of our problem [@LasBOOK]. Given a sequence of moments, $\mathbf{m}^{\left( 2s+1\right) }=\left(
m_{1},...,m_{2s+1}\right) $, our localizing matrix is a Hankel matrix defined as:$$H_{s}\left( c\right) \triangleq R_{2s+1}-c~R_{2s}\text{,}
\label{Localizing matrix}$$where $R_{2s}$ and $R_{2s+1}$ are the Hankel matrices of moments defined  as$$\begin{aligned}
R_{2s}& =\left[ 
\begin{array}{cccc}
1 & m_{1} & \cdots & m_{s} \\ 
m_{1} & m_{2} & \cdots & m_{s+1} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
m_{s} & m_{s+1} & \cdots & m_{2s}%
\end{array}%
\right] ,  \label{Hankel matrices} \\
R_{2s+1}& =\left[ 
\begin{array}{cccc}
m_{1} & m_{2} & \cdots & m_{s+1} \\ 
m_{2} & m_{3} & \cdots & m_{s+2} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
m_{s+1} & m_{s+2} & \cdots & m_{2s+1}%
\end{array}%
\right] .  \notag\end{aligned}$$Hence, for a given sequence of moments, the entries of $H_{s}\left( c\right) 
$ depend affinely on the variable $c$. We can compute $\alpha _{s}$ and $%
\beta _{s}$ as the solution to the following semidefinite programs [Las11]{}:

\[Lasserre bound\]Let $\mathbf{m}^{\left( 2s+1\right) }=\left(
m_{1},...,m_{2s+1}\right) $ be the truncated sequence of moments of a positive Borel measure $\mu $. Then,$$\begin{aligned}
a& \leq \alpha _{s}\triangleq \max_{\alpha }\left\{ \alpha :H_{s}\left(
\alpha \right) \succcurlyeq 0\right\} ,  \label{Bound min eigenval} \\
b& \geq \beta _{s}\triangleq \min_{\beta }\left\{ \beta :-H_{s}\left( \beta
