interval $\left[
a,b\right] $ containing the support of $\mu _{\mathcal{G}}\left( x\right) $ is equal to $\left[ \lambda _{n},\lambda _{1}\right] $, by the definition in (\[Spectral Measure\]).

Lasserre also proposed in [@Las11] a numerical scheme to compute tight bounds on the values of $a$ and $b$ when a *truncated* sequence of moments $\left( m_{r}\right) _{0\leq r\leq k}$ is known. This numerical scheme involves a series of semidefinite programs (SDP) in one variable. As we show below, at step $s$ of this series of SDP’s, we are given a sequence of moments $\left( m_{1},...,m_{2s+1}\right) $ and solve two SDP’s whose solution provides an inner approximation $\left[ \alpha _{s},\beta _{s}%
\right] \subseteq \left[ a,b\right] $. In our case, since we have expressions for the first five spectral moments, $\left( m_{1}\left( A_{%
\mathcal{G}}\right) ,...,m_{5}\left( A_{\mathcal{G}}\right) \right) $, we can solve the first two steps of this series of SDP’s to find inner approximations $\left[ \alpha _{s},\beta _{s}\right] \subseteq \left[
\lambda _{n},\lambda _{1}\right] $. In other words, the solution to the SDP’s provide us with the bounds $\alpha _{s}\geq \lambda _{n}$ and $\beta
_{s}\leq \lambda _{1}$.

In order to formulate the series of SDP’s proposed in [@Las11], we need to define the so-called localizing matrix 