Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 2H (Topics in Analysis)

The Legendre polynomial $p_n$ is orthogonal on $[-1,1]$ to every polynomial of degree below $n$. Suppose it changes sign at only $r<n$ distinct interior points $x_1,\dots ,x_r$, and putchoosing its overall sign so that $p_{n}q\ge 0$ on $[-1,1]$. This product is positive except at finitely many points, socontradicting orthogonality because $\mathrm{deg}q=r<n$. Thus $p_n$ has at least $n$ sign-changing roots in $(-1,1)$; its degree is $n$, so these are all its roots and they are distinct.

For arbitrary distinct nodes $x_1,\dots ,x_n$, letbe the Lagrange basis polynomials. Every polynomial $P$ of degree below $n$ satisfies $P(t)={\sum }_{i}P(x_i)L_i(t)$, so the required weights exist and areThey are unique because applying any such formula to $P=L_i$ recovers $A_i$.

Now assume exactness for every $P$ of degree below $2n$ and define $Q(t)={\prod }_i(t-x_i)$. For every polynomial $R$ of degree below $n$, the product $QR$ has degree below $2n$ and vanishes at every node. ThereforeThere is a unique monic polynomial of degree $n$ orthogonal to all lower-degree polynomials, so $Q$ is the monic normalization of $p_n$ and the $x_i$ are precisely the roots of $p_n$. Taking $P=1$ givesFinally $L_i^2$ has degree $2n-2$, so exactness givesThese are the positive weights of Gaussian quadrature.