In this paper, pseudo-differential operators on \(\mathbb {Z}^{n+1}_{+}\) involving the Fourier Bessel transform are introduced and proved that these operators map the Schwartz space \(S_{*}(\mathbb {Z}^{n+1}_{+})\) into itself. Using this theory, we discussed the boundedness result of Friedrich’s operators on \(L^{2}_{\alpha }(\mathbb {T}_{+}^{n+1})\) -type Sobolev space \({\mathcal {H}}^{s,2}_{\alpha }(\mathbb {Z}_{+}^{n+1})\) . These results are applied to get the solution of the heat equation on the frequency space \(\mathbb {T}^{n+1}_{+}\) .
