Best approximations for the Laguerre-type Weierstrass transform on [0, ∞[×ℝ

For α = n− 1, n ∈ N\{0}, the operator D2 is the radial part of the sub-Laplacian on the Heisenberg groupHn (see [2, 4]). These operators have gained considerable interest in various fields of mathematics (see [1, 4]). They give rise to generalizations of many two-variable analytic structures like the Laguerre-Fourier transform L, the Laguerre-convolution product, the dispersion and the Gaussian distributions (see [1, 2, 4]). In this paper, we consider the Laguerre-type Weierstrass transform Lr associated with D1 and D2: