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\newtheorem{exercise}{Exercise}
\newtheorem{defn}{Definition}
\newtheorem{note}{Note}
\newtheorem*{objectives}{Objectives}
\newtheorem*{requirements}{Requirements}

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\begin{document}

\mytitle[Very slowly oscillating functions on $(0,+\infty)$]%
{Very slowly oscillating functions\\on the positive half-line}

\medskip

\bigskip\noindent
This text is a rough draft.

\bigskip

\begin{objectives}
Study the $C^\ast$-algebra of the bounded functions
$f\colon\PositiveRealNumbers\to\ComplexNumbers$
that are uniformly continuous with respect to the \emph{logarithmic distance}
\[
\rho(x,y)\eqdef\bigl|\ln(x)-\ln(y)\bigr|.
\]
\end{objectives}

\begin{requirements}
Bounded uniformly continuous functions on a metric space,
logarithmic distance $\rho$ and its properties,
uniform deviation $\Omega_{\rho,f}$ with respect to the logarithmic distance.
\end{requirements}

\bigskip

\section*{Bounded uniformly continuous functions\\
on a metric space (review)}

\medskip\indent
In these exercises we suppose that $(M,\rho)$ is a metric space.

\bigskip

\begin{exercise}[uniform norm $=$ supremum-norm]
Let $f\colon M\to\ComplexNumbers$.
Recall the definition of the supremum-norm:
\par\smallskip
\[
\|f\|_\infty\eqdef\hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise}[uniform deviation of a function defined on a metric space]
Let $(M,\rho)$ be a metric space and $f\colon M\to\ComplexNumbers$.
Recall the definition:
\par\smallskip
\[
\Omega_{\rho,f}(\delta)\eqdef\hsp{20em}
\]
\end{exercise}

\bigskip

\begin{exercise}[bounded uniformly continuous functions on a metric space]
Recall the definition:
\par\smallskip
\[
\BUC(M,\rho)\eqdef\Bigl\{f\in\ComplexNumbers^M\colon \hsp{20em}\Bigr\}.
\]
\end{exercise}

\bigskip

\begin{exercise}[bounds for arithmetic operations]
Let $f,g\in\BUC(M,\rho)$ and $\lambda\in\ComplexNumbers$.
Recall identities or upper bounds for the supremum-norms
and uniform deviations of the functions $f+g$, $\lambda f$, $fg$ and $\overline{f}$:
\begin{align*}
\|f+g\|_\infty&\le\hsp{10em}
&
\Omega_{\rho,f+g}(\delta)&\le\hsp{10em}
\\[2ex]
\|\lambda f\|_\infty&=\hsp{10em}
&
\Omega_{\rho,\lambda f}(\delta)&=\hsp{10em}
\\[2ex]
\|fg\|_\infty&
&
\Omega_{\rho,fg}(\delta)&
\\[2ex]
\|\overline{f}\|_\infty&
&
\Omega_{\rho,\overline{f}}(\delta)&
\end{align*}
\end{exercise}

\bigskip

\begin{exercise}
Let $f,g\colon M\to\ComplexNumbers$ and $\delta>0$.
Write an upper bound for $\Omega_{\rho,f}(\delta)$
in terms of $\Omega_{\rho,g}(\delta)$ and $\|f-g\|_\infty$.
That inequality can be used to prove that $\BUC(M,\rho)$
is a closed subset of the set of all bounded functions $M\to\ComplexNumbers$.
\par\smallskip
\[
\Omega_{\rho,f}(\delta)\le\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
$\BUC(M,\rho)$ has the following properties.
Indicate with arrows logical relations between of them.
\begin{enumerate}
\item[1)] $\BUC(M,\rho)$ is closed with respect to the linear operations.
\item[2)] $\|\cdot\|_\infty$ is a norm.
\item[3)] $\BUC(M,\rho)$ is a normed vector space.
\item[4)] $\BUC(M,\rho)$ with the distance induced by the norm $\|\cdot\|_\infty$ is a complete metric space.
\item[5)] $\BUC(M,\rho)$ is a Banach space.
\item[6)] $\BUC(M,\rho)$ is closed with respecto to the multiplication.
\item[7)] $\|\cdot\|_\infty$ is submultiplicative.
\item[8)] The constant function $1$ belongs to $\BUC(M,\rho)$, and $\|1\|_\infty=1$.
\item[9)] $\BUC(M,\rho)$ is a unital Banach algebra.
\item[10)] $\BUC(M,\rho)$ is closed with respect to the conjugation $f\mapsto\overline{f}$.
\item[11)] $\|\overline{f}\,f\|_\infty=\|f\|_\infty^2$.
\item[12)] $\BUC(M,\rho)$ is a $C^\ast$-algebra.
\end{enumerate}
\end{exercise}

\clearpage
\section*{Logarithmic distance on the positive half-line (review)}

\medskip

\begin{exercise}[logarithmic distance on the positive half-line]
Recall the definition:
\[
\rho(x,y)\eqdef\hsp{10em}
\]
\end{exercise}

\bigskip

\begin{exercise}
Let $x,y,z>0$. Simplify:
\[
\rho(zx,zy)=\hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise}[$\delta$-neighborhood of $1$]
\[
\rho(x,1)<\delta \qquad\Longleftrightarrow\qquad \hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise}[$\delta$-neighborhood of $x$]
\[
\rho(x,y)<\delta \qquad\Longleftrightarrow\qquad
\ubrace{5em}<y<\ubrace{5em}
\]
\end{exercise}

\bigskip

\begin{exercise}
\[
\rho(x,1)\le\delta \qquad\Longleftrightarrow\qquad \hsp{15em}
\]
\end{exercise}

\bigskip

\begin{exercise} \label{exer:rho_x_y_le_delta}
\[
\rho(x,y)\le\delta \qquad\Longleftrightarrow\qquad
\ubrace{5em}\le y\le\ubrace{5em}
\]
\end{exercise}

\bigskip

\clearpage
\section*{Uniform deviation of functions\\
with respect to the logarithmic distance (review)}

\bigskip

\begin{exercise}
Using the result of the Exercise \ref{exer:rho_x_y_le_delta}
write $\Omega_{\rho,f}(\delta)$ as a double supremum:
\par\medskip
\[
\Omega_{\rho,f}(\delta)=\quad\sup_{\substack{\ubrace{4em}}}\quad\sup_{\substack{\ubrace{8em}}}\quad\ubrace{8em}.
\]
\end{exercise}

\bigskip

\begin{exercise}
Write $\Omega_{\rho,f}(\delta)$ in terms of the quotient $y/x$:
\par\medskip
\[
\Omega_{\rho,f}(\delta)=\quad\sup_{\substack{x,y>0\\\ubrace{3em}\le\frac{y}{x}\le\ubrace{3em}}}\quad\ubrace{8em}.
\]
\end{exercise}

\bigskip

\begin{defn}[standard $\delta$-deviation of a function $f$ near a point $x$]
Let $f\colon\PositiveRealNumbers\to\ComplexNumbers$ and $\delta>0$.
The $\Omega_{d,f,x}(\delta)$ is defined by
\medskip
\[
\Omega_{d,f,x}(\delta)\eqdef\sup\bigl\{|f(x)-f(y)|\colon\ d(x,y)\le\delta\bigr\}.
\]
\end{defn}

\bigskip

\begin{exercise}
Let $f\colon\PositiveRealNumbers\to\ComplexNumbers$, $\delta>0$ and $x>0$.
Compare $\Omega_{d,f,x}(\delta)$ with $\Omega_{\rho,f}(\delta)$.
\end{exercise}

\bigskip

\clearpage
\section*{Definition of very slowly oscillating function}

\medskip

\begin{exercise}
Prove that
\par\medskip
\[
\BUC(\PositiveRealNumbers,\rho)
=\bigl\{g\circ\exp\colon\ g\in\BUC(\RealNumbers,d)\bigr\}.
\]
\end{exercise}

\bigskip

\begin{defn}[very slowly oscillating functions]
Denote by $\VSO(\PositiveRealNumbers)$ the following set:
\[
\VSO(\PositiveRealNumbers)
\eqdef\BUC(\PositiveRealNumbers,\rho)
=\bigl\{g\circ\exp\colon\ g\in\BUC(\RealNumbers,d)\bigr\}.
\]
\end{defn}

\bigskip

\begin{exercise}
Prove that $\VSO(\PositiveRealNumbers)$ is a subset of $C_b(\PositiveRealNumbers)$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip


\section*{Examples}

\medskip

\begin{exercise}
Determine whether $f\in\VSO(\PositiveRealNumbers)$, where
\[
f\colon\PositiveRealNumbers\to\ComplexNumbers,\qquad
f(x)\eqdef\cos\left(\ln(x)\right).
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Determine whether $f\in\VSO(\PositiveRealNumbers)$, where
\[
f\colon\PositiveRealNumbers\to\ComplexNumbers,\qquad
f(x)\eqdef\cos\left(\sqrt{x}\right).
\]
\end{exercise}

\bigskip\bigskip\bigskip\bigskip\bigskip\bigskip


\clearpage
\section*{Comparison with the $\boldsymbol{C^\ast}$-algebra\\
of the continuous functions\\
having finite limits at $\boldsymbol{0}$ and $\boldsymbol{+\infty}$}

\medskip

\begin{defn}[a distance on ${[0,+\infty]}$ inducing the standard topology]
The standard topology on $[0,+\infty]$ can be induced by the distance
\par\medskip
\[
d_{[0,+\infty]}(x,y)\eqdef \bigl|\zeta(x)-\zeta(y)\bigr|,
\]
where
\[
\zeta(x)\eqdef
\begin{cases}
\frac{x}{x+1}, & x\in[0,+\infty); \\
1, & x=+\infty.
\end{cases}
\]
\end{defn}

\bigskip

\begin{exercise}
Let $x,y\in\PositiveRealNumbers$.
Compare $d_{[0,+\infty]}(x,y)$ with $\rho(x,y)$.
\end{exercise}

\bigskip\bigskip\bigskip\bigskip

\begin{exercise}
Let $f\colon\PositiveRealNumbers\to\ComplexNumbers$ and $\delta>0$.
Compare:
\par\medskip
\[
\Omega_{d_{[0,+\infty]},f}(\delta)\ \ubrace{3em}\ \Omega_{\rho,f}(\delta).
\]
\end{exercise}

\bigskip

\begin{defn}
Given a function $f\colon\PositiveRealNumbers\to\ComplexNumbers$,
the following conditions are equivalent:
\begin{itemize}
\item[(a)] $f\in C(\PositiveRealNumbers)$ and $f$ have finite limits at $0^+$ and $+\infty$.
\item[(b)] $f$ can be prolonged to a continuous function $[0,+\infty]\to\ComplexNumbers$.
\item[(c)] $f$ can be prolonged to a uniformly continuous function $[0,+\infty]\to\ComplexNumbers$.
\item[(d)] $f$ is uniformly continuous with respect to the distance $d_{[0,+\infty]}$.
\end{itemize}
The set of these functions will be denoted by $C([0,+\infty])$.
\end{defn}

\bigskip

\begin{exercise}
Compare $\VSO(\PositiveRealNumbers)$ with $C([0,+\infty])$:
\par\medskip
\[
\VSO(\PositiveRealNumbers)\qquad\ubrace{3em}\qquad C([0,+\infty]).
\]
\end{exercise}

\bigskip

\label{endlabel}
\end{document}
