Convergence of Fourier Series - Summability

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 at t, then the Fourier series of ƒ cannot converge to a value different from ƒ(t). It may either converge to ƒ(t) or diverge. This is because, if converges to some value x, it is also summable to it, so from the first summability property above, x = ƒ(t).