Abel]Tauber Theorems for Fourier]Stieltjes Coefficients

We prove 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. The results in the cosine case can be applied to stationary time series with long-time memory. The analogues for Fourier]Stieltjes transforms are also given.