  We investigate necessary and/or sufficient conditions for the pointwise and
uniform convergence of the weighted Hankel transforms
$$\mathcal{L}^\alpha_{\nu,\mu}f(r) = r^\mu\int_0^\infty (rt)^\nu f(t)
j_\alpha(rt)\, dt, \quad \alpha\geq -1/2, \quad r\geq 0, $$ where $\nu,\mu\in
\mathbb{R}$ are such that $0\leq \mu+\nu\leq \alpha+3/2$. 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.
