Multivariate and abstract 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 Fréchet dierentiable functions to the unit operator, as well as other basic approximations including those under convexity. 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 for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming FrØchet dierentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fullls a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.
Anastassiou, G. (2017). Multivariate and abstract approximation theory for Banach space valued functions. Demonstratio Mathematica, 50 (1), 208-222. https://doi.org/10.1515/dema-2017-0020