Abel-Tauber theorems for Hankel and Fourier transforms and a problem of Boas (Q1962717)

The authors prove five interesting theorems on the subject determined by the title of the paper. A more specified characterization of the results is given in the abstract as follows: We prove Abel-Tauber theorems for Hankel and Fourier transforms. For example, let \(f\) be a locally integrable function on \([0,\infty)\) which is eventually decreasing to zero at infinity. Let \(\rho= 3, 5, 7, \dots\) and \(\ell\) be slowly varying at infinity. We characterize the asymptotic behavior \(f(t) \sim \ell(t)t^{-\rho}\) as \(t\to \infty\) in terms of the Fourier cosine transform of \(f\). Similar results for sine and Hankel transforms are also obtained. As an application, we give an answer to a problem of R. P. Boas on Fourier series.