A unified class of integral transforms related to the Dunkl transform

WITHDRAWN: Generalized translation operator and Heat equation for the generalized canonical Fourier Bessel transform

“…In the last few years authors built a class of linear integral transform with a kernel involving a Bessel function and matrix parameter m ∈ SL(2, R). In [9], the authors introduced the Dunkl linear canonical transform (DLCT) which is a generalization of the LCT. DLCT includes many well-known transforms such as the Dunkl transform [5], the fractional Dunkl transform [7], the linear canonical Fourier Bessel transform [4] and the generalized linear canonical Fourier Bessel transform [1].…”

Prolate spheroidal wave functions associated with the canonical Fourier–Bessel transform and uncertainty principles

“…Definition The canonical Fourier–Bessel transform of a function $$$ f\in {\mathcal{L}}_{1,\nu } $$$ is defined by Ghazouani et al [5] $$$$ {\mathcal{F}}_{\nu}^{\mathbf{m}}f(x)=\frac{c_{\nu }}{(ib)^{\nu +1}}{\int}_0^{+\infty}\kern0.1em {K}_{\nu}^{\mathbf{m}}\left(x,y\right)f(y){y}^{2\nu +1} dy, $$$$ where $$$$ {K}_{\nu}^{\mathbf{m}}\left(x,y\right)={e}^{\frac{i}{2}\left(\frac{d}{b}{x}^2+\frac{a}{b}{y}^2\right)}\kern0.1em {j}_{\nu}\left(\frac{xy}{b}\right). $$$$ …”

Generalized translation operator and the Heat equation for the canonical Fourier Bessel transform

“…Throughout this paper, we denote by m = a b c d an arbitrary matrix in SL(2, R). For m ∈ SL(2, R) such that b = 0, the canonical Fourier Bessel transform of a function f ∈ L 1,ν is defined by [6,9]…”

“…Special cases of this transformation are the Hankel transform and the fractional Hankel transform [13]. In [9], the authors introduced the Dunkl linear canonical transform (DLCT) which is a generalization of the LCT in the framework of Dunkl transform [4]. DLCT includes many well-known transforms such as the Dunkl transform [4,7], the fractional Dunkl transform [10,11] and the canonical Fourier Bessel transform [6,9].…”

“…In [9], the authors introduced the Dunkl linear canonical transform (DLCT) which is a generalization of the LCT in the framework of Dunkl transform [4]. DLCT includes many well-known transforms such as the Dunkl transform [4,7], the fractional Dunkl transform [10,11] and the canonical Fourier Bessel transform [6,9]. In [6], the authors developed a harmonic analysis related the canonical Fourier Bessel transform F m ν .…”