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.
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
$$
\widetilde{h}(b)=\big\{(\widetilde{h}(b))_{n}\big\}_{n\in \mathbb Z},\quad \big(\widetilde{h}(b)\big)_{n}=\sum_{m\in \mathbb Z}\frac{b_{m}}{n-m-1/2},\quad n\in \mathbb Z,\quad b=\{b_{n}\}_{n\in \mathbb Z} \in l_{1}.
$$
Theorem 2. For any $\delta >0$ and $u\in L_{p} \left(R\right)$, $1<p<\infty$, the following inequality holds:
\[H_{\delta } \left(H_{\delta } u\right)\left(t\right)=-u\left(t\right).\]
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 $u\in L_{2} \left(R\right)$:
$$
\lim\limits_{n\to \infty}\|H_{\delta/n} u-Hu\|_{L_{2}(R)}=0.
$$
Keywords: Hilbert transform, singular integral, approximation, discrete Hilbert transform
Received April 8, 2019
Revised May 6, 2019
Accepted May 13, 2019
Rashid Avyazaga oglu Aliev, Dr. Math. Sci., Baku State University, Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan, e-mail: aliyevrashid@mail.ru
Chinara Arif kizi Gadjieva, doctoral student, Baku Engineering University, Baku, Azerbaijan,
e-mail: hacizade.chinara@gmail.com
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Cite this article as: R.A.Aliev, Ch.A.Gadjieva. On the approximation of the Hilbert transform. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 30–41.