Fourier and Fourier-Stieltjes Algebras for Locally Compact Groups

Abstract

We show Abel-Tauber theorems for Fourier cosine transforms. We treat the boundary cases of the Abel-Tauber theorem of Pitman, and Soni. For locally compact groups, Fourier algebras and Fourier-Stieltjes algebras have proven to be useful dual objects. We encode the representation theory of the group, that is, positive definite functions on the group. Because groupoids and their representation appear in studying operator algebras, ergodic theory, geometry , and the representation theory of groups ,it would be useful to have a duality theory for them. We show Abel-Tauber theorems which link the asymptotics of a function and its Fourier-Stieltjes coefficients under a weak Tauberian condition .Both cosine and sine coefficients are studied