Fractional Hankel and Bessel wavelet transforms of almost periodic signals

Banu Ünalmış Uzun

doi:10.1186/s13660-015-0909-8

Abstract

The main objective of this paper is to study the Hankel, fractional Hankel, and Bessel wavelet transforms using the Parseval relation. We construct a generalized frame and write new relations and inequalities using almost periodic functions, strong limit power signals, and these transform methods.

Highlights

The Hankel transform (HT) is a well-known integral transform which uses the νth order Bessel function of the first kind, Jν, as a kernel
Since the HT is equivalent to the twodimensional Fourier transform (FT) of a circularly symmetric function, it plays an important role in a number of applications including optical data processing, digital filtering, etc. [ – ]
The conventional HT is extended to the fractional Hankel transform (FrHT) by Kerr [ ] and Namias [ ]

Summary

Introduction

The Hankel transform (HT) is a well-known integral transform which uses the νth order Bessel function of the first kind, Jν , as a kernel. Its properties are discussed in detail by several authors [ – ] and its applications in many areas (such as optics, signal processing, quantum mechanics) are given in [ – ]. Using the theory of Hankel translation, Pathak and Dixit defined continuous and discrete Bessel wavelet transforms (BWT) and studied their properties [ ]. The fractional Hankel transformation and the continuous fractional Bessel wavelet transformation, some of their basic properties and applications are studied in [ ]. In [ ], the relation between the Bessel wavelet transformation and the Hankel-Hausdorff operator is discussed. We first of all introduce the FrHT, the BWT, and almost periodic functions and some of their properties in brief. 1.1 Hankel and fractional Hankel transforms We define the Fourier transform of an integrable function f as f(t) = (F f )(t) = √

The general HT is given as

The FrHT is defined by

Lνn y einπ xν

AP lim

Using the substitution u