Akhilesh Prasad

The generalized continuous wavelet transform associated with the fractional Fourier transform

The main objective of this paper is to study the fractional Fourier transform (FrFT) and the generalized continuous wavelet transform and some of their basic properties. Applications of the FrFT in solving generalized nth-order linear nonhomogeneous ordinary differential equations and a generalized wave equation are given. The generalized continuous wavelet transform, and its inversion formula, and the Parseval relation are also studied using the fractional Fourier transform.

The continuous fractional Bessel wavelet transformation

The main objective of this paper is to study the fractional Hankel transformation and the continuous fractional Bessel wavelet transformation and some of their basic properties. Applications of the fractional Hankel transformation (FrHT) in solving generalized n th order linear nonhomogeneous ordinary differential equations are given. The continuous fractional Bessel wavelet transformation, its inversion formula and Parseval’s relation for the continuous fractional Bessel wavelet transformation are also studied.MSC:46F12, 26A33.

The fractional wavelet transform on spaces of type S

The fractional wavelet transform is defined and its properties are studied. The continuity of the fractional wavelet transform on Gelfand–Shilov spaces of type S is shown.

Composition of pseudodifferential operators associated with fractional Hankel–Clifford integral transformations

Two versions of pseudodifferential operators (pdo) involving fractional powers Hankel–Clifford integral transformations are defined. The composition of first and second fractional pdo is defined. We show that the pdo and composition of pdo are bounded in a certain Sobolev-type space associated with the fractional powers of Hankel–Clifford integral transformations. Some special cases are also discussed.

Pseudo-differential operators associated with the Jacobi differential operator and Fourier-cosine wavelet transform

Using the inverse of Fourier–Jacobi transform a symbol is defined, and the pseudo-differential operator (p.d.o.) 𝒫α, β (x,D) associated with Jacobi-differential operator in terms of this symbol is defined. It is shown that the p.d.o. is bounded in a certain Sobolev type space associated with the Fourier–Jacobi transform. Continuous Jacobi wavelet transform (JWT) and Fourier-cosine wavelet transform are defined and a reconstruction formula is obtained for Fourier-cosine wavelet transform. Properties of Fourier-cosine wavelet transform are investigated.

The continuous fractional wave packet transform

Based on the idea of fractional Fourier transform (FrFT), fractional mother wavelet ψb,a,θ and wave packet transform (WPT) the concept of continuous fractional wave packet transform (CFrWPT) is introduced. Properties of CFrWPT are investigated and a reconstruction formula is obtained for CFrWPT.

Boundedness of pseudo-differential operators involving Kontorovich–Lebedev transform

ABSTRACT A pair of pseudo-differential operators (p.d.o.) involving the Kontorovich–Lebedev transform (KL-transform) as well as its adjoint is defined. Estimates for translation and convolution function associated with the Kontorovich–Lebedev transform are obtained. Continuity of KL-transform and p.d.o. on and are discussed.

The continuous fractional wavelet transform on generalized weighted Sobolev spaces

In this paper, the continuous fractional wavelet transform is defined and boundedness results of this transform on generalized weighted Sobolev spaces are obtained. Characterization of the range of continuous fractional wavelet transform is also discussed.

Continuity of pseudo-differential operator h μ, a involving Hankel translation and Hankel convolution on some Gevrey spaces

The pseudo-differential operator (p.d.o.) h μ, a associated with the Bessel operator involving the symbol a(x, y) whose derivatives satisfy certain growth conditions depending on some increasing sequences is studied on certain Gevrey spaces. The p.d.o. h μ, a on Hankel translation τ and Hankel convolution of Gevrey functions is a continuous linear mapping into another Gevrey space.

Wavelet transforms associated with the Kontorovich–Lebedev transform

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.