AN EXTREMAL PROBLEM OF ERDOS IN INTERPOLATION THEORY

One of the intriguing problems of interpolation theory posed by Erdos in 1961 is the problem of finding a set of interpolation nodes in [-1, 1] minimizing the integral I-n of the sum of squares of the Lagrange f undamental polynomials. The guess of Erdos that the optimal set corres ponds 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 val ue I-n of the integral. It was conjectured by Erdos, Szabados, Varma and Vertesi in 1994 that asymptotically I-n-I-n(F) = o(1/n). In the p resent paper, we use a numerical approach in order to find the solutio n 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 Richardso n's extrapolation method, we found the first two terms in the asymptot ic expansion of I-n, and thus, disproved the above-mentioned conjectu re. Moreover, by using some heuristic arguments, we give an analytic d escription of nodes which are, for all practical purposes, as useful a s the optimal nodes.