)(x)-\tau(x)|$
$\displaystyle=\sup_{|x|\leq
R}\left|\int\delta_{n}(t)\tau(x-t)dt-\tau(x)\int\delta_{n}(t)dt\right|$
$\displaystyle=\sup_{|x|\leq R}\left|\int_{|t|\leq
1/n}\delta_{n}(t)[\tau(x-t)-\tau(x)]dt\right|$

By the mean value theorem for integrals, there exists $|t_{n}|\leq 1/n$ such
that

$=\sup_{|x|\leq R}\left|[\tau(x-t_{n})-\tau(x)]\int\delta_{n}(t)dt\right|$

and by the extreme value theorem there exists $|x_{n}|\leq R$ such that

$=\left|\tau(x_{n}-t_{n})-\tau(x_{n})\right|$

This last expression vanishes as $n\to\infty$ since $\tau$ is uniformly
continuous. The case $\alpha\neq 0$ is similar. $\blacktriangle$

For the proof of the next lemma see (Folland [5] p.318):

###### Lemma 11.3

Suppose $F$ is a distribution and $\phi$ and $\psi$ are test functions. Then
$(F\ast\phi)[\psi]=F[\tilde{\phi}\ast\psi]$, where $\tilde{\phi}(x)=\phi(-x)$.

_Proof of the theorem_ We must show that for any distribution $T$ and any test
function $\tau$,

$(T\ast\delta_{n})[\tau(x)]\to T[\tau(x)]$

Using (Lemma 11.3) and remembering that $\delta_{n}$ is symmetric for all $n$,

$(T\ast\delta_{n})[\tau(x)]=T[(\delta_{n}\ast\tau)(x)]\to T[\tau(x)]$

by (Lemma 11.2) and the continuity of $T$. $\blacktriangle$

##  12 Embedding of Schwartz Distributions in
${}^{*}\mathcal{E}(\mathbb{R}^{d})$

Finally, we are ready to define the embedding $\Sigma$ of
$\mathcal{D}^{\prime}(\mathbb{R}^{d})$ into
${}^{*}\mathcal{E}(\mathbb{R}^{d})$ as follows:

$\Sigma(T)=\langle T\ast\varphi\rangle$

By (theorem 11.1), $\Sigma(T)\in\,^{*}\mathcal{E}(\mathbb{R}^{d})$. From the
definition of the convolution, it is clear that $\Sigma$ is linear. It remains
to prove that $\Sigma$ is injective.

###### Lemma 12.1

$\Sigma$ is injective.

_Proof_ Since $\Sigma$ is linear, it is enough to show that $\Sigma(T)=0$
implies $T=0$.
If $\Sigma(T)=0$, we have that $T\ast\varphi=0$ a.e. That is,
$\Phi=\\{\varphi\,|\,T\ast\varphi=0\\}\in\mathcal{U}$. Thus
$\varnothing\neq\Phi\cap\mathcal{B}_{n}\in\mathcal{U}$ for each $n$, where
$\mathcal{B}_{n}$ are the basic sets. Therefore we can construct a sequence
$\\{\varphi_{n}\\}$ such that $\varphi_{n}\in\Phi\cap\mathcal{B}_{n}$ for each
$n$. Then by (Theorem 11.2), we have that $T=0$ since $T\ast\varphi_{n}=0$ for
every $n$. $\blacktriangle$

###### Theorem 12.1

  * (i)

$\langle P\ast\varphi\rangle=\,^{*}P$ for every polynomial
$P\in\mathbb{C}[x_{1},\ldots,x_{d}]$

  * (ii)

$\langle f\ast\varphi\rangle\overset{\rho}{=}\,^{*}f$ for all
$f\in\mathcal{E}(\mathbb{R}^{d})$

_Proof_ (i) Let $P\in\mathbb{C}[x_{1},\ldots,x_{d}]$ be a polynomial of degree
$p$. By the Taylor formula,

$P(x-t)=P(x)+\sum_{|\alpha|=1}^{p}\frac{(-1^{|\alpha|})\partial^{\alpha}P(x)}{\alpha!}t^{\alpha}$

It follows that for every test function $\varphi$ and $x\in\mathbb{R}^{d}$

$(P\ast\varphi)(x)=\int
P(x-t)\varphi(t)dt=P(x)\int\varphi(t)dt+\sum_{|\alpha|=1}^{p}\frac{(-1^{|\alpha|})\partial^{\alpha}P(x)}{\alpha!}\int
t^{\alpha}\varphi(t)dt$

Notice that if $\varphi\in\mathcal{B}_{n}$ for some $n\geq p$, then
$\int\varphi(t)dt=1$ and $\int t^{\alpha}\varphi(t)dt=0$,
$|\alpha|=1,2,\ldots,p$. Thus we have

$\mathcal{B}_{n}\subseteq\\{\varphi\,|\,P\ast\varphi=P\\}$

implying that $P\ast\varphi=P$ a.e. as required.

(ii) Let $\xi\in\mathcal{F}(^{*}\mathbb{R}^{d})$, $n\in\mathbb{N}$, and
$\alpha$ be a multi-index. We have to show that
$|\partial^{\alpha}(f\ast\varphi)(\xi)-\partial^{\alpha}f(\xi)|<\rho^{n}$. We
will show this for the case $\alpha=0$, the general result will follow.
Since $st(\xi)\in\mathbb{R}^{d}$ we can find an open relatively compact set
$\mathcal{O}\subset\mathbb{R}^{d}$ such that $st(\xi)\in\mathcal{O}$ and by
(Robinson [12], p.90 Theorem 4.1.4) $\xi\in\,^{*}\mathcal{O}$ and hence
$\xi\in\,^{*}\overline{\mathcal{O}}$.
As before, the Taylor formula gives

$f(x-t)=f(x)+\sum_{|\alpha|=1}^{n}\frac{(-1^{|\alpha|})\partial^{\alpha}f(x)}{\alpha!}t^{\alpha}+\sum_{|\alpha|=n+1}\frac{(-1)^{|\alpha|}\partial^{\alpha}f(\eta(x,t))}{\alpha!}t^{\alpha}$

where $\eta(x,t)$ is a point in $\mathbb{R}^{d}$ ”between $x$ and $t$”. It
follows that for every $\varphi\in\mathcal{D}(\mathbb{R}^{d})$

$(f\ast\varphi)(x)=f(x)\int\varphi(t)dt+\sum_{|\alpha|=1}^{n}\frac{(-1)^{|\alpha|}\partial^{\alpha}f(x)}{\alpha!}\int
t^{\alpha}\varphi(t)dt+$
$\sum_{|\alpha|=n+1}\int\frac{(-1)^{|\alpha|}\partial^{\alpha}f(\eta(x,t))}{\alpha!}t^{\alpha}\varphi(t)dt$

Letting

$M\overset{def}{=}2\sum_{|\alpha|=n+1}\sup_{x\in
K}\sup_{t\in{\rm{supp}}(\varphi)}\left|\frac{\partial^{\alpha}f(\eta(x,t))}{\alpha!}\right|$

we have that $M\rho^{n+1}<\rho^{n}$ since $M\in\mathbb{R}$ and $\rho$ is a
positive infinitesimal. In other words, if $\rho=\langle R_{\varphi}\rangle$,
then $MR_{\varphi}^{n+1}<R_{\varphi}^{n}$ a.e.
By the properties of $\mathcal{B}_{n}$, it follows that for a.e. $\varphi$,
$x\in K$,

$\displaystyle|(f\ast\varphi)(x)-f(x)|<\sup_{x\in K}|(f\ast\varphi)(x)-f(x)|$
$\displaystyle<\sum_{|\alpha|=n+1}\sup_{x\in
K}\sup_{t\in{\rm{supp}}(\varphi)}\left|\frac{\partial^{\alpha}f(\eta(x,t))}{\alpha!}\right|\left(\sup_{t\in{\rm{supp}}(\varphi)}\|t\|^{n-1}\right)\int|\varphi(t)|dt$
$\displaystyle\leq MR_{\varphi}^{n+1}<R_{\varphi}^{n}$

Finally, since $\xi\in\,^{*}K$, we have that
$|(f\ast\varphi)(\xi)-f(\xi)|<\rho^{n}$, as required. The general result
follows from the case $\alpha=0$ and the fact that
$\partial^{\alpha}(f\ast\varphi)=(\partial^{\alpha}f)\ast\varphi$.
$\blacktriangle$

##  13 Conservation Laws in ${}^{*}\mathcal{E}(\Omega)$ and the Hopf Equation

The embedding in the previous section is done deliberately, with the intent of
showing that ${}^{*}\mathcal{E}$ is a natural extension of
$\mathcal{D}^{\prime}$ and an appropriate setting for the study of weak
solutions to non-linear partial differential equations, an abundance of which
arise from the conservation law of physics.

######  Theorem 13.1 (Conservation Laws in ${}^{*}\mathcal{E}(\Omega)$)

Let $L\in\mathbb{R}_{+}\cup\\{\infty\\}$,
$F\in\mathcal{C}^{\infty}(\mathbb{C})$ and let ${}^{*}F$ be the non-standard
extension of $F$. Let $u\in{{}^{*}\mathcal{E}(\Omega)}$, where
$\Omega=(0,L)\times(0,\infty)$. Then the following are equivalent:

(i) $u_{t}(x,t)+\left[{}^{*}F(u(x,t))\right]_{x}=0$ for all
$x,t\in{{}^{*}\mathbb{R}},\;0<x<L,\;t>0$.

(ii) $u_{t}(x,t)+{{}^{*}F^{\prime}\left(u(x,t)\right)}\,u_{x}=0$ for all
$x,t\in{{}^{*}\mathbb{R}},\;0<x<L,\;t>0$.

(iii)
$\frac{d}{dt}\int_{a}^{b}\,u(x,t)\,dx={{}^{*}F(u(a,t))}-{{}^{*}F(u(b,t))}$ for
every $a,b,t\in{{}^{*}\mathbb{R}},\;0<a<b<L,\;t>0$.

###### Remark 13.1

The term “conservation law” is due to (iii) which in a classical setting is
given by

$\frac{d}{dt}\int_{a}^{b}\,u(x,t)\,dx={F(u(a,t))}-{F(u(b,t))}.$

Here $u(x,t)$ stands for the density of a physical quantity (the density of
the mass of a fluid, the density of the heat energy, etc.) in a rod of length
$L$ and $F(u(x,t))$ stands for the flux of the quantity from left to right
through the $x$-cross section. Then the above equality expresses the
conservation of this quantity in any $(a,b)$-segment of the rod. Recall that,
according to the classical theory, (i)-(iii) are equivalent for solutions $u$
in the class $\mathcal{C}^{2}(\Omega)$ and for all $x,t,a,b\in\mathbb{R}$ in
the corresponding intervals. The proof which follows can be generalized
(without new complications) in the case of more complicated flux $F(u,u_{x})$
or even $F(u,u_{x},u_{xx})$.

_Proof_ (i) $\Leftrightarrow$ (ii): The equivalency between (i) and (ii)
follows immediately from the fact that the partial differentiation and
extension mapping $*$ commute in ${}^{*}\mathcal{E}(\Omega)$ and the fact that
${}^{*}\mathcal{E}(\Omega)$ is a differential algebra (with Leibniz rule for
differentiation of products and chain rule). So, we have

$\left[{}^{*}F(u(x,t))\right]_{x}={(^{*}F)^{\prime}(u(x,t))\,u_{x}(x,t)}={{}^{*}F^{\prime}(u(x,t))\,u_{x}(x,t)},$

as required.

(i) $\Rightarrow$ (iii): We have
$a=\left<a_{\varphi}\right>,b=\left<b_{\varphi}\right>$ and
$u=\left<u_{\varphi}\right>$ for some families of real numbers
$(a_{\varphi}),(b_{\varphi})\in\mathbb{R}^{\mathcal{D}(\mathbb{R}^{2})}$ and
some family of smooth functions
$(u_{\varphi})\in\mathcal{E}(\Omega)^{\mathcal{D}(\mathbb{R}^{2})}$. We have
$\Phi=\\{\varphi\mid a_{\varphi}<b_{\varphi}\\}\in\mathcal{U}$ since $a<b$ in
${}^{*}\mathbb{R}$, by assumption. Thus (involving the classical arguments in
the framework of $\mathcal{E}(\Omega)$) we have $\Phi\subseteq\Phi_{1}$, where

$\Phi_{1}=\\{\varphi\mid\frac{d}{dt}\int_{a_{\varphi}}^{b_{\varphi}}\,u_{\varphi}(x,t)\,dx={F(u_{\varphi}(a_{\varphi},t))}-{F(u_{\varphi}(b_{\varphi},t))}\text{\;
for all\;}t\in\mathbb{R}_{+}\\}.$

The latter implies $\Phi_{1}\in\mathcal{U}$ which implies (iii), as required,
after transferring the result from representatives to the corresponding
equivalence classes.

(i) $\Leftarrow$ (iii): Suppose (on the contrary) that there exist
$\xi,\tau\in{{}^{*}\mathbb{R}},\,0<\xi<L,\,\tau>0$, such that
$u_{t}(\xi,\tau)+\left[{}^{*}F(u(\xi,\tau))\right]_{x}\not=0$ in
${}^{*}\mathbb{C}$. We have $\xi=\left<\xi_{\varphi}\right>$ and
$\tau=\left<\tau_{\varphi}\right>$ for some
$(\xi_{\varphi}),(\tau_{\varphi})\in\mathbb{R}^{\mathcal{D}(\mathbb{R}^{2})}$.
We denote
$\Phi=\\{\;\varphi\mid(u_{\varphi})_{t}(\xi_{\varphi},\tau_{\varphi})+\left[F(u_{\varphi}(\xi_{\varphi},\tau_{\varphi}))\right]_{x}\not=0\;\\}$
and observe that $\Phi\in\mathcal{U}$ (by our assumption). Also, we let

$\displaystyle\Phi_{\varphi}=\\{\;(\alpha,\beta,\gamma)\in\mathbb{R}^{3}\mid$
$\displaystyle\;\frac{d}{dt}\int_{\alpha}^{\beta}u_{\varphi}(x,\gamma)\,dx\not=F(u_{\varphi}(\alpha,\gamma))-F(u_{\varphi}(\beta,\gamma)),$
$\displaystyle\;0<\alpha<\beta<L,\;\gamma>0\;\\},$

and observe that $\Phi_{\varphi}\not=\varnothing$ for all $\varphi\in\Phi$ (by
the classical theory in the framework of $\mathcal{E}(\Omega)$). By axiom of
choice, there exist families
$(a_{\varphi}),(b_{\varphi}),(\gamma_{\varphi})\in\mathbb{R}^{\Phi}$ such that
$(a_{\varphi},b_{\varphi},\gamma_{\varphi})\in\Phi_{\varphi}$ for all
$\varphi\in\Phi$. If $\varphi\in\mathcal{D}(\mathbb{R}^{2})\setminus\Phi$, we
define $(a_{\varphi},b_{\varphi},\gamma_{\varphi})$ anyhow (say, by
$a_{\varphi}=b_{\varphi}=\gamma_{\varphi}=1$). These families (of real
numbers) determine the non-standard real numbers
$a=\left<a_{\varphi}\right>,b=\left<b_{\varphi}\right>$ and
$t=\left<\gamma_{\varphi}\right>$. Next, we observe that $\Phi\subseteq\Psi$
(by the definition of $\Phi_{\varphi}$), where

$\displaystyle\Psi=\\{\;\varphi\mid\;$
$\displaystyle\frac{d}{dt}\int_{a_{\varphi}}^{b_{\varphi}}u_{\varphi}(x,\gamma_{\varphi})\,dx\not=F(u_{\varphi}(a_{\varphi},\gamma_{\varphi}))-F(u_{\varphi}(b_{\varphi},\gamma_{\varphi})),$
$\displaystyle 0<a_{\varphi}<b_{\varphi}<L,\;\gamma_{\varphi}>0\\}.$

Next, $\Phi\in\mathcal{U}$ implies $\Psi\in\mathcal{U}$ which implies

$\frac{d}{dt}\int_{a}^{b}u(x,t)\,dx\not=F(u(a,t))-F(u(b,t)),\;0<a<b<L,\;t>0,$

in the framework of ${}^{*}\mathbb{C}$, contradicting (iii). The proof is
complete. $\blacktriangle$

######  Example 13.1 (Hopf Equation)

To appreciate the result of Theorem 13.1 we recall that (i)-(iii) might or
might not be equivalent in classes of classical functions and Schwartz
distributions larger than $\mathcal{C}^{2}(\Omega)$, where the (important for
the theory and applications) fundamental solutions and shock wave solutions
belong. For a discussion we refer to (J. David Logan [9], p. 309-310). Here is
an example: Let $F(u)=\frac{1}{2}u^{2}$ and $L=\infty$, so we have
$\Omega=\mathbb{R}^{2}_{+}$. In this case (i)-(iii) become:

(i) $u_{t}(x,t)+[\frac{1}{2}u(x,t)^{2}]_{x}=0$ for all
$x,t\in{\mathbb{R}},\;x>0,\;t>0$.

(ii) $u_{t}(x,t)+u(x,t)u_{x}(x,t)=0$ (Hopf equation) for all
$x,t\in{\mathbb{R}},\;x>0,\;t>0$.

(iii)
$\frac{d}{dt}\int_{a}^{b}\,u(x,t)\,dx=\frac{1}{2}\left[u^{2}(a,t)-u^{2}(b,t)\right]$
for every $a,b,t\in{\mathbb{R}},\;0<a<b,\;t>0$,

respectively. Let $v\in\mathbb{R}_{+}$, $H$ be the Heaviside step function and
let $u(x,t)=2vH(x-vt)$ be a shock wave. The next analysis shows that (i), (ii)
and (iii) are not equivalent in the spaces of classical functions and Schwartz
distributions:

(i) Since $u=2vH(x-vt)\notin\mathcal{C}^{2}(\mathbb{R}^{2}_{+})$ this function
can not be a classical solution of (i). However, $u=2vH(x-vt)$ is a
(generalized) solution of (i) in the framework of the class of Schwartz
distributions $\mathcal{D}^{\prime}(\mathbb{R}^{2}_{+})$. Indeed, for the
first term of (i) we have $u_{t}(x,t)=-2v^{2}\delta(x-vt)$, where $\delta(x)$
is the Dirac delta function. For the second term we have
$[\frac{1}{2}u(x,t)^{2}]_{x}=[\frac{4v^{2}}{2}H(x-vt)^{2}]_{x}=[2v^{2}H(x-vt)]_{x}=2v^{2}\delta(x-vt)$.
Thus $u(x,t)=2vH(x-vt)$ is a (generalized) solution of (i). We should notice
that $u(x,t)=2vH(x-vt)$ is also a weak solution of (i) in the framework of
$\mathcal{L}_{loc}(\mathbb{R}^{2}_{+})$ (see the remark below).

(ii) $u(x,t)=2vH(x-vt)$ is clearly not a solution of (ii) in classical sense.
Neither is it a (generalized) solution of (ii) in the class
$\mathcal{D}^{\prime}(\mathbb{R}^{2}_{+})$ because the term
$uu_{x}=4v^{2}H(x-vt)\delta(x-vt)$ does not make sense within
$\mathcal{D}^{\prime}(\mathbb{R}^{2}_{+})$ (recall that there is no
multiplication in $\mathcal{D}^{\prime}(\mathbb{R}^{2}_{+})$).

(iii) For the LHS of (iii) we have
$\frac{d}{dt}\int_{a}^{b}\,u(x,t)\,dx=\frac{d}{dt}\int_{a}^{b}\,2vH(x-vt)\,dx=2v\frac{d}{dt}\int_{a-vt}^{b-vt}\,H(x)\,dx=-2v^{2}H(vt-a)$.
For the RHS of (iii) we have
$\frac{1}{2}\left[u^{2}(a,t)-u^{2}(b,t)\right]=2v^{2}\left[H(a-vt)-H(b-vt)\right]=-2v^{2}H(vt-a)$.
Thus $u(x,t)=2vH(x-vt)$ is a solution of (iii).

######  Remark 13.2 (Weak Solution)

Suppose the $u\in\mathcal{L}_{loc}(\Omega)$ is a solution of
$u_{t}(x,t)+\left[F(u(x,t))\right]_{x}=0$ in the framework of
$\mathcal{D}^{\prime}(\Omega)$ (that means that both $u_{t}(x,t)$ and
$\left[F(u(x,t))\right]_{x}$ are in $\mathcal{D}^{\prime}(\Omega)$). These
solutions are often called weak solutions because they satisfy the weak
equality:

$\iint_{\Omega}\left[u(x,t)\tau_{t}(x,t)+F(u(x,t))\tau_{x}(x,t)\right]\,dx\,dt=0,$

for all test functions $\tau\in\mathcal{D}(\Omega)$. In
$\mathcal{D}^{\prime}(\Omega)$ we have:

$\iint_{\Omega}\left[u(x,t)\tau_{t}(x,t)+F(u(x,t))\tau_{x}(x,t)\right]\,dx\,dt=\left<u_{t}(x,t)+\left[F(u(x,t))\right]_{x},\tau(x,t)\right>,$

where $\left<\;,\;\right>$ stands for the pairing between
$\mathcal{D}^{\prime}(\Omega)$ and $\mathcal{D}(\Omega)$. Thus every
(generalized) solution in $\mathcal{D}^{\prime}(\Omega)$ is also weak solution
in $\mathcal{L}_{loc}(\Omega)$. In particular the function $u(x,t)=2vH(x-vt)$
in the example above is both a generalized and a weak solution of (i).

##  14 Generalized Delta-like Solution of the Hopf Equation

In this section we will prove the existence of a delta-like weak solution to
the Hopf equation of the type $\overset{\rho}{\simeq}$. That is, we are
looking for a function of the form

$u(x,t)=u_{0}+\frac{A}{\rho}\Theta\left(\frac{x-vt}{\rho}\right)$

where $\Theta\in\,^{*}\mathcal{S}(\mathbb{R})$ ($\mathcal{S}(\mathbb{R})$ is
the class of rapidly decreasing functions, such as $e^{-x^{2}}$,
${}^{*}\mathcal{S}$ is its non-standard extension, defined similarly to
${}^{*}\mathcal{E}$), $\int\Theta(x)dx=1$,
$u_{0},A,v\in\mathcal{M}_{\rho}(^{*}\mathbb{R})$ (we consider A to be the
amplitude of the soliton and $v$ its velocity), and for all $t>0$,

$u_{t}+uu_{x}\overset{\rho}{\simeq}0$

That is, for all $t>0$, $\tau\in\mathcal{D}^{\prime}(\mathbb{R})$,

$\int[u_{t}+uu_{x}]\tau(x)dx\overset{\rho}{=}0$

In addition, we would like this function to satisfy the conservation law, so
that for all $a,b\in\mathbb{R}$, $t>0$,

$\frac{d}{dt}\int_{a}^{b}u(x,t)dx\overset{\rho}{=}\frac{1}{2}[u^{2}(a,t)-u^{2}(b,t)]$

Calculating $u_{t}+uu_{x}$ we get

$u_{t}+uu_{x}=-\frac{Av}{\rho^{2}}\Theta^{\prime}\left(\frac{x-vt}{\rho}\right)+\frac{u_{0}A}{\rho^{2}}\Theta^{\prime}\left(\frac{x-vt}{\rho}\right)+\frac{A^{2}}{\rho^{3}}\Theta\left(\frac{x-vt}{\rho}\right)\Theta^{\prime}\left(\frac{x-vt}{\rho}\right)$

Simplifying and letting
$\Theta\left(\frac{x-vt}{\rho}\right)\Theta^{\prime}\left(\frac{x-vt}{\rho}\right)=\frac{\rho}{2}\left(\Theta^{2}\left(\frac{x-vt}{\rho}\right)\right)_{x}$
gives us

$\displaystyle\int[u_{t}+uu_{x}]\tau(x)dx$
$\displaystyle=\frac{(u_{0}-v)A}{\rho^{2}}\int\Theta^{\prime}\left(\frac{x-vt}{\rho}\right)\tau(x)dx+\frac{A^{2}}{2\rho^{2}}\int\left(\Theta^{2}\left(\frac{x-vt}{\rho}\right)\right)_{x}\tau(x)dx$

Integrating by parts and making the substitution $y=\frac{x-vt}{\rho}$ gives

$=\int\left[(v-u_{0})A\Theta(y)-\frac{A^{2}}{2\rho}\Theta^{2}(y)\right]\tau^{\prime}(vt+\rho
y)dy$

Finally, using the Taylor formula for $\tau^{\prime}(vt+\rho y)$, we have that
for each $m\in\mathbb{N}$,

$=\sum_{n=0}^{m}\int\left[(v-u_{0})A\Theta(y)-\frac{A^{2}}{2\rho}\Theta^{2}(y)\right]y^{n}\frac{\tau^{(n+1)}(vt)}{n!}\rho^{n}\,dy+R_{m}(\tau)$

where the remainder term is

$R_{m}(\tau)=\rho^{m+1}\int\left[(v-u_{0})A\Theta(y)-\frac{A^{2}}{2\rho}\Theta^{2}(y)\right]\frac{\tau^{(m+2)}(\eta(y,t))}{(m+1)!}y^{m+1}dy$

We would like to find a function $\Theta$ such that for every $m$,

$\int\left[(v-u_{0})A\Theta(y)-\frac{A^{2}}{2\rho}\Theta^{2}(y)\right]y^{n}dy=0,\,0\leq
n\leq m$

and $|R_{m}(\tau)|<\rho^{m+k}$ (for some fixed k).
When $m=0$, we have that

$A=\frac{2\rho(v-u_{0})}{\int\Theta^{2}(y)dy}$

(remembering that $\int\Theta(y)dy=1$) Replacing this value of A, we have that
for every $m$,

$\int\Theta^{2}(y)dy\int\Theta(y)y^{n}dy=\int\Theta^{2}(y)y^{n}dy,\,0\leq
n\leq m$

Define

$S_{m}=\\{f\in\mathcal{S}\,:\,\int f(x)x^{n}dx=\frac{\int
f^{2}(x)x^{n}dx}{\int f^{2}(x)dx},\,0\leq n\leq m\\}$

For each $m$, $S_{m}$ is non-empty by (M. Radyna [11] p. 275).

Now let

$\displaystyle\overline{S}_{m}=\\{f\in\,^{*}S_{m}\,:$ $\displaystyle f(0)=0$
$\displaystyle|\ln\rho|^{-1}\int|f(x)x^{n}|<1$
$\displaystyle|\ln\rho|^{-1}\int|f^{2}(x)x^{n}|dx<1,\,0\leq n\leq m\\}$

The standard functions in ${}^{*}S_{m}$ certainly satisfy the second and third
conditions, since their integrals will be standard (and therefore finite) and
$|\ln\rho|^{-1}$ is infinitely small. As for the first condition, we can say
with certainty that if a function $f\in\,^{*}S_{m}$ has at least one zero, say
$f(-k)=0$, then $g(x)\overset{def}{=}f(x-k)\in\,^{*}S_{m}$ and $g(0)=0$ by the
following lemma:

###### Lemma 14.1

Suppose $f(x)$ satisfies

$\int f^{2}(x)dx\int f(x)x^{n}dx=\int f^{2}(x)x^{n}dx$

Then $g(x)=f(x-k)$ also satisfies

$\int g^{2}(x)dx\int g(x)x^{n}dx=\int g^{2}(x)x^{n}dx$

_Proof_ Substituting $y=x-k$, we get

$\displaystyle\int g^{2}(x)dx\int g(x)x^{n}dx$ $\displaystyle=\int
f^{2}(y)dy\int f(y)(y+k)^{n}dy$
$\displaystyle=\sum_{j=0}^{n}\begin{pmatrix}n\\\ j\end{pmatrix}k^{n-j}\int
f^{2}(y)dy\int f(y)y^{j}dy$ $\displaystyle=\sum_{j=0}^{n}\begin{pmatrix}n\\\
j\end{pmatrix}k^{n-j}\int f^{2}(y)y^{j}dy$ $\displaystyle=\int
f^{2}(y)\sum_{j=0}^{n}\begin{pmatrix}n\\\ j\end{pmatrix}y^{j}k^{n-j}dy$
$\displaystyle=\int f^{2}(y)(y+k)^{n}dy$ $\displaystyle=\int
f^{2}(x-k)x^{n}dx$ $\displaystyle=\int
g^{2}(x)x^{n}dx\quad\quad\blacktriangle$

Thus, if at least one function in ${}^{*}S_{m}$ has at least one zero, then
$\overline{S}_{m}$ will be non-empty. In addition, the sets $\overline{S}_{m}$
are internal and $\overline{S}_{0}\supset\,\overline{S}_{1}\supset\ldots$.
Therefore, by the saturation principle, there exists a function
$\Theta(x)\in\bigcap_{n=0}^{\infty}\overline{S}_{n}$.
Noting that $\int|\Theta(x)x^{n}|dx$ and $\int|\Theta^{2}(x)x^{n}|dx$ are at
most $\mathcal{C}_{\rho}(\mathbb{{}^{*}C})$, we have that for this $\Theta$,

$|R_{m}(\tau)|<\frac{\rho^{m+1}}{(m+1)!}\sup_{x\in\mathbb{R}}|\tau^{(m+2)}(x)|\int\left|\left[(v-u_{0})A\Theta(y)-\frac{A^{2}}{2\rho}\Theta^{2}(y)\right]y^{n}\right|dy<\rho^{m+k}$

where $k$ is some real constant that depends on $v,u_{0},$ and $A$. Therefore
$u(x,t)=u_{0}+\frac{A}{\rho}\Theta\left(\frac{x-vt}{\rho}\right)$ satisfies
the Hopf equation weakly, in the sense of $\overset{\rho}{\simeq}$.
If, in addition, it is true that $\Theta(0)=0$, then $u(x,t)$ will also
satisfy the conservation law:
for all $a,b\in\mathbb{R}$, $t>0$,

$\frac{d}{dt}\int_{a}^{b}u(x,t)dx\overset{\rho}{=}\frac{1}{2}[u^{2}(a,t)-u^{2}(b,t)]$

Let us prove this by first calculating the left side:

$\displaystyle\frac{d}{dt}\int_{a}^{b}\left[u_{0}+\frac{A}{\rho}\Theta\left(\frac{x-vt}{\rho}\right)\right]dx$
$\displaystyle=\frac{d}{dt}\left[u_{0}(b-a)+\frac{A}{\rho}\int_{a}^{b}\Theta\left(\frac{x-vt}{\rho}\right)dx\right]$
$\displaystyle=\frac{d}{dt}\left[A\int_{\frac{a-vt}{\rho}}^{\frac{b-vt}{\rho}}\Theta(y)dy\right]$
$\displaystyle=A\left[\Theta\left(\frac{b-vt}{\rho}\right)\left(-\frac{v}{\rho}\right)-\Theta\left(\frac{a-vt}{\rho}\right)\left(-\frac{v}{\rho}\right)\right]$
$\displaystyle=A\frac{v}{\rho}\left[\Theta\left(\frac{a-vt}{\rho}\right)-\Theta\left(\frac{b-vt}{\rho}\right)\right]$

Since $\Theta\in\,^{*}\mathcal{S}$, if $a\neq vt$ and $b\neq vt$ then this
quantity vanishes. However, if $a=vt$ or $b=vt$ (but not both) then we have
that the left side equals $\pm\frac{Av}{\rho}\Theta(0)$, respectively. If
$\Theta(0)=0$, this also vanishes.
Calculating the right side, we have:

$\displaystyle\frac{1}{2}\left[\left[u_{0}+\frac{A}{\rho}\Theta\left(\frac{a-vt}{\rho}\right)\right]^{2}-\left[u_{0}+\frac{A}{\rho}\Theta\left(\frac{b-vt}{\rho}\right)\right]^{2}\right]$
$\displaystyle=\frac{1}{2}u_{0}^{2}+\frac{u_{0}A}{\rho}\Theta\left(\frac{a-vt}{\rho}\right)+\frac{A^{2}}{2\rho^{2}}\Theta^{2}\left(\frac{a-vt}{\rho}\right)$
$\displaystyle-\frac{1}{2}u_{0}^{2}-\frac{u_{0}A}{\rho}\Theta\left(\frac{b-vt}{\rho}\right)-\frac{A^{2}}{2\rho^{2}}\Theta^{2}\left(\frac{b-vt}{\rho}\right)$
$\displaystyle=\frac{u_{0}A}{\rho}\left[\Theta\left(\frac{a-vt}{\rho}\right)-\Theta\left(\frac{b-vt}{\rho}\right)\right]+\frac{A^{2}}{2\rho^{2}}\left[\Theta^{2}\left(\frac{a-vt}{\rho}\right)-\Theta^{2}\left(\frac{b-vt}{\rho}\right)\right]$

Again, we have that if $a\neq vt$ and $b\neq vt$ then this quantity vanishes.
If $a=vt$ or $b=vt$ (but not both) then the right side equals
$\pm\left(\frac{u_{0}A}{\rho}\Theta(0)+\frac{A^{2}}{2\rho^{2}}\Theta^{2}(0)\right)$,
respectively. Here also, if $\Theta(0)=0$ the right side vanishes, and so the
conservation law holds.

In conclusion, we may make some conjectures based on the relation

$A=\frac{2\rho(v-u_{0})}{\int\Theta^{2}(y)dy}$

There are many possibilities here, but if we assume (for simplicity) that
$\int\Theta^{2}(y)dy$ is finite (and not infinitesimal) and $u_{0}=0$, then
there are at least the following two particular cases:

  * (i)

u has infinitesimal amplitude with finite or infinitely large velocity,
resembling a small signal, or

  * (ii)

u has non-infinitesimal, finitely large amplitude, and infinitely large
velocity, resembling an explosion.

######  Remark 14.1 (Connection with Perturbation Theory)

The closest to our result is the work by M. Radyna [11] in the framework of
perturbation theory. M. Radyna proves the following result: For every
$n\in\mathbb{N}$ there exists a function
$\Theta_{n}\in\mathcal{S}(\mathbb{R})$ such that the function
$u(x,t)=\frac{A}{\epsilon}\Theta_{n}(\frac{x-vt}{\epsilon})$ satisfies:

$\left|\int_{\mathbb{R}}[u_{t}(x,t)+u(x,t)u_{x}(x,t)]\tau(x)\,dx\right|<\epsilon^{n},$

for every test function $\tau\in\mathcal{D}(\mathbb{R})$, every
$t\in\mathbb{R}$ and all sufficiently small $\epsilon\in\mathbb{R}$. For
comparison we mention the following:

(a) Instead of a small real parameter $\epsilon$ we use a proper positive
infinitesimal $\rho$. Our framework is, of course, quite different from M.
Radyna’s theory.

(b) In contrast to M. Radyna’s result, we have proved the existence of a
function $\Theta\in{{}^{*}\mathcal{S}(\mathbb{R})}$ (not depending on $n$)
such that the function $u(x,t)=\frac{A}{\rho}\Theta(\frac{x-vt}{\rho})$
satisfies:

$\left|\int_{\mathbb{R}}[u_{t}(x,t)+u(x,t)u_{x}(x,t))\tau(x)\,dx\right|<\rho^{n},$

for every test function $\tau\in\mathcal{D}(\mathbb{R})$, every
$t\in\mathbb{R}$ and for all $n\in\mathbb{N}$.

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