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 Trudy Inst. Mat. i Mekh. UrO RAN, 2019, Volume 25, Number 2, Pages 30–41 (Mi timm1621)

On the approximation of the Hilbert transform

R. A. Alievab, Ch. A. Gadjievac

a Baku State University
b Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku
c Baku Engineering University

Abstract: The article is devoted to the approximation of the Hilbert transform $(Hu)(t)=\displaystyle\frac{1}{\pi } \int _{R}\displaystyle\frac{u(\tau )}{t-\tau } d\tau$ of functions $u\in L_{2} (R)$ 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 $L_{p} (R)$, $1<p<\infty$, and
$$\| H_{\delta } \| _{L_{p} (R)\to L_{p} (R)} \le \| \tilde{h}\| _{l_{p} \to l_{p} },$$
where $\tilde{h}$ is the modified discrete Hilbert transform defined by the equality

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

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

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

Keywords: Hilbert transform, singular integral, approximation, discrete Hilbert transform.

DOI: https://doi.org/10.21538/0134-4889-2019-25-2-30-41

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UDC: 517.518.85+519.651
MSC: 44A15, 42A50, 41A35, 65D30

Citation: R. A. Aliev, Ch. A. Gadjieva, “On the approximation of the Hilbert transform”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 2, 2019, 30–41

Citation in format AMSBIB
\Bibitem{AliGad19} \by R.~A.~Aliev, Ch.~A.~Gadjieva \paper On the approximation of the Hilbert transform \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2019 \vol 25 \issue 2 \pages 30--41 \mathnet{http://mi.mathnet.ru/timm1621} \crossref{https://doi.org/10.21538/0134-4889-2019-25-2-30-41} \elib{http://elibrary.ru/item.asp?id=38071597}