# Correction to: A Generalized Fejér’s Theorem for Locally Compact Groups

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## 1 Correction to: J Geom Anal (2018) 28:909–920 https://doi.org/10.1007/s12220-017-9847-7

As pointed out by Hanfeng Li, we fix a gap in Theorem 1.2.

- (1)
Let

*G*be a locally compact group with the unit \(e_G\) and a fixed left Haar measure \(\mu \). Let \(\nu \) be the right Haar measure given by \(\nu (A)=\mu (A^{-1})\) for every Borel subset*A*of*G*.

### Theorem

(A generalized Fejér’s theorem)

Consider a locally compact group *G* with a fixed left Haar measure \(\mu \) and the corresponding right Haar measure \(\nu \). Let \(\{F_\theta \}_{\theta \in \Theta }\) be an approximate identity of \(L^1(G)\). Assume that there exists a local partition \(\{A_1,A_2,\ldots , A_k\}\) of *G* such that \(\displaystyle \lim _{\theta }\int _{A_j}F_\theta (y)\,d\mu (y)=\lambda _j\) for every \(1\le j\le k\).

*f*in \(L^\infty (G)\), if there exists

*x*in

*G*such that \(\displaystyle \lim _{\begin{array}{c} y\rightarrow e_G \\ y\in A_j \end{array}} f(y^{-1}x)\) (denoted by \(f(x,A_j)\)) exists for every \(1\le j\le k\), then

*f*in \(L^1(G,\nu )\) (or \(L^\infty (G)\)) such that each \(f(x,A_j)\) exists for some

*x*in

*G*, we have

- (a)
The sentence “Now assume that \(\displaystyle \lim _{\theta }\sup _{y\in \mathcal {N}^c}|F_\theta (y)|=0\) for any neighborhood \(\mathcal {N}\) of \(e_G\) and

*f*is in \(L^1(G)\) such that each \(f(x,A_j)\) exists for some*x*in*G*” in the line -5 on page 5 is changed to “Now assume that \(\displaystyle \lim _{\theta }\sup _{y\in \mathcal {N}^c}|F_\theta (y)|=0\) for any neighborhood \(\mathcal {N}\) of \(e_G\) and*f*is in \(L^1(G,\nu )\) such that each \(f(x,A_j)\) exists for some*x*in*G*”. - (b)the last identities are changed to:$$\begin{aligned} \limsup _{\theta }\Big |\int _{\mathcal {N}^c} F_\theta (y)f(y^{-1}x)\,d\mu (y)\Big |\le \Big [\lim _{\theta }\sup _{y\in \mathcal {N}^c}|F_\theta (y)|\Big ] \Vert f\Vert _{{L^1(G,\nu )}}=0. \end{aligned}$$

- (2)
Statement of Corollary 2.2 is changed to the following:

### Corollary

*G*, if for an

*f*in \(L^\infty (G)\), every \(f(x, A_j)\) exists, then there exists a subnet \(\Theta _1\) of \(\Theta \) such that every \(\displaystyle \lim _{\theta \in \Theta _1}\int _{A_j}F_\theta (y)\,d\mu (y)\) exists (denoted by \(\lambda _j(\Theta _1)\)) and

*f*in \(L^1(G,\nu )\) (or \(L^\infty (G)\)) such that every \(f(x,A_j)\) exists for some

*x*in

*G*, we have

- (3)
Before Corollary 3.5, add a definition.