## 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 $\Gamma$ of the real line ${\bf R} \equiv (-\infty, \infty)$, which means the interval $\Gamma$ may be a finite or semi-infinite interval, or the entire real line ${\bf R}$. Given a column vector ${\bf f}$ consisting of $m$ values of a function $f$ defined on $m$ sinc points of $\Gamma$, we obtain a column vector ${\bf g} = {\bf Af}$ 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 ${\bf A}$.

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### Key words

sinc methods, Hilbert transform, Cauchy principal value integral

65R10

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