# a Kernel free asymptotic for a sampling operator

Let $$\Pi=\left\{ t_{k}\right\} _{k\in\mathbb{Z}}$$ a sequence of real numbers such that $$-\infty for every $$k\in\mathbb{Z}$$, $$\lim_{k\rightarrow\pm\infty}t_{k}=\pm\infty$$ and such that there are two positive constants $$\Delta,\,\delta$$ with $$\delta\leq t_{k+1}-t_{k}\leq\Delta$$. We also define $$\Delta_{k}:=t_{k+1}-t_{k}$$, for every $$k\in\mathbb{Z}$$. We call a function $$\chi:\,\mathbb{R}\rightarrow\mathbb{R}$$ a kernel if it belongs to $$L^{1}\left(\mathbb{R}\right)$$, is bounded in a neighbourhood of the origin, and satisfies the following conditions :

$$(\chi_{1})$$ for every $$u\in\mathbb{R}$$ $$\sum_{k\in\mathbb{Z}}\chi\left(u-t_{k}\right)=1;$$

$$(\chi_{2})$$ $$m_{1,\,\Pi}\left(\chi\right)=\sup_{u\in\mathbb{R}}\sum_{k\in\mathbb{Z}}\left|\chi\left(u-t_{k}\right)\right|\left|u-t_{k}\right|<+\infty.$$ Let us define the following operator: $$\left(K_{w}^{\chi}f\right)\left(x\right):=\sum_{k\in\mathbb{Z}}\chi\left(wx-t_{k}\right)\left[\frac{w}{\Delta}_{k}\int_{t_{k}/w}^{t_{k+1}/w}f\left(u\right)du\right],\,x\in\mathbb{R}$$where $$f:\,\mathbb{R}\rightarrow\mathbb{R}$$ is locally integrable function such that the above series is a convergent for every $$x\in\mathbb{R}$$ and $$\chi$$ and $$w \geq \overline{w}>0$$. Assume that exists a kernel belonging to $$C^{1}(\mathbb{R})$$ such that $$\left\Vert K_{w}^{\chi}f-f\right\Vert _{\infty}=O\left(w^{-1}\right)$$ as $$w \rightarrow +\infty$$, where the implicit constant depends only on $$f$$ and $$\chi$$ and $$\left\Vert \cdot\right\Vert _{\infty}$$ is the classical sup norm.

I would like to prove that exists some $$s \neq 0$$ such that $$\left\Vert K_{w}^{\chi_{s}}f-f\right\Vert _{\infty}=O\left(w^{-1}\right)$$ as $$w \rightarrow +\infty$$, where $$\chi_{s}(\cdot)=\chi(\cdot+s)$$ (which is obviously a kernel).

I'm quite sure that it is true but I'm not able to prove it. I tried to use $$\left\Vert K_{w}^{\chi_{s}}f-f\right\Vert _{\infty}\leq\left\Vert K_{w}^{\chi}f-f\right\Vert _{\infty}+\left\Vert K_{w}^{\chi_{s}}f-K_{w}^{\chi}f\right\Vert _{\infty}$$ but it not seems very helpful. Thank you.