Orthogonal polynomials for the absolute-value weight (${\varphi }_0=1,{\varphi }_1=x,{\varphi }_2=x^2-\frac{1}{2}$)

For the inner product obtained by integrating with weight $|x|$ on $[-1,1]$, odd moments vanish and the moment of $x^{2m}$ is $1/(m+1)$. The first monic orthogonal polynomials are $1$, $x$ and $x^2-1/2$. Positive definiteness follows because the weight is positive almost everywhere and a nonzero polynomial cannot vanish almost everywhere.