Hankel Transforms of General Monotone Functions

Abstract

We show that the Hankel transform of a general monotone function converges uniformly if and only if the limit function is bounded. To this end, we rely on an Abel-Olivier test for realvalued functions. Analogous results for cosine series are derived as well. We also show that our statements do not hold without the general monotonicity assumption in the case of cosine integrals and series.

The Boas Problem on Hankel Transforms

“…Proof. The proof just consists on rewriting that of [9] in the context of functions, with the difference that in the mentioned work the parameter r = 2 is fixed (see also [14], where this idea was originally carried out for sequences). Assume (3.3) does not hold for n ∈ Z.…”

“…The parameter r will be arbitrarily chosen at each point according to our convenience. In contrast with [9,14], here we consider a slightly different definition of good numbers by incorporating the parameter r > 0 (in the cited papers r = 2 is fixed). The reason to do this is that every power function x ρ (which is a GM function for any ρ) will have an infinite amount of good numbers if r is chosen appropriately according to ρ.…”