On quadrature convergence of extended Lagrange interpolation

Abstract:

Quadrature convergence of the extended Lagrange interpolant $L_{2n+1}f$ for any continuous function $f$ is studied, where the interpolation nodes are the $n$ zeros $\tau _i$ of an orthogonal polynomial of degree $n$ and the $n+1$ zeros $\hat {\tau }_j$ of the corresponding “induced” orthogonal polynomial of degree $n+1$. It is found that, unlike convergence in the mean, quadrature convergence does hold for all four Chebyshev weight functions. This is shown by establishing the positivity of the underlying quadrature rule, whose weights are obtained explicitly. Necessary and sufficient conditions for positivity are also obtained in cases where the nodes $\tau _i$ and $\hat {\tau }_j$ interlace, and the conditions are checked numerically for the Jacobi weight function with parameters $\alpha$ and $\beta$. It is conjectured, in this case, that quadrature convergence holds for $| \alpha | \leq \frac {1}{2}, ~ | \beta | \leq \frac {1}{2}$.