Kergin approximation in Banach spaces

We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges. Kergin interpolation is a generalization of both Lagrange interpolation in the one dimensional case, and the Taylor polynomial in the case where all interpolation points coincide. In several variables, interpolation polynomials are not unique. However, Kergin [K] proved that interpolation polynomials enjoying several natural properties exist and are unique: Theorem 1 (Kergin). Let N ∈ N,K ∈ N, and x0, . . . , xK ∈ R , not necessarily distinct. There is a unique χ : C(R )→ P(R ) satisfying: 1. χ is linear. 2. For every f ∈ C(R ), every q ∈ Q(R ), where k ∈ {0, . . . ,K}, and every J ⊂ {0, . . . ,K} with card J = k + 1, there exists x ∈ [xj ]j∈J such that q(∂/∂x)(χ(f)− f)(x) = 0. Here, C(R ) is the set of functions withK continuous derivatives, Q is the set of differential operators of degree k with constant coefficients, and P(R ) is the set of polynomials of degree K. It fell to Micchelli [Mi] and Milman [MM] to discover a formula for these polynomials. This formula also extends to the Banach space case, see [F, P]. In this case, the potential unboundedness of continuous functions, even on bounded sets bounded away from the boundary of the domain, presents new difficulties in proving convergence results. Filipsson [F] proved a convergence result for holomorphic functions bounded on a ball. We give the formula for the Kergin polynomial below. Let X,Y be complex Banach spaces, U ⊂ X open and f : U → Y . Define df = f and df : U ×X → Y , df(x; ξ1, . . . , ξk+1) = lim t→0 1 t df(x+ tξk+1; ξ1, . . . , ξk), if this limit exists. This is just the k+1 iteration of the directional derivative of f , see, e.g., [L]. Let p0, . . . pn,∈ X. Suppose f is an n-times differentiable