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%********************************************************
\title
    {Weighted approximation of continuous positive functions }
 %TEMA\thanks{Agradecimentos por auxílio; Trabalho apresentado no Congresso... (se for o caso).}
\author
    {M. S. KASHIMOTO %
     \thanks{kaxixi@unifei.edu.br, mskashim@gmail.com; }\,,
     Departamento de Matem\'atica e Computação,
     IMC,
     UNIFEI, Universidade Federal de Itajub\'a, 37500-903 Itajub\'a, MG, Brasil.}

%VERSÃO ENVIADA PARA TEMA

\criartitulo

\runningheads {Kashimoto}{Approximation of  positive functions}

\begin{abstract}
{\bf Abstract}.
We investigate the density of convex cones of continuous positive functions in weighted spaces and present some applications.

{\bf keywords}. convex cone, weighted space, Bernstein's Theorem
.
\end{abstract}





%********************************************************
\newsec{Introduction and Preliminaries }

\hspace*{0.3 cm}
 Throughout this paper we shall assume, unless stated otherwise, that
$X$ is a locally compact Hausdorff space. We shall denote by
$C(X;\cal{R})$  the space of all  continuous real-valued functions on $X$
and by $C_b(X;\cal{R})$ the space of  continuous and bounded real-valued
functions on $X.$ The vector subspace of all functions in $C(X;{\cal R})$ with compact support is denoted by $C_c(X;{\cal R}).$

An  upper semicontinuous real-valued function $f$ on $X$ is said to
{\it vanish at infinity} if, for every $\varepsilon > 0$, the closed
subset $\{x \in X: |f(x)| \ge \varepsilon\}$ is compact.

In what follows, we shall present the concept of {\it weighted
spaces} as developed by Nachbin  in \cite{Nachbin}.
\smallskip
We introduce a set $V$ of non-negative upper semicontinuous
functions on $X,$ whose elements are called {\it weights}. We assume
that $V$ is directed,
 in the sense that, given $v_1, v_2 \in V,$ there exist $\lambda > 0 $ and $v\in V$
such that $v_1 \le \lambda v $ and $v_2 \le \lambda v.$

 Let $V$ be a directed set of weights.
The vector subspace of $C(X;\cal{R})$ of all functions $f $ such that
$vf$ vanishes at infinity for each  $v \in V$ will be denoted by
$CV_ \infty(X;\cal{R})$.

When $CV_ \infty(X;\cal{R})$ is equipped with the
 locally convex topology $\omega_V$ generated by the seminorms
\begin{eqnarray*}
 p_v:CV_ \infty(X;\cal{R})&\to& \cal{R}^+ \\
 f &\mapsto& sup \ \{v(x)|f(x)|: x \in X\}
\end{eqnarray*}
for each $v \in V,$ we call $CV_ \infty(X;\cal{R})$ a {\it weighted
space}.

We assume that for each $x \in X$, there is $v \in V$ such that $v(x)>0.$

In the following we present some examples of weighted spaces.

\medskip

\noindent (a) If $V$ consists of the constant function ${\bf 1}$,
defined by ${\bf 1}(x)=1$ for all $x\in X,$ then $CV_ \infty(X;\cal{R})$
is $C_{0}(X;\cal{R})$, the vector subspace of all functions in $C(X;\cal{R})$
that vanish at infinity. In particular, if $X$ is compact then $CV_ \infty(X;{\cal R})=C(X;\cal{R})$. The corresponding weighted topology is the
topology of uniform convergence on $X$.
\smallskip

\noindent (b) Let $V$ be the set of characteristic functions of all
compact subsets of $X$. Then the weighted space $CV_ \infty(X;\cal{R})$
is $C(X;\cal{R})$ endowed with the compact-open topology.
\smallskip

\noindent (c) If $V$ consists of  characteristic functions of all
finite subsets of $X$, then $CV_ \infty(X;\cal{R})$ is $C(X;\cal{R})$ endowed
with the topology of pointwise convergence.
\smallskip

\noindent (d) If $V=\{v \in $ $C_ {0}(X;\cal{R}):$ $ v \ge 0 \}$, then $CV_ \infty(X;\cal{R})$ is the vector space $C_b(X;{\cal R})$. The corresponding
weighted topology is the strict topology $\beta$ (see Buck
\cite{Buck}).

\medskip



For more information on weighted spaces we refer the reader to
\cite{Nachbin, Prolla1}.



%If $S\subset C(X;{\cal R})$, we set $S^+= \{ f \in S: f \ge 0 \}.$

We set $CV^+_ \infty(X;{\cal R})= \{ f \in CV_ {\infty}(X;{\cal R}): f \ge 0 \}.$

A subset $W \subset CV^+_ \infty(X;{\cal R})$ is a convex cone if $\lambda W \subset W$, for each $\lambda \ge 0$ and $W+W \subset W$.


We denote by $ CV_ \infty^{+}(X;{\cal R}) \bigotimes CV_ \infty^{+}(Y;{\cal R})$ the subset of
$CV_ \infty^{+}(X \times Y;{\cal R})$ consisting of all functions of the form
 \[ \sum_{i=1}^n g_i(x)h_i(y), \hspace{1cm} x\in X,\,\,  y\in Y \]

\noindent where $g_i \in CV_ \infty^{+}(X;{\cal R}),\,h_i\in CV_ \infty^{+}(Y;{\cal R}) ,\,  i=1,...,n, \, n\in {\cal N}$.




 Let$W \subset CV^+_ \infty(X;{\cal R})$ be a nonempty
subset. A function $\phi \in C(X;{\cal R}),$ $0\le \phi \le 1$,  is called
a {\it multiplier} of $W$ if $\phi f + (1-\phi)g \in W$
for every pair $f$ and $g$ of elements of $W$.
The set of all multipliers of $W$ is denoted by $M(W)$.
The notion of a multiplier of  $W$ is due to Feyel and De La Pradelle \cite{Feyel} and
Chao-Lin \cite{Lin}.

 For any $x \in X, [x]_{M(W)}$ denotes the equivalence class of $x$,
when one defines the following equivalence
relation on $X$: $x\equiv t(mod M(W))$ if, and only if, $\phi(x)=\phi(t)$
for all $\phi \in M(W)$.

 A subset $A  \subset C(X;{\cal R})$ separates the
points of $X$ if, given any two distinct points $s$ and
$t$ of $X$, there is a function $\phi \in A$
such that $\phi(s) \ne \phi(t)$.



Weierstrass' first theorem states that any real-valued continuous
function $f$ defined on the closed interval [0,1] is the limite of
a uniformly convergent sequence of algebraic polynomials. One of
the most elementary proofs of this classic result is that which
uses the Bernstein polynomials of $f$

\[ (B_nf,x):= \sum_{k=0}^n \left (\matrix{ n \cr k} \right )
f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}, \hspace{1cm} x\in
[0,1] \]

\noindent for each natural number $n.$ Bernstein's theorem states that
$B_n(f)\rightarrow f$ uniformly on [0,1] and, since each $B_n(f)$
is a polynomial, we have as a consequence the Weierstrass approximation theorem. The operator $B_n$ defined on the space
$C([0,1])$ with values in the vector subspace of all polynomials
of degree at most $n$ has the property that $B_n(f) \ge 0$
whenever $f \ge 0$. Thus Bernstein's theorem also establishes the
fact that each positive continuous real-valued function on $[0,1]$
is the limit of a uniformly convergent sequence of positive
polynomials.


Consider a compact Hausdorff space $X$ and the convex cone
\[ C^+(X;{\cal R} )=\left\{ f\in C(X;{\cal R}): f\ge 0 \right\}. \]
 A generalized Bernstein's theorem would be a theorem stating when a convex cone
contained in $C^+(X;{\cal R})$ is dense in it.




Prolla  \cite{Prolla2} proved the following result of uniform density of convex cones in $C^+(X;\cal{R})$.


\bigskip


\begin{thmTEMA}
{\it Let $X$ be a compact Hausdorff space.
Let $W\subset C^+(X;\cal{R})$ be a convex cone satisfying the following
conditions:
\begin{description}
\item[(a)] given any two distinct points $x$ and $y$ in $X$, there
is a multiplier $\phi$ of $W$ such that $\phi(x)\ne \phi(y);$
\item[(b)] given any $x\in X$, there is $g\in W$ such that
$g(x)>0$.
\end{description}
Then $W$ is uniformly dense in $C^+(X;\cal{R}).$ }
\end{thmTEMA}

\bigskip

The purpose of this note is to present an extension of this
result to  weighted spaces and give some applications.
The  main tool is a Stone-Weierstrass-type theorem
for subsets of weighted spaces.

\newsec{The results}

\hspace*{0.3 cm}
We need the following lemma, whose proof  can be found in \cite{Prolla3}.

\begin{lemmaTEMA}\label{lema}
Let $W$ be a nonempty subset of $CV_ \infty(X;\cal{R})$. Given any $f\in CV_ \infty(X;\cal{R})$, $v\in V$ and $\varepsilon > 0$, the following statements are equivalent:
\begin{description}
\item[1.] there exists $h\in W$ such that $v(x)\|f(x)-h(x)\|< \varepsilon$ for all $x\in X$;
\item[2.] for each $x\in X$, there exists $g_x \in W$ such that $v(t)\|f(t)-g_x(t)\| < \varepsilon $ for all $ t\in [x]_{M(W)}.$
\end{description}
\end{lemmaTEMA}

Now we state the main result.

\begin{thmTEMA}\label{the}
Let $W\subset CV_ \infty^{+}(X;\cal{R})$ be a convex cone satisfying the following conditions:
\begin{description}
\item[(a)] given any two distinct points $x$ and $y$ in $X$, there exists a multiplier $\phi$ of $W$ such that $\phi(x)\ne \phi(y)$;
\item[(b)] given any $x\in X$, there exists $g\in W$ such that $g(x)>0$.
\end{description}
    Then $W$ is $\omega_V$-dense in $CV_ \infty^{+}(X;\cal{R})$.
\end{thmTEMA}

\begin{proof}
Let $x$ be an arbitrary element of $X.$ Condition (a) implies that $[x]_{M(W)}=\{x\}.$
By condition (b), there exists $g\in W$ such that $g(x)>0.$ Then, for any $f\in CV^+_ \infty(X;\cal{R})$, $v\in V$ and $\varepsilon >0$, we have

\[ v(x)\left \|f(x)-\frac{f(x)}{g(x)}g(x)\right \|=0<\varepsilon. \]

Since $W$ is a convex cone, $\frac{f(x)}{g(x)}g \in W$.
Then, it follows from Lemma \ref{lema} that there exists $h\in W$ such that
$v(t)\|f(t)-h(t)\| < \varepsilon$ for  all $t\in X$.
\end{proof}



\begin{coroTEMAi}
Let $X$ and $Y$ be locally compact Hausdorff spaces.
Then
\[ CV_ \infty^{+}(X;{\cal R}) \bigotimes CV_ \infty^{+}(Y;{\cal R}) \] is dense in $CV_ \infty^{+}(X \times Y;{\cal R})$.
\end{coroTEMAi}
\begin{proof}
 It follows from Urysohn's Lemma \cite{Rudin} that for any two distinct elements $(s,t)$ and $(u,v)$ of $X \times Y$, there exist functions $h_1 \in C_c(X;{\cal R})$  and $h_2 \in C_c(Y;{\cal R}),$ $0\le h_1, h_2 \le 1,$ such that $\varphi(x,y):=h_1(x)h_2(y)$ is a multiplier of $CV_ \infty^{+}(X;{\cal R}) \bigotimes CV_ \infty^{+}(Y;{\cal R})$ and $\varphi(s,t)=1$ and $\varphi(u,v)=0.$ Hence, condition (a) of Theorem \ref{the} is satisfied.

By using Urysohn's Lemma again, given $(x,y)  \in X \times Y$, there exist $\phi \in C_c(X;{\cal R})$ and $\psi \in C_c(Y;{\cal R})$ such that $\phi(x)=1$ and $\psi(y)=1$ so that $\phi(x)\psi(y)>0$, $\phi \psi \in CV_ \infty^{+}(X;{\cal R}) \bigotimes CV_ \infty^{+}(Y;{\cal R})$. Then, condition (b) of Theorem \ref{the} is satisfied.
Hence, the assertion follows by Theorem \ref{the}.
\end{proof}

\medskip

\noindent {\bf Example 2.1.}
 Consider  $CV_ \infty^{+}(\cal{R};\cal{R}),$ where  $V$ is the set of characteristic functions of all
compact subsets of $\cal{R}$.
Let $\psi \in C(\cal{R};{\cal R}),$  $0\le \psi \le 1,$ be a one-to-one function. Let $W$ be the set of all functions $g$ of the form

\[ g(x)= \sum_{i+j\le n} b_{ij}\psi(x)^i(1-\psi(x))^j, \hspace{1cm} x\in \cal{R} \]

\noindent where each  $b_{ij}$  is a non-negative real number and $i, j,n$ are non-negative integers numbers.
Note that $W \subset CV_ \infty^{+}(\cal{R};\cal{R})$ is a convex cone.

 Since $\psi \in M(W)$ and $W$ contains positive constant functions, it follows from Theorem \ref{the}  that $W$ is  dense in $CV_ \infty^{+}(\cal{R};\cal{R})$.

\bigskip

\noindent {\bf Example 2.2.}
 Let $a$ be  a fixed positive real number. Let $W$ be the set of all functions of the form
 \[ f(x)e^{-ax}, x\in [0, \infty), \,\, f\in C_b^+([0, \infty);\cal{R}). \]
 Clearly, $W$ is a convex cone contained in $C_0^+([0, \infty);\cal{R})$.
 The function $e^{-ax}$, $x\in [0, \infty),$ belongs to $W$ and is a multiplier of $W$ that separates the points of $X$. Hence, by Theorem \ref{the} $W$ is dense in $C_0^+([0, \infty);\cal{R})$.


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\end{thebibliography}



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