The problem under discussion is the relation between the behaviour of a function ɸ ( t ) near a particular point t = x and the behaviour of the partial sums s n of its Fourier series at that point. We start from Hardy and Littlewood’s theorem † that if ɸ(t) tends to a limit in the Ceaáro sense as t → x , then the Fourier series at that point is summable in the Cesàro sense, and conversely . The exact order of the Cesàro means in each case is unspecified.