On an asymptotic expansion of the inverse Kontorovich-Lebedev transform (Q2718030)

The author studies the asymptotic behaviour of the Kontorovich-Lebedev transform NEWLINE\[NEWLINE F(s) = \int_0^\infty f(x) K_{is}(x) {{dx}\over{x}}, \qquad s\in{\mathbb R}, NEWLINE\]NEWLINE where \(K_{is}\) denotes the modified Bessel function of the third kind of purely imaginary order. In [Appl. Anal. 39, No.~4, 249-263 (1990; Zbl 0692.44024; \textit{D. Naylor}, Analysis, München 18, No.~3, 269-283 (1998)] the author gave an asymptotic expansion of the function \(F(s)\) if \(f(x)\) satisfies some growth condition for \(s\to\infty\). In the paper under review, this relation is somehow reversed. The growth condition of the function \(f\) is deduced from assumptions concerning the asymptotic behaviour of it's transform \(F\) under some conditions.