Complex Mean-Value Interpolation and Approximation of Holomorphic Functions

Abstract

We show that complex mean-value interpolation, a generalization of Lagrange Hermite interpolation, may be defined in any domain that is C-convex, whereas the original definition required ordinary, real convexity. We also show that C-convex domains are the natural ones in which to perform mean-value interpolation, in the sense that any Runge domain which admits mean-value interpolation must in fact be C-convex. Finally, we obtain an integral formula for the error and give some applications concerning approximation of holomorphic functions. 1997Academic Press

Kergin interpolation in Banach spaces

“…This is a contradiction so P F ( n F ) is dense in P( n F ). This also implies that P F ( n F ) is dense for σ(P( n F ), H Nb (E)) since by formula (6), f, Q = n! f n , Q n for every f ∈ H Nb (E) and Q ∈ P( n F ) ⊆ Exp(F ).…”

“…The Kergin operator on function spaces with finite-dimensional domains is studied in [1,6]. Many of its properties depend only on the properties of the functions under consideration restricted to finite-dimensional subspaces and are therefore easily extended to Gateaux holomorphic functions.…”

On the convergence of Kergin and Hakopian interpolants at Leja sequences for the disk

“…There is an explicit but rather complicated formula for mean-value interpolation, see [10,Theorem 1] or [9,Theorem 4.3]. Here we summarize a few basic properties of mean-value interpolation.…”

“…The general definition is recalled below. Approximation properties of Kergin and Hakopian interpolation polynomials have been deeply investigated, see e.g., [1,[3][4][5]9]. Elegant results were in particular obtained in the two-dimensional case when the nodes form a complete set of roots of unity (viewed as a subset of R 2 ).…”

Lagrange approximation in Banach spaces

“…Lagrange interpolation has been generalized to several variables by the work of Kergin [10], Andersson and Passare [1], and the approximation problem has been studied by Bloom [2], Filipsson [7] and others.…”