88 Chapter 1 Fourier Series and Integrals
uous, it is certain that the differentiated series off(x)will fail to converge, at
some points at least.
Example.
Letfbe the function that is periodic with period 2πand has the formula
f(x)=|x|, −π<x<π.This function is indeed continuous and sectionally smooth and is equal to its
Fourier series,
f(x)=π
2 −4
π(
cos(x)+cos( 3 x)
9 +cos( 5 x)
25 +···)
.
According to Theorem 5, the differentiated series
4
π(
sin(x)+sin(^3 x)
3+sin(^5 x)
5+···
)
converges tof′(x)at any pointxwheref′′(x)exists. Now, the derivative of the
sawtooth functionf(x)(see Fig. 10) is the square-wave function
f′(x)={ 1 , 0 <x<π,
− 1 , −π<x< 0 (9)(see Fig. 9). Moreover, we know that the foregoing sine series is the Fourier
seriesofthesquarewavef′(x)andthatitconvergestothevaluesgivenby
Eq.(9),exceptatthepointswheref′(x)has a jump. These are precisely the
points wheref′′(x)does not exist.
Later on, it will frequently happen that we know a function only through its
Fourier series. Thus, it will be important to obtain properties of the function
by examining its coefficients, as the next theorem does.
Theorem 7.If f is periodic, with Fourier coefficients an,bn,andiftheseries
∑∞
n= 1(∣∣
nkan∣∣
+
∣∣
nkbn∣∣)
converges for some integer k≥ 1 , then f has continuous derivatives f′,...,f(k)
whose Fourier series are differentiated series of f.
Example.
Consider the function defined by the series
f(x)=∑∞
n= 1e−nαcos(nx),