Uniform convergence of Hankel transforms

We investigate necessary and/or sufficient conditions for the pointwise and uniform convergence of the weighted Hankel transforms $${\mathcal{L}}_{\nu ,\mu }^{\alpha }f(r)=r^{\mu }{\int }_0^{\mathrm{\infty }}(rt{)}^{\nu }f(t)j_{\alpha }(rt)dt,\alpha \ge -1\mathrm{/}2,r\ge 0,$$ where $\nu,\mu\in $\mathbb{R}$$aresuchthat$0\leq \mu+\nu\leq \alpha+3/2$. We subdivide these transforms into two classes in such a way that the uniform convergence criteria is remarkably different on each class. In more detail, we have the transforms satisfying $\mu +\nu =0$ (such as the classical Hankel transform), that generalize the cosine transform, and those satisfying $0<\mu+\nu\leq \alpha+3/2$, generalizing the sine transform.