Convergence of multivariate polynomials interpolating on a triangular array

Abstract:

Given a triangular array of complex numbers, it is well known that for any function $f$ smooth enough, there is a unique polynomial ${G_n}f$ of degree $\leq n$ such that on each of the first $n + 1$ rows of the array the divided difference of ${G_n}f$ coincides with that of $f$. This result has recently been generalized to give a unique polynomial ${\mathcal {G}_n}f$ in $k$ variables $(k > 1)$ of total degree $\leq n$ which interpolates a given function $f$ on a triangular array in ${C^k}$. In this paper we extend some results of A. O. Gelfond and derive formulas for ${\mathcal {G}_n}f$ and $f - {\mathcal {G}_n}f$ to prove some results on convergence of ${\mathcal {G}_n}f$ to $f$ as $n \to \infty$ under various conditions on $f$ and on the triangular array.