  We introduce a modified version of the Whitney extension operators for
collections of functions from a closed subset of $\mathbb{R}^n$ into scales of
Banach spaces with smoothing operators. We prove an extension theorem for
collections whose elements take values in different spaces of the scale. A
motivation for considering this kind of collections comes from very basic
observations on the composition of functions of more than one real variable.
The idea at the base of the proof is rather natural in the context of scales of
Banach spaces, and consists in introducing smoothing operators in the
construction of the extension, with smoothing parameters related to the
diameter of each Whitney dyadic cube. Classical examples of scales of Banach
spaces with smoothing operators are also given, and some new related
observations are proved.
