Kergin Interpolants of Holomorphic Functions

Abstract.

Let D be a C-convex domain in C ⁿ. Let $\{A_{dj}\}, \ j = 0,\ldots,d$ , and d = 0,1,2, ..., be an array of points in a compact set $K \subset D$ . Let f be holomorphic on $\overline D$ and let K _d (f) denote the Kergin interpolating polynomial to f at A _d0 ,... , A _dd . We give conditions on the array and D such that $\lim_{d\to\infty} \|K_d (f) - f\|_K = 0$ . The conditions are, in an appropriate sense, optimal.

This result generalizes classical one variable results on the convergence of Lagrange—Hermite interpolants of analytic functions.