Wavelet transforms associated with the Kontorovich–Lebedev transform

Abstract

The main objective of this paper is to study continuous wavelet transform (CWT) using the convolution theory of Kontorovich–Lebedev transform (KL-transform) and discuss some of its basic properties. Plancherel’s as well as Parseval’s relation and Reconstruction formula for CWT are obtained and some examples are also given. The discrete version of the wavelet transform associated with KL-transform is also given and reconstruction formula is derived.

Wavelet transforms associated with the index Whittaker transform

“…Since translation, dilation, and convolution operators have been defined for various integral transforms, using these operators and following the above notion, many authors defined wavelets and wavelet transforms in terms of fractional Fourier transform 12,14,15 and studied their theory and properties. In addition, some more wavelet transforms have also been established for positive half line by using the transforms of special functions, namely, the Bessel wavelet transform, 16–18 Laguerre wavelet transform, 19 Mehler–Fock wavelet transform, 20 and Kontorovich–Lebedev wavelet transform, 21 and discussed their properties and applications.…”

“…Many fundamental results of this transform are already known, but study of wavelets and wavelet transforms involving index Whittaker transform is still missing. Motivated from previous studies, 16–21 our main aim is to define the wavelet and wavelet transform in terms of index Whittaker transform and further develop their theory and basic properties by this correspondence.…”

Composition of wavelet transforms and wave packet transform involving Kontorovich-Lebedev transform

“…Further, construction of wavelet transforms by using various kind of integral transforms have been carried out by the authors of the field [11,13,17,18,20,30,31]. In [21,25], Prasad et al have constructed and studied key properties of wavelet transform associated with index transforms like Kontorovich-Lebedev transform (KL-transform) and Mehler-Fock transform. Wavelet transform is categorized into two types; one is a continuous wavelet transform, and another is a discrete wavelet transform.…”

The Kontorovich-Lebedev-Clifford transform

“…[33,34], Prasad and Mandal [22,23] and many more [1,5,15,16,30,40]. Analogue theories and investigations for different types of KL-transforms, other integral transforms, pseudo-differential operators as well as wavelet transforms may also be viewed in [15,17,21,23,31,32,37,39]. The Hankel transform was first introduced by a German mathematician H. Hankel by using the Bessel function of first kind J ν (x) of order ν and then studied by many authors [14,19,38,43].…”