On mean convergence of extended Lagrange interpolation

AbstractLagrange interpolation to any continuous function on [-1,1] at the zeros of orthogonal polynomials is known to converge in the mean. Here, following Bellen, we study mean convergence of Lagrange interpolation on an extended set of nodes that includes, in addition to the n zeros of the orthagonal (relative to some positive weight function w) polynomial πn of degree n, other n+1 nodes, which in turn are zeros of an orthogonal polynomial \̂gpn+1 of degree n+1 corresponding to the weight function ŵn=π2nw. A sufficient criterion of Bellen for mean convergence (as n→∞) of such extended Lagrange interpolation, for arbitrary continuous functions, is shown to fail for Chebyshev weight functions of the first, third and fourth kind. (It holds trivially for Chebyshev weights of the second kind.) Based on extensive computations, it is conjectured, on the other hand, that the criterion is satisfied for certain Jacobi weights with parameters α and β suitably restricted. Necessary conditions for mean convergence, due to Erdős and Turán, are shown to be violated for the three kinds of Chebyshev weights mentioned above. For smooth functions, a comparison is made of the speed of convergence of simple vs. extended Lagrange interpolation