Volume 23, pp. 320-328, 2006.

Approximation of the Hilbert transform via use of sinc convolution

Toshihiro Yamamoto

Abstract

This paper derives a novel method of approximating the Hilbert transform by the use of sinc convolution. The proposed method may be used to approximate the Hilbert transform over any subinterval $\mathrm{\Gamma }$ of the real line $\mathbf{R}\equiv (-\mathrm{\infty },\mathrm{\infty })$, which means the interval $\mathrm{\Gamma }$ may be a finite or semi-infinite interval, or the entire real line $\mathbf{R}$. Given a column vector $\mathbf{f}$ consisting of $m$ values of a function $f$ defined on $m$ sinc points of $\mathrm{\Gamma }$, we obtain a column vector $\mathbf{g}=\mathbf{A}\mathbf{f}$ whose entries approximate the Hilbert transform on the same set of $m$ sinc points. The present paper describes an explicit method for the construction of such a matrix $\mathbf{A}$.

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Key words

sinc methods, Hilbert transform, Cauchy principal value integral

AMS subject classifications

65R10