An extremal problem of Erdős in interpolation theory

AbstractOne 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 In 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 − In(F) = o(1n).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

On the Divergence of Lagrange Interpolation to |x|

AbstractIt is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In the present paper we show that the case of equally spaced nodes is not an exceptional one in this sense. Namely, we prove that the divergence everywhere in 0 < |x| < 1 of the Lagrange interpolation to |x| takes place for a broad family of nodes, including in particular the Newman nodes, which are known to be very efficient for rational interpolation