We prove Abel-Tauber theorems for Hankel and Fourier transforms. For example, let f be a locally integrable function on [O, oo) which is eventually decreasing to zero at infinity. Let p = 3, 5, 7, · · · and £ be slowly varying at infinity. We characterize the asymptotic behavior f(t) l(t)t-P as t -+ oo 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
