282 CHAPTER 12. SERIES AND TRANSFORMS

It might be of interest to note that in the argument for convergence given earlier inLemma 10.2.7, h was determined by |g(t)−g(0+)| small enough. If you have the caseof a continuous function defined on a closed and bounded interval, |g(t)−g(s)| would besmall enough whenever |t− s| is suitably small independent of the choice of s thanks touniform continuity. When the above corollary is applied to convergence of Fourier series,one massages things as above to reduce to the kind of thing given in Lemma 10.2.7 asshown above and a single h will then suffice for all the points at once. Once h has beendetermined, the convergence of the other terms in Lemma 10.2.7 for such a continuousperiodic function of bounded variation will not depend on the point and so this argumentends up showing that one has uniform convergence of the Fourier series to the function ifthe periodic function is of bounded variation and continuous on every interval.

12.3 Integrating and Differentiating Fourier SeriesFirst here is a review of what it means for a function to be piecewise continuous.

Definition 12.3.1 Let f be a function defined on [a,b] . It is called piecewise contin-uous if there is a partition of [a,b] ,{x0, · · · ,xn} such that on [xk−1,xk] there is a continuousfunction gk such that f (x) = gk (x) for all x ∈ (xk−1,xk).

You can typically integrate Fourier series term by term and things will work out accord-ing to your expectations. More precisely, if the Fourier series of f is ∑

∞k=−∞

akeikx then itwill be true that

F (x)≡∫ x

−π

f (t)dt = limn→∞

n

∑k=−n

ak

∫ x

−π

eiktdt

= a0 (x+π)+ limn→∞

n

∑k=−n,k ̸=0

ak

(eikx

ik− (−1)k

ik

).

I shall show this is true for the case where f is an arbitrary 2π periodic function which ispiecewise continuous according to the above definition. However, with a better theory ofintegration, it all works for much more general functions than these. It is limited here toa simpler case because we don’t have a very sophisticated theory of integration. With theLebesgue theory of integration, all restrictions vanish. It suffices to consider very generalfunctions with no assumptions of continuity. This is still a remarkable result however, evenwith the restriction to piecewise continuous functions.

Note that it is not necessary to assume anything about the function f being the limit ofits Fourier series. Let

G(x)≡ F (x)−a0 (x+π) =∫ x

−π

( f (t)−a0)dt

Then G equals 0 at−π and π because 2πa0 =∫

π

−πf (t)dt. Therefore, the periodic extension

of G, still denoted as G, is continuous. Also

|G(x)−G(x1)| ≤∣∣∣∣∫ x

x1

Mdt∣∣∣∣≤M |x− x1|

where M is an upper bound for | f (t)−a0|. Thus the Dini condition of Corollary 12.2.1holds. Therefore for all x ∈ R,

G(x) =∞

∑k=−∞

Akeikx (12.10)