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Important problems of real algebraic geometry are representations of
non-negative polynomials on closed semialgebraic sets. Recall the
17th Hilbert problem (solved by E. Artin (1927)): if $f\in
\mathbf{\mathbb{R}}[x]$ is non-negative on $\mathbf{\mathbb{R}}^n$,
then $fh^2=h_1^2+\cdots+h_m^2 \hbox{ for some $h,h_1,\ldots,h_m\in
\mathbf{\mathbb{R}}[x]$, $h\ne 0$,}$ that is, $f$ is a sum of
squares of rational functions. If $f$ is homogeneous and $f(x)>0$
for $x\ne 0$, B. Reznick (1995) proved that the polynomial
$(x_1^2+\cdots+x_n^2)^Nf(x)$ is a sum of even powers of linear
functions provided $N\in\mathbb{Z}$ is sufficiently large.
Let $X\subset \mathbf{\mathbb{R}}^n$ be a {closed basic
semialgebraic set} defined by $g_1,\ldots,g_r\in
\mathbf{\mathbb{R}}[x]$, i.e., $X=\{x\in
\mathbf{\mathbb{R}}^n:g_1(x) \ge 0,\ldots,g_r(x)\ge 0\}$. The
{preordering} generated by $g_1,\ldots,g_r$ denoted by
$T(g_1,\ldots,g_r)$ is defined to be the set of polynomials of the
form $\sum_{e\in\{0,1\}^r}\sigma_e g_1^{e_1}\cdots g_r^{e_r}$, where
$\sigma_e\in \sum \mathbf{\mathbb{R}}[x]^2$ for $e\in\{0,1\}^r$ and
$\sum \mathbf{\mathbb{R}}[x]^2$ denotes the set of sums of squares
(s.o.s.) of polynomials from $\mathbf{\mathbb{R}}[x]$. Natural
generalizations of the above theorem of Artin are the Stellensätze
of J.-L. Krivine (1964), D. W. Dubois (1969), and J.-J. Risler
(1970). When the set $X$ is compact, a very important result was
obtained by K.Schmüdgen (1991): every strictly positive polynomial
$f$ on $X$ belongs to $T(g_1,\ldots,g_r)$. C. Berg,
J. P. R. Chris\-ten\-sen and P. Ressel (1976) and J. B. Lasserre and
T. Netzer (2007) proved that any polynomial $f$ which is
non-negative on $[-1,1]^n$ can be approximated in the $l_1$-norm by
sums of squares of polynomials. In this connection J. B. Lasserre
(2008) obtained a result on approximation in the $l_1$-norm of
convex polynomials provided that $g_1,\ldots,g_r$ are concave.
We show that a polynomial $f\in \mathbf{\mathbb{R}}[x]$ is
non-negative on the set $X$, if and only if $f$ can be approximated
uniformly on compact sets by polynomials of the form
$\sigma_0+\varphi(g_1)\cdot g_1+\cdots +\varphi(g_r)\cdot g_r$,
where $\sigma_0\in \mathbf{\mathbb{R}}[x]^2$ and
$\varphi\in\mathbf{\mathbb{R}}[t]^2$. Moreover, if $X$ is a convex
set such that $0\not\in X$, and $d$ is a positive even number such
that $d>\deg f$, then the above conditions are equivalent to: for
any $a>0$ there exists $N_0\in\mathbf{\mathbb{N}}$ such that for any
integer $N\ge N_0$ the polynomial
$\varphi_N(x)=(1+|x|^2)^N(f(x)+a|x|^{d})$ is a strictly convex
function on $X$.
Additionally, we give necessary and sufficient conditions for the
existence of an exponent $N\in\mathbb{N}$ such that
$(1+|x|^2)^Nf(x)$ is a convex function on $X$.
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