An extremal problem of Erd s in interpolation theory

One of the intriguing problems of interpolation theory posed by Erd s in 1961 is the problem of finding a set of interpolation nodes in [−1, 1] minimizing the integral In of the sum of squares of the Lagrange fundamental polynomials. The guess of Erd s that the optimal set corresponds to the set F of the Fekete nodes (coinciding with the extrema of the Legendre polynomials) was disproved by Szabados in 1966. Another aspect of this problem is to find a sharp estimate for the minimal value I n of the integral. It was conjectured by Erd s, Szabados, Varma and Vertesi in 1994 that asymptotically I n − In(F) = o(1/n). In the present paper, we use a numerical approach in order to find the solution of this problem. By applying an appropriate optimization technique, we found the minimal values of the integral with high precision for n from 3 up to 100. On the basis of these results and by using Richardsonʹs extrapolation method, we found the first two terms in the asymptotic expansion of I*n, and thus, disproved the above-mentioned conjecture. Moreover, by using some heuristic arguments, we give an analytic description of nodes which are, for all practical purposes, as useful as the optimal nodes