The discrete Fourier transform data sequence need not be circularly defined

Three key discrete Fourier transform (DFT) theorems, namely, those on inversion, shift, and convolution, are considered without assumptions on the data sequence. It is shown that the three DFT theorems can be proved without the usual assumption that the data sequence is circular and that the circularity of DFT shift and convolution is a consequence of the DFT properties, not necessarily of those of the data sequence. The advantage of this alternative viewpoint is that puzzling circularity assumptions with respect to nonperiodic data sequences are avoided.<>