Upper bounds and asymptotic expansion for Macdonald's function and the summability of the Kontorovich-Lebedev integrals

Uniform upper bounds and the asymptotic expansion with an explicit remainder term are established for Macdonald function Kiτ(x). The results can be applied, instance, to study summability of divergent Kontorovich-Lebedev integrals in sense Jones. Namely, we answer affirmatively a question (cf. [Ehrenmark U. Summability experiments class inverse transforms. Comput Math Appl. 2018;76(1):141–154.]) whether these converge even entire functions exponential type weak sense.