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
then there exists \(t_0>0\) such that \(M_0(t,\varphi ) \le t^{i_\varphi +1}\) for all \(0<t < t_0\), where
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
Proof
If \(n=0\), it is obvious by [1, formula (20)]. Assume \(n>0\) and let
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
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
Then
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
for all \(x \in B(x_0,\epsilon )\), \(0< \epsilon < \epsilon _0\). Whence by integrating on \(B(x_0,\epsilon )\) we can deduce that
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”.
Reference
Ferreyra, D.E., Levis, F.E., Roldán, M.V.: A new concept of smoothness in Orlicz spaces. Collect. Math. (2021)
Acknowledgements
The authors wants to acknowledge Federico Kovac (Universidad Nacional de La Pampa) for the useful comments and suggestions for improving the erratum.
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Ferreyra, D.E., Levis, F.E. & Roldán, M.V. Correction to: A new concept of smoothness in Orlicz spaces. Collect. Math. 74, 501–502 (2023). https://doi.org/10.1007/s13348-022-00355-8
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DOI: https://doi.org/10.1007/s13348-022-00355-8