On the approximation of the Hilbert transform

R. A. Aliev ; Ch. A. Gadjieva

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 2, pp. 30-41

Résumé

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 $L_{p} \left(R\right)$, $1$, 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 $u\in L_{p} \left(R\right)$, $1$, 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 $u\in L_{2} \left(R\right)$: $$ \lim\limits_{n\to \infty}\|H_{\delta/n} u-Hu\|_{L_{2}(R)}=0. $$

Détail

Mots-clés : Hilbert transform
Keywords: singular integral, approximation, discrete Hilbert transform.