1 Correction to: Collectanea Mathematica https://doi.org/10.1007/s13348-021-00331-8

The goal of this erratum is to correct a mistake that appears in the proof of Corollary 3. In the second line of the proof we claim that if \(\varphi \) is a superadditive function and

$$\begin{aligned} M_0(t,\varphi )=\sup \limits _{\epsilon >0}\frac{\varphi (\epsilon ^n t)}{\varphi (\epsilon ^n)}, \end{aligned}$$

then there exists \(t_0>0\) such that \(M_0(t,\varphi ) \le t^{i_\varphi +1}\) for all \(0<t < t_0\), where

$$\begin{aligned} i_\varphi =\lim \limits _{t \rightarrow 0^+} \frac{\ln (M_0(t,\varphi ))}{\ln (t)}. \end{aligned}$$

This affirmation is not correct. For example the superadditive function \(\varphi (t)=t^{3/2}\) satisfies \(M_0(t,\varphi )=t^{3/2}\), \(i_\varphi =\frac{3}{2}\), and \(M_0(t,\varphi ) > t^{i_\varphi +1}\) for all \(0<t <1\).

However, Corollary 3 is still true. A proof is provided below.

Corollary 3

Assume \(x_0\in {\mathbb {R}}\) and \(f\in c_n^{\varPhi }(x_0)\) such that \(\varphi \) is a superadditive function. Let \(\{P_{B(x_0, \epsilon )}(f)\}\) be a net of best \(\varPhi \)-approximation of f from \(\varPi ^n\) on \(B(x_0, \epsilon )\). Then

$$\begin{aligned} \frac{1}{\epsilon \varphi (\epsilon ^n)} \int _{B(x_0,\epsilon )}\varphi (|P_{B(x_0, \epsilon )}(f)-C_{x_0,n}(f)|)dx=o(1), \quad \text {as} \quad \epsilon \rightarrow 0. \end{aligned}$$

Proof

If \(n=0\), it is obvious by [1, formula (20)]. Assume \(n>0\) and let

$$\begin{aligned} M_0(t,\varphi )=\sup \limits _{\epsilon >0}\frac{\varphi (\epsilon ^n t)}{\varphi (\epsilon ^n)}. \end{aligned}$$

Since \(M_0(\cdot ,\varphi )\) is non-decreasing and non-negative, the limit \(i=\lim \limits _{t \rightarrow 0^+} M_0(t,\varphi )\) exists. We claim that \(i=0\). In fact, as \(\varphi \) is a superadditive function, it is easy to see that

$$\begin{aligned} \varphi \left( \frac{u}{2^k}\right) \le \frac{\varphi (u)}{2^k}, \quad \text {for all} \quad u\ge 0 \quad \text {and} \quad k \in {\mathbb {N}}. \end{aligned}$$

Consequently \(0\le M_0\left( \frac{1}{2^k},\varphi \right) \le \frac{1}{2^k}\), for all \(k \in {\mathbb {N}}\), which gives \(i=0\).

Now, let \(\beta >0\) and let \(0<\eta <1\) be such that

$$\begin{aligned} M_0(\eta ,\varphi )< \frac{\beta }{2}. \end{aligned}$$

Then

$$\begin{aligned} \frac{\varphi (\epsilon ^n \eta )}{\varphi (\epsilon ^n)} < \frac{\beta }{2} \quad \text {for all} \quad \epsilon >0. \end{aligned}$$
(1)

From [1, formula (20)], there exists \(\epsilon _0=\epsilon _0(\eta )>0\), such that \(\epsilon ^{-n}|(P_{B(x_0, \epsilon )}(f)-C_{x_0,n}(f))(x)| \le \eta \), for all \(x \in B(x_0,\epsilon )\) and \(0<\epsilon < \epsilon _0\). According to (1) we have

$$\begin{aligned} \frac{\varphi (|(P_{B(x_0, \epsilon )}(f)-C_{x_0,n}(f))(x)|)}{\varphi (\epsilon ^n)} \le \frac{\varphi (\epsilon ^{n}\eta )}{\varphi (\epsilon ^n)}\le \frac{\beta }{2}, \end{aligned}$$

for all \(x \in B(x_0,\epsilon )\), \(0< \epsilon < \epsilon _0\). Whence by integrating on \(B(x_0,\epsilon )\) we can deduce that

$$\begin{aligned} \frac{1}{\epsilon \varphi (\epsilon ^{n})} \int _{B(x_0,\epsilon )} \varphi (|P_{B(x_0, \epsilon )}(f)-C_{x_0,n}(f)|)dx < \beta , \end{aligned}$$

for all \(0<\epsilon < \epsilon _0\). This completes the proof. \(\square \)

Finally, we observe that if \(\varphi \in {\mathcal {F}}\) then \(\varphi (u)>0\) for all \(u>0\). Thus, if \(\varphi \in {\mathcal {F}}\) is also superadditive then, for all \(0\le t < s\), \(\varphi (s-t)>0\), whence \(\varphi (t) < \varphi (t) + \varphi (s-t) \le \varphi (s)\), that is, \(\varphi \in {\mathcal {F}}\) is a strictly increasing function. So, in Theorem 5 and Corollary 4, we replace “\(\varphi \) is a superadditive strictly increasing function” by “\(\varphi \) is a superadditive function”.