An extremal problem of Erdős in interpolation theory
This paper follows a numerical approach to study the Erdős' optimization problem \[ I^*_n= \min_{X} I_n(X):= \min_X \int^1_{-1} \Biggl[\sum^n_{k= 1} l^2_k(X; x)\Biggr]dx,\tag{1} \] where \[ l_k(X; x)= \prod^n_{j= 1,j\neq k} {x- x_j\over x_k- x_j}. \] The minimal values of \(I_n(X)\) are found by applying optimization techniques with at least 25 correct digits. These numerical results suggest that the solution of the problem (1) is unique, the optimal nodes are symmetric on \([-1,1]\), and include the end points of the interval. The value of \(c_1\) in the asymptotic expansion \[ I^*_n= 2-{c_1\over n}-{c_2\over n^2}-\cdots\tag{2} \] was calculated by means of Richardson's extrapolation method. The result \(c_1= 1.094219\dots\), with at least seven significant digits, disproves the conjecture \(c_1= 1\) of Erdős, Szabados, Varma and Vertesi. Heuristic arguments are used to describe configurations of near-optimal nodes. In this report a set \(A\) of nodes, for which the values of \(I_n(A)\) are very close to \(I^*_n\), is given. A conjecture is made by comparing the expansion of \(I_n(A)\) with (2): The following asymptotic relationship holds \[ I^*_n- I_n(A)= o\Biggl({1\over n}\Biggr),\quad n\to\infty. \] {}.
scientific article; zbMATH DE number 1192649