On the Analytical Relationship between the Fractional Hilbert Transform and Fourier Transform

Page No.: 56-60

Abstract

The fractional Hilbert transform is a generalization of the classical Hilbert transform and plays an important role in various branches of engineering and science. In this paper, new analytical relationships between the fractional Hilbert transform and the classical Fourier transform are established. The results are derived using the definition of the fractional Hilbert transform and properties of the Fourier transform in the generalized function framework. Several fundamental transform identities are obtained, and the corresponding classical Hilbert transform relations are recovered as special cases. The derived theorems provide a useful theoretical framework for the analysis of fractional integral transforms and may find applications in solving differential equations, signal processing, and related areas of applied mathematics.