Math 172: Convergence of the Fourier Series

We now discuss convergence of the Fourier series on compact intervals I. ‘Convergence’ depends on the notion of convergence we use, such as (i) L: uj → u in L if ‖uj − u‖L2 → 0 as j →∞. (ii) uniform, or C: uj → u uniformly if ‖uj−u‖C0 = supx∈I |uj(x)−u(x)| → 0. (iii) uniform with all derivatives, or C∞: uj → u in C∞ if for all non-negative integers k, supx∈I |∂uj(x)− ∂u(x)| → 0. (iv) pointwise: uj → u pointwise if for each x ∈ I, uj(x) → u(x), i.e. for each x ∈ I, |uj(x)− u(x)| → 0. Note that pointwise convergence is too weak for most purposes, so e.g. just because uj → u pointwise, it does not follow that ∫ I uj(x) dx → ∫ u(x) dx. This would follow, however, if one assumes uniform convergence, or indeed L (or L) convergence, since ∣∣∣ ∫ I uj(x) dx− ∫