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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

We show that Kergin interpolation, a generalized Lagrange-Hermite polynomial interpolation, may be defined on mappings between general Banach spaces. Like its finitedimensional counterpart, Kergin interpolation in this setting is an affine-invariant projector. We obtain an error formula which we use to approximate holomorphic mappings. As an application we give a convergence theorem applicable to, for instance, operators on Banach algebras, such as the algebra of square matrices with complex coefficients. r

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