Direct computation of the simultaneous stone-weierstrass approximation of a function and its partial derivatives in banach spaces, and combination with hermite interpolation

We present a new variant of the simultaneous Stone-Weierstrass approximation of a function and its partial derivatives, when the function takes its values in a Banach space, and provide an explicit and direct computation of this approximation. In the particular case of approximation by means of polynomials, we show that the simultaneous approximation can be required to be exact at a finite number of prescribed points.