Integrating the Fourier series term by term

Question

How to prove that if $f$ is a $2 \pi$ periodic integrable function, then the Fourier series of it can be integrated term by term?

I do know the case when $f$ is piecewise continuous, but cannot figure out the proof when $f$ is merely integrable.

Answer 1

While the result claimed is true (Fourier series can be integrated term by term), there are a lot of subtleties so one has to be very careful how the result is stated.

Assume $F$ ($2\pi$) periodic and absolutely continuous.

Then (we know by general Lebesgue theory) $F'=f$ exists a.e., $f$ is integrable (and here periodic with same period), and the fundamental theorem of calculus holds in the sense that $F(x)-F(a)=\int_{a}^{x}f(t)dt$ for any $a,x$.

Theorem: Assuming the above ($F$ absolutely continuous periodic, $f=F'$ a.e. ), then the Fourier series of $f$ (which exists as $f$ integrable) is the formal derivative of the Fourier series of $F$

Proof: integration by parts holds for absolutely continuous functions, so just do the usual computations.

Corollary: If $f$ is integrable periodic , $c_0$ its constant Fourier term (so the normalized integral on $f$ on the period), $F$ an antiderivative of $f$ (or if you want define, $F(x)=\int_{a}^{x}f(t)dt$, for some fixed $a$) then $F(x)-c_0x$ is periodic and absolutely continuous, while its Fourier series is then obtained by the term by term integration of the Fourier series of $f-c_0$ with an appropriate constant term corresponding to the choice of the initial point $a$

So indeed Fourier series can be integrated term by term but in the above sense as of course the original Fourier series may not converge even at any point when $f$ is just integrable, while the integrated series will converge everywhere by Dirichlet-Jordan theorem say since an absolutely continuous function is bounded variation.

More generally the same result is true if we start with Fourier-Stieltjes series of (finite) measures on the unit circle (or $[0, 2\pi]$ - though of course here care must be taken if they have singular mass points at the period ends), when the corresponding $F$ is only bounded variation and could be discontinuous (with at most countable jumps - so there the Fourier series converges to the average as usual) at mass points of the measure (and is a.c. iff the measure is a.c with respect to the Lebesgue measure, or if you want it has no singular part)