1.4 Uniform Convergence 79
- The functionf(x)is periodic with period 2. Its graph for− 1 <x<1isa
semicircle with radius 1 centered at the origin.
a.Find the equation off(x)for− 1 <x<1.
b.Determine the value of the coefficienta 0 in its Fourier series. (This is the
only cosine coefficient that can be found in closed form.)
c.Isf(x)sectionally smooth?
d.What does the theorem tell us about the convergence of the Fourier se-
ries off(x)?
1.4 Uniform Convergence
The theorem of the preceding section treats convergence at individual points
of an interval. A stronger kind of convergence is uniform convergence in an
interval. Let
SN(x)=a 0 +∑N
n= 1ancos(nπx
a)
+bnsin(nπx
a)
be the partial sum of the Fourier series of a functionf.Themaximumdevia-
tion between the graphs ofSN(x)andf(x)is
δN=max∣∣
f(x)−SN(x)∣∣
, −a≤x≤a,where the maximum^2 is taken over allxin the interval, including the end-
points. If the maximum deviation tends to zero asNincreases, we say that the
seriesconverges uniformlyin the interval−a≤x≤a.
Roughly speaking, if a Fourier series converges uniformly, then the sum of a
finite numberNof terms gives a good approximation — to within±δN—of
the value off(x)atanyandeverypoint of the interval. Furthermore, by taking
alargeenoughN, one can make the error as small as necessary.
There are two important facts about uniform convergence. If a Fourier series
converges uniformly in a period interval, then (1) it must converge to a con-
tinuous function, and (2) it must converge to the (continuous) function that
generates the series. Thus, a function that has a nonremovable discontinuity
cannothave a uniformly convergent Fourier series. (And not all continuous
functions have uniformly convergent Fourier series.)
Figure 9 presents graphs of some partial sums of a square-wave function. It
is easy to see that for everyNthere are points nearx=0andx=±πwhere
(^2) Iffis not continuous, the maximum must be replaced by the supremum, or least upper
bound.