Past exam of the approximation theory and finite element methods course of the University of Oxford / 2009 / A / 1 / b / Solution

With $w_s=1/{\pi }_{n,s}(x_s)$, the Lagrange interpolation polynomial can be written, away from its nodes,Apply the same formula to the constant data $u_s=1$. Uniqueness of polynomial interpolation gives ${\sum }_s{\ell }_s=1$, and henceDivide the two identities to obtain the barycentric interpolation formulaAt $x=x_r$, use the continuous limiting value $u_r$. Multiplying every weight by the same nonzero constant cancels between numerator and denominator, so only relative weights are needed in computation.