On the approximation of the Hilbert transform

The article is devoted to the approximation of the Hilbert transform $\left(Hu\right)\left(t\right)=\displaystyle\frac{1}{\pi } \int _{R}\displaystyle\frac{u\left(\tau \right)}{t-\tau } d\tau $ of functions $u\in L_{2} \left(R\right)$ by operators of the form $(H_{\delta}u)(t)=\displaystyle\frac{1}{\pi}\sum_{k=-\infty}^{\infty}\displaystyle \frac{u(t+(k+1/2)\delta)}{-k-1/2}$,  $\delta >0$. The main results are the following statements.
$\bf{Theorem~1.}$  For any $\delta >0$ the operators $H_{\delta } $ are bounded in the space $L_{p} \left(R\right)$, $1<p<\infty $, and
$$\left\| H_{\delta } \right\| _{L_{p} \left(R\right)\to L_{p} \left(R\right)} \le \left\| \tilde{h}\right\| _{l_{p} \to l_{p} },$$
where $\tilde{h}$ is the modified discrete Hilbert transform defined by the equality 

$$ \widetilde{h}(b)=\big\{(\widetilde{h}(b))_{n}\big\}_{n\in \mathbb Z},\quad  \big(\widetilde{h}(b)\big)_{n}=\sum_{m\in \mathbb Z}\frac{b_{m}}{n-m-1/2},\quad n\in \mathbb Z,\quad b=\{b_{n}\}_{n\in \mathbb Z} \in l_{1}. $$

$\bf {Theorem~2.}$  For any $\delta >0$ and $u\in L_{p} \left(R\right)$, $1<p<\infty$, the following inequality holds:
$$H_{\delta } \left(H_{\delta } u\right)\left(t\right)=-u\left(t\right).$$

$\bf {Theorem~3.}$  For any $\delta >0$ the sequence of operators $\{H_{\delta/n}\}_{n\in \mathbb N}$  strongly converges to the operator $H$ in $L_{2} \left(R\right)$; i.e., the following inequality holds for any $u\in L_{2} \left(R\right)$:
$$ \lim\limits_{n\to \infty}\|H_{\delta/n} u-Hu\|_{L_{2}(R)}=0. $$