if a = 0. 4 The last restriction is unnecessary in the case of a function integrable in the Lebesgue sense. It arises from the fact that the nth Fourier coefficient is o (n), and examples show that this cannot be improved, even in the case of a function integrable in the Cauchy-Lebesgue sense. "Paley (15). See also Hardy and Littlewood (10) and Bosanquet and Linfoot (6). 15Hardy and Littlewood (9). 00