Formulas for best extrapolation (Q2544892)

We consider \(n\)-point Lagrange-Hermite extrapolation for \(f(x)\), \(x>1\), based upon \(f(x_i)\), \(i=1(1)n\), \(-1\le x_i\le 1\) including non-distinct points \(x_i\) in confluent formulas involving derivatives. The problem is to find the points \(x_i\) that minimize ) in the remainder the factor \(P_n(x) \equiv \prod_{i=1}^n (x - x_i)\), in the remainder \(P_n(x)f^{(n)}(\xi)/n!\), \(-1 <\xi < x\), subject to the condition \(\vert P_n(x)\vert \le M\), \(-1\le x_i\le 1\), \(2^{-n+1} \le M\le 2^n\). The solution is significant only when a single set of points \(x_i\) suffices for every \(x>1\). The problem is here completely solved for \(n =1(1)4\). For \(n>4\) it may be conjectured that there is a single minimal \(P_n(x) = (x - 1)^r \prod_{I=1}^{n-r} (x - x_i)\), \(0\le r\le n\), where \(\Gamma\equiv \Gamma(M)\) is a non-decreasing function of \(M\), \(P_n(-1) = (-1)^nM\), and for \(0\le r\le n-2\), the \(j\)th extremum \(P_n(x_{0,j})= (-1 )^{n-j}M\), \(j=1\, (1)\, n - r - 1\) (except for \(M = M_r\), \(r= 1\,(1)\,n-1\), when \(j=1\,(1)\,n-r\).