The Fractional Hilbert Transform on the Real Line

Abstract

This paper examines some special properties and important results of the fractional Hilbert transform (FHT) on the real line $ℝ$ . In this rigorous study, we modify certain theorems of classical Hilbert transform for FHT and develop new theorems. Moreover, FHT of some common functions is given and eigenfunctions are also studied. We prove that FHT, denoted by $H_{\alpha }$ , is an isomorphism on $L^2\left(ℝ\right)$ . We also show that in $L^2\left(ℝ\right)$ , FHT is an isometry. Furthermore, we investigate Riesz inequality on $L^p\left(ℝ\right)$ for $p>1$ to establish Hilbert formulae for FHT.