An extremal problem of Erdős in interpolation theory

Abstract

One of the intriguing problems of interpolation theory posed by Erdős 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 fundamental polynomials. The guess of Erdős 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 Erdős, Szabados, Varma and Vertesi in 1994 that asymptotically I★ n − I n(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 Richardson's 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.