Uniform convergence of Hankel transforms

Abstract

We investigate necessary and/or sufficient conditions for the pointwise and uniform convergence of the weighted Hankel transforms $${\mathcal{L}}_{\nu ,\mu }^{\alpha }f(r)=r^{\mu }{\int }_0^{\infty }(rt{)}^{\nu }f(t)j_{\alpha }(rt)dt,\alpha \ge -1/2,r\ge 0,$$ where $\nu ,\mu \in \mathbb{R}$ are such that $0\le \mu +\nu \le \alpha +3/2$. 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 $0<\mu +\nu \le \alpha +3/2$, generalizing the sine transform.

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Cited by in corpus (2)

• Hankel transforms of general monotone functions
• Weighted norm inequalities for generalized Fourier-type transforms and applications