High order approximation theory for Banach space valued functions

Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued di§erentiable functions to the unit operator. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators whose we study their approximation properties. We derive pointwise and uniform estimates which imply the approximation of these operators to the unit assuming di§erentiability of functions. At the end we study the special case where the high order derivative of the on hand function fulÖlls a convexity condition resulting into sharper estimates. MR3724631