A piecewise polynomial lacunary interpolation method (Q1094614)

The following general lacunary interpolation problem is solved: let \(\Delta =\{x_ 1<x_ 2<...<x_{n+1}\}\) and let \(\nu_ 1,...,\nu_ p\) be integers with \(0\leq \nu_ 1<...<\nu_ p\). Suppose \(\{f_ i^{\nu_ j}\}\), \(j=1,...,p\); \(i=1,...,n+1\) are given real numbers. Find a function f defined on \([x_ 1,x_{n+1}]\) such that \(D^{\nu_ j}f(x_ i)=f_ i^{\nu_ j}\), \(j=1,...,p\) and \(i=1,...,n+1\). Special piecewise polynomial methods are developed for the solution. It is further shown that for typical smooth classes of functions the methods deliver optimal-order approximations.