As a complement: suitably interpreted, your computation is perfectly correct. That is, while the resulting series will not converge pointwise, there are many other possible (useful!) ways a Fourier series may converge. And, suitably interpreted, term-wise differentiation is always correct. [Edit: typo'd "pointwise" earlier, when I meant "termwise". Sorry!]
Recall that, for two Fourier expansions of "nice" functions f(x)=∑naneinx and g(x)=∑nbneinx, we have the Parseval-Plancherel theorem that 12π∫2π0f(x)¯g(x)dx=∑nan¯bn. And, yes, Fourier series do converge pointwise for infinitely-differentiable functions, and can be differentiated termwise...
Slightly changing your example, observe that the Fourier series δ(x)=∑n1⋅einx (the Dirac comb) certainly does not converge point-wise, because the terms do not go to 0. Nevertheless, thinking in terms of Parseval-Plancherel, for nice function f
f(0)=∑nanein⋅0=∑nan⋅1
which has the same form as
12π∫2π0f(x)⋅¯δ(x)dx if the latter were to make sense. That is, the evaluation-at-
0 functional
can be represented as a sort of inner product, under some constraints.
More precisely, for s∈R, the s-th Sobolev space Hs here consists of Fourier series ∑nbneinx with ∑n|bn|2/(1+|n|)s<∞. The Fourier series for the Dirac comb is in H−12−ϵ for every ϵ>0. For an "ordinary" function f whose Fourier coefficients an have sufficient decay to put f in H12+ϵ, e.g., some smoothness of f itself,
|f(0)|=|∑nan⋅1|=|∑nan⋅(1+|n|)12+ϵ⋅1(1+|n|)12+ϵ|
and by Cauchy-Schwartz the square of this is dominated by
∑n|an|2(1+|n|)1+2ϵ⋅∑n1(1+|n|)1+2ϵ
That is, the evaluation functional
f→f(0) is continuous in the
H12+ϵ metric topology, and the Dirac comb is a continuous linear functional on it. (This argument actually shows an instance of Sobolev imbedding, namely, that
H12+ϵ consists of continuous functions.)
Thus, a Fourier series in H−s (with s>0), even though not pointwise convergent at all, directly gives a (continuous) linear functional on Hs by extending Parseval-Plancherel:
(∑nbneinx)((∑naneinx))=∑nan⋅bn
That is, distributions have legitimate Fourier series expansions, though typically not converging pointwise at all. Termwise differentiation of not-very-convergent Fourier series is completely justifiable if construed as distributional derivatives, via integration by parts: after all, differentiation in the Fourier series is termwise multiplication by in.
Indeed, Sobolev and others looked at this sort of situation in the 1930s. A discussion in this direction, beginning more-or-less from scratch, is at http://www.math.umn.edu/~garrett/m/mfms/notes/09_sobolev.pdf
It bears repeating that there are many other types of convergence than pointwise.
This post imported from StackExchange Mathematics at 2014-06-01 19:37 (UCT), posted by SE-user paul garrett