was first done by Andrey Kolmogorov (see below). For infinity, the result is a corollary of the uniform boundedness principle. If the partial summation operator SN is replaced by a suitable summability kernel (for example the Fejér sum obtained by convolution with the Fejér kernel), basic functional analytic techniques can be applied to show that norm convergence holds for 1 ≤ p < ∞. Convergence almost everywhere The problem whether the Fourier series of any continuous function converges almost everywhere was posed by Nikolai Lusin in the 1920s. It was resolved positively in 1966 by Lennart Carleson. His result, now known as Carleson's theorem, tells the Fourier expansion of any function in L2 converges almost everywhere. Later on, Richard Hunt generalized this to Lp for any p > 1. Contrariwise, Andrey Kolmogorov, as a student at the age of 19, in his very first scientific work, constructed an example of a function in L1 whose Fourier series diverges almost everywhere (later improved to diverge everywhere). Jean-Pierre Kahane and Yitzhak Katznelson proved that for any given set E of measure zero, there exists a continuous function ƒ such that the Fourier series of ƒ fails to converge on any point of E. Summability Does the sequence 0,1,0,1,0,1,... (the partial sums of Grandi's series) converge to ½? This does not seem like a very unreasonable generalization of the notion of convergence. Hence we say that any sequence is Cesàro summable to some a if It is not difficult to see that if a sequence converges to some a then it is also Cesàro summable to it. To discuss summability of Fourier series, we must replace with an appropriate notion. Hence we define and ask: does converge to f? is no longer associated with Dirichlet's kernel, but with Fejér's kernel, namely where is Fejér's kernel, The main difference is that Fejér's kernel is a positive kernel. Fejér's theorem states that the above sequence of partial sums converge uniformly to ƒ. This implies much better convergence properties If ƒ is continuous at t then the Fourier series of ƒ is summable at t to ƒ(t). If ƒ is continuous, its Fourier series is uniformly summable (i.e. converges uniformly to ƒ). For any integrable ƒ, converges to ƒ in the norm. There is no Gibbs phenomenon. Results about summability can also imply results about regular convergence. For example, we learn that if ƒ is continuous