Kergin interpolation in Banach spaces
The author extends the notion of Kergin interpolation [\textit{P. Kergin}, J. Approximation Theory 29, 278--293 (1980; Zbl 0492.41008)] to infinite dimensional, real or complex Banach spaces \(X\) and \(Y\). For a sequence \(p = (p_0,p_1,\dots p_j)\) of points in \(X\) and for any continuous function \(g\) on the convex hull of \(p\), simplex functional w.r.. \(p\) is defined by \(g \rightarrow \int g(p_0+s_1(p_1-p_0) +\dots +s_j(p_j -p_0))\,ds_1\dots ds_j\) where the integration is over the standard \(j\)-simplex in \(\mathcal{R}^{j}\). When the components \(p_i\) are in an open convex set \(U\), for a \(f \in C^k(U,Y)\), the Kergin polynomial of \(f\) w.r.t. \(p\) is defined by \(K_pf(x) = f(p_0) + \int D_{x-p_0} f + \dots + \int D_{x-p_k-1}\dots D_{x-p_0} f\). With these notations the main result here states that \(K_p f\) is a polynomial of degree at most \(k\) such that \(K_p f(p_j) = f(p_j)\) for \( j = 0,1,\dots ,k\). Further \(K_p\) is independent of the order of the points in \(p\), is associative , affine invariant and is a projection. An error formula and estimate for the rate of convergence in the case of functions holomorphic on balls are also given.
scientific article; zbMATH DE number 2081951