In the literature, I've found that DFS and DFT are one and the same. If they are one and the same why to use two different names for them? If there is really a difference what is it and what is the significance of discrete Fourier series?
we have had this fight many, many, many times at comp.dsp. the DFT is the same thing as the DFS. the DFT maps a discrete and periodic sequence of numbers with period length of $N$ to another discrete and periodic sequence of numbers with period length of $N$ and the iDFT (which has the same form as the DFT) maps it back.
some people don't like anthropomorphizing algorithms or procedures, but i do. the DFT "assumes" that the $N$ samples passed to it are one period of a periodic sequence. the DFT periodically extends the data passed to it.
it is clear in the math, both in the definition of the DFT (and iDFT), and in any theorem applicable to the DFT other than linearity (the periodic nature of the DFT is not evident in the linearity property, but it is evident in anything that causes shifting or convolution in one domain or multiplication by a non-constant in the other domain).
this is why, if periodicity is not assumed (a better word would be "recognized"), then people need to use this clunky modulo notation in the indices, like $x[ ((n))_N ]$ (this is the notation that O&S use) and that, in my opinion, is a pathetic confession from the periodicity deniers that, when it comes down to the bottom line, even they recognize that the DFT is inherently periodic.
I think part of the problem is an awkward and inconsistent naming convention. There are 4 flavors of Fourier Transforms depending on which domain is continuous or discrete (which maps to being aperiodic or perodic in the other domain). So we have
Name Time Frequency Fourier Transform continous/aperiodic continous/aperiodic Fourier Series continous/periodic discrete/aperiodic Discrete Time FT discrete/aperiodic continous/periodic DFT or DFS discrete/periodic discrete/periodic
A better naming would have been
Name Time Frequency Fourier Transform continous/aperiodic continous/aperiodic Fourier Series continous/periodic discrete/aperiodic Discrete FT discrete/aperiodic continous/periodic Discrete FS discrete/periodic discrete/periodic
so that discrete refers to "discrete in time and periodic in frequency" and "series" refers "discrete in frequency and periodic in time". In other words "series" means sums and "transform" means integrals. Discrete mean sums and continuous means integrals.