On Abel-Tauber Theorems for Fourier Cosine Transforms

We prove Abel-Tauber theorems for Fourier cosine transforms. They treat the boundary cases of the Abel-Tauber theorem of Pitman. and Soni and Soni. A similar result for Fourier cosine series is obtained as a corollary. The latter gives an answer to an open problem in Boas′ book on Fourier series. Applications to probability distributions and stationary processes are given.