$\phi_{\sigma} (x) = O(|x|^{−α})$, as $x→±∞$.

I am currently studying a paper published in the "Journal of Approximation Theory" on Neural Networks. We have the following definitions:

Definition(Sigmoidal functions):A function $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is said to be sigmoidal function if it satisfies $$\lim_{x\rightarrow-\infty} σ(x) = 0 ~\text{and}~\lim_{x\rightarrow+\infty} σ(x) = 1.$$Further, we consider a sigmoidal function that is non-decreasing and satisfying three conditions:$$~(S1)~ σ(x) − 1/2~ \text{is an odd function};\\(S2)~ σ∈ C^2(\mathbb{R})~ \text{is concave for}~ x ≥ 0;\\(S3)~ σ(x) = O(|x|^{−α}) ~\text{as}~ x →−∞,~\text{ for some}~ α> 1,$$

Defintion($\phi_{\sigma}$):Now we define the kernal function $\phi_{\sigma}$ as follows:$\phi_{\sigma} :=1/2[σ(x + 1) −σ(x − 1)],~ x ∈ \mathbb{R}$.

Now i have to prove the following lemma:

Lemma 2.2. Let $\sigma$ be as sigmoidal function as in the Definition above, and let $\alpha$ be as in condition (S3). Then

• (i) $\phi_{\sigma} (x) ≥ 0$ for every $x ∈ \mathbb{R}$, with $\phi_{\sigma}(1) > 0$, and moreover $\lim_{x→±∞}\phi_{\sigma}(x) = 0;$
• (ii) The function $\phi_{\sigma}(x)$ is even;
• (iii) The function $\phi_{\sigma} (x)$:is non-decreasing for $x < 0$ and non-increasing for $x ≥ 0$;
• (iv) $\phi_{\sigma} (x) = O(|x|^{−α})$, as $x→±∞$.

Furthermore, $\phi_{\sigma} ∈ L^1(R)$;

• (v) For every $x ∈ \mathbb{R}$,$\sum_{k∈\mathbb{Z}}\phi_\sigma(x − k) = 1,$and$∥\phi_{\sigma} ∥_1 =\int_{\mathbb{R}}\phi_{\sigma} (x) dx =1$

In part (iv) we need to take help of condition (S3), i found its proof in some other paper but not satisfied with the way they have proved,the proof is as follows:

"Proof. By assumption (S3), there exist $C>0,M > 0$ such that$σ(x) ≤ C|x|^{−α}$, for every $x < −M$, for some given $α> 0$. Then,$\phi_σ (x) ≤σ(x + 1) ≤ C |x|^{−α}$,for every $x < −M − 1$. Moreover, using assumptions (S1) and(S3), we have for every $x > M + 1$$\phi_σ (x) ≤ 1 −σ(x − 1) = σ(1 − x) ≤ C |x|^{−α}$,and then the proof follows."

My Doubt:

What I don't understand here is:

Knowing $σ(x) ≤ C|x|^{−α}$, for every $x < −M$, for some given $α> 0$,Then,$\phi_σ (x) ≤σ(x + 1) ≤ C |x|^{−α}$,for every $x < −M − 1$

while i think it should be like ;

$σ(x) ≤ C|x|^{−α}$, for every $x < −M$, for some given $α> 0$. Then,$\phi_σ (x) ≤σ(x + 1) ≤ C |x+1|^{−α}$,for every $x < −M − 1$

How author has directly use $|x|^{−α}$ while if we change the input x to x+1 then it should be changed both sides.

similarly for the case of $\sigma(1-x)$ he has used same logic i.e., "we have for every $x > M + 1$$\phi_σ (x) ≤ 1 −σ(x − 1) = σ(1 − x) ≤ C |x|^{−α}$, while it should be like "$σ(1 − x) ≤ C |1-x|^{−α}$please help me to understand this doubt.

Thanks!!