arbitrary, we can set it equal to 0. Hence , which completes the proof. Modifications and Generalisations of Fejér's Theorem In fact, Fejér's theorem can be modified to hold for pointwise convergence. Sadly however, the theorem does not work in a general sense when we replace the sequence with . This is because there exist functions whose Fourier series fails to converge at some point. However, the set of points at which a function in diverges has to be measure zero. This fact, called Lusins conjecture or Carleson's theorem, was proven in 1966 by L. Carleson. We can however prove a corrollary relating which goes as follows: A more general form of the theorem applies to functions which are not necessarily continuous . Suppose that f is in L1(-π,π). If the left and right limits f(x0±0) of f(x) exist at x0, or if both limits are infinite of the same sign, then Existence or divergence to infinity of the Cesàro mean is also implied. By a theorem of Marcel Riesz, Fejér's theorem holds precisely as stated if the (C, 1) mean σn is replaced with (C, α) mean of the Fourier series . References . Fourier series Theorems in approximation theory