Simultaneous Lagrange interpolating approximation need not always be convergent

Recently, several publications have been devoted to investigation of simultaneous Lagrange interpolating approximation. In this paper we carefully construct a counterexample with a system of nodes showing that the simultaneous Lagrange interpolating approximation need not always be convergent. It is especially interesting to note that the system of nodes behaving “badly” in this case is exactly the “near optimal choice” in the ordinary Lagrange interpolating case, the zeros of the Chebyshev polynomials.