Criteria for absolute convegence of Fourier series of functions of bounded variation

The usual criteria for establishing that a function of bounded variation or an absolutely continuous function has an absolutely convergent Fourier series are given in terms of the modulus of continuity, the integrated modulus of continuity or conditions on the derivative. The relations between these criteria are investigated. A class of functions is constructed to provide counterexamples which show to what extent the existing theorems are best possible. In the case of absolutely continuous functions a few new criteria are given involving the variation of the given function. A couple of necessary and sufficient conditions are given for a class of absolutely continuous functions to have absolutely convergent Fourier series.