Harmonic analysis associated to the canonical Fourier Bessel transform

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

“…In the present work, we continue the analysis begun in [6] by studying the translation operator and the convolution product related to this transformation. Following the framework of Delsarte [3] and Levitan [15], we introduce a generalized translation T ν,m x f (y) = u(x, y) (x, y ≥ 0, m ∈ SL(2, R)) of a function f ∈ C 2 ([0, +∞[) as the solution to the following Cauchy problem ∆ m ν,x u(x, y) = ∆ m ν,y u(x, y), u(x, 0) = f (x), ∂ ∂x u(x, 0) = 0,…”

“…The aim of this section is to give a brief review of the theory of canonical Fourier Bessel transform that is relevant to the succeeding sections [6]. Throughout this paper, ν denotes a real number such that ν > − 1 2 .…”

“…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]…”

“…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 .…”

“…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 . Several properties, such as a Riemann-Lebesgue lemma, inversion formula, operational formulas, Plancherel theorem and Babenko inequality and several uncertainty inequalities are established.…”