One of the oldest problems in the theory of Fourier series is that of looking for a criterion that a Fourier series shall converge. No one, however, has been able to find a simple, necessary and sufficient condition for this. Thus, for instance, bounded variation of the function is sufficient but not necessary. Continuity is neither necessary nor sufficient. That is to say, there are functions whose Fourier series converge at points of discontinuity, and others whose Fourier series diverge at points of continuity. If we consider the same problem for Cesàro summability of any particular order, similar difficulties arise.