  We give a detailed study of the discrete Fourier transform (DFT) of $r$-even
arithmetic functions, which form a subspace of the space of $r$-periodic
arithmetic functions. We consider the DFT of sequences of $r$-even functions,
their mean values and Dirichlet series. Our results generalize properties of
the Ramanujan sum. We show that some known properties of $r$-even functions and
of the Ramanujan sum can be obtained in a simple manner via the DFT.
