The weight function $W$ is non-negative in the integration interval $x\in [a,b]$ such that for any $n \ge 0$, the integral $\int_a^b |x|^n W(x) dx$ is integrable. The naming weight function arises from the fact that it may be used to give more emphasis to one part of the interval than another. A quadrature formula $$\begin{equation} \int_a^b W(x)f(x)dx \approx \sum_{i=1}^N\omega_if(x_i), \tag{8} \end{equation}$$ with $N$ distinct quadrature points (mesh points) is a called a Gaussian quadrature formula if it integrates all polynomials $p\in P_{2N-1}$ exactly, that is $$\begin{equation} \int_a^bW(x)p(x)dx =\sum_{i=1}^N\omega_ip(x_i), \tag{9} \end{equation}$$ It is assumed that $W(x)$ is continuous and positive and that the integral $$\begin{equation*} \int_a^bW(x)dx \end{equation*}$$ exists. Note that the replacement of $f\rightarrow Wg$ is normally a better approximation due to the fact that we may isolate possible singularities of $W$ and its derivatives at the endpoints of the interval.