Errata Corrige to: "Approximation by means of nonlinear integral operators in the space of functions with bounded $\varphi-$variation"

Abstract

Here the authors want to point out that the term $I_2,$ of Proposition 2 of the original paper ([2]), has to be estimated in a different way. Moreover, now the proof of Theorem 4 of the original paper holds with the new assumption (2) (involving $K_w.3)'$ mentioned below), instead of (6.2) of [2] (involving $K_w.3)$ of the original paper), while, since Theorem 3 (convergence theorem) can be proved with both assumptions $K_w.3)$ and $K_w.3)'$, we prefer here to use directly $K_w.3)'$, in analogy with condition (2). Let us notice that it is easy to see that the two conditions $K_w.3)$ and $K_w.3)'$ cannot be compared. Here we want also to point out that in the convergence theorem of [3] as well as in Lemma 2 of [3], a similar problem occurs and it is solved in the same way proving that $V_{\varphi}[\lambda (H_w \circ f-f)]{\mbox{$\rightarrow$}} 0$, as $w{\mbox{$\rightarrow$}} +\infty$ for sufficiently small $\lambda>0,$ using assumption $K_w.3)$ (see Remark below).