Uniform convergence of Hankel transforms

Abstract

We investigate necessary and/or sufficient conditions for the pointwise and uniform convergence of the weighted Hankel transforms $$\mathcal{L}^α_{ν,μ}f(r) = r^μ\int_0^\infty (rt)^νf(t) j_α(rt)\, dt, \quad α\geq -1/2, \quad r\geq 0, $$ where $ν,μ\in \mathbb{R}$ are such that $0\leq μ+ν\leq α+3/2$. 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 $μ+ν=0$ (such as the classical Hankel transform), that generalize the cosine transform, and those satisfying $0<μ+ν\leq α+3/2$, generalizing the sine transform.