Generalized Hankel transform on the line

Since 2012, in harmonic analysis on the line with a power-law weight, the two-parameter $(k,a)$-generalized Fourier transform proposed by S. Ben Saïd, T. Kobayashi, B. Orsted has been intensively studied. It generalizes the Dunkl transform depending on only one parameter $k\ge 0$. Together with an increase in the variety of unitary transforms, the presence of a parameter $a>0$ for $a\neq 2$ leads to the appearance of deformation properties, for example, for functions from the Schwartz space, the generalized Fourier transform may not be infinitely differentiable or rapidly decreasing at infinity. The fast decay is preserved only for the sequence $a=2/n$, $n\in$ $\mathbb{N}$. Some change of variable in this case also improves other properties of the generalized Fourier transform. The generalized Dunkl transform obtained after changing the variable at $a=2/(2r+1)$, $r\in$ $\mathbb{Z}_+$ is devoid of deformation properties and, to a large extent, has already been studied. In this paper, we study the generalized Hankel transform obtained after a change of variable for $a=1/r$, $r\in$ $\mathbb{N}$. An invariant subspace of functions rapidly decreasing at infinity is described for it, and a differential-difference operator is found for which the kernel of the generalized Hankel transform is an eigenfunction. On the basis of a new multiplication theorem for the Bessel functions Boubatra–Negzaoui–Sifi, two generalized translation operators are constructed, and their $L^p$-boundedness and positivity are investigated. A simple proof is given for the multiplication theorem. Two convolutions are defined for which Young's theorems are proved. With the help of convolutions, generalized means are defined, for which sufficient conditions for $L^p$-convergence and convergence almost everywhere are proposed. Generalized analogs of the Gauss-Weierstrass, Poisson and Bochner–Riesz means are investigated.