  We consider the problem of minimizing a given $n$-variate polynomial $f$ over
the hypercube $[-1,1]^n$. An idea introduced by Lasserre, is to find a
probability distribution on $[-1,1]^n$ with polynomial density function $h$ (of
given degree $r$) that minimizes the expectation $\int_{[-1,1]^n}
f(x)h(x)d\mu(x)$, where $d\mu(x)$ is a fixed, finite Borel measure supported on
$[-1,1]^n$. It is known that, for the Lebesgue measure $d\mu(x) = dx$, one may
show an error bound $O(1/\sqrt{r})$ if $h$ is a sum-of-squares density, and an
$O(1/r)$ error bound if $h$ is the density of a beta distribution. In this
paper, we show an error bound of $O(1/r^2)$, if $d\mu(x) = \left(\prod_{i=1}^n
\sqrt{1-x_i^2} \right)^{-1}dx$ (the well-known measure in the study of
orthogonal polynomials), and $h$ has a Schm\"udgen-type representation with
respect to $[-1,1]^n$, which is a more general condition than a sum of squares.
The convergence rate analysis relies on the theory of polynomial kernels, and
in particular on Jackson kernels. We also show that the resulting upper bounds
may be computed as generalized eigenvalue problems, as is also the case for
sum-of-squares densities.
