Absolute Convergence of Some Trigonometric Series, I

Let f be an integrable function of period 2n. Our main concern is with the absolute convergence of the Fourier series of some functions derived from f , particularly when f itself has an absolutely convergent Fourier series. For the sake of simplicity we confine our discussion to cosine series, although most of our results hold mutatis mutandis for sine series or general Fourier series.