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How to show that integral of density(kernal) function $\phi_{\sigma}$ over $\mathbb{R}$is equal to 1.

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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$∥\phi_{\sigma} ∥_1 =\int_{\mathbb{R}}\phi_{\sigma} (x) dx =1$

I have proved all parts of lemma but is not able the prove that integral of density function is 1 over reals i.e how to prove that $$ ||\phi_{\sigma} ||_1 =\int_{\mathbb{R}}\phi_{\sigma} (x) dx =1$$ i think it follows directly from $\sum_{k∈\mathbb{Z}}\phi_\sigma(x − k) = 1,$ but still not able to compplete.Please help me to complete the proof.Also i do not understand $\phi_{\sigma} ∈ L^1(R)$ knowing $\phi_{\sigma} (x) = O(|x|^{−α})$

Thanks in advance!!


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