1.3 Convergence of Fourier Series 77
sense, the value offatx=+ais unimportant. But because of the averaging
features of the Fourier series, it is reasonable to define
f(a)=f(−a)=^12(
f(a−)+f(−a+))
.
That is, the value offat the endpoints is the average of the one-handed limits
at the endpoints, each limit taken from the interior. For instance, iff(x)=
1 +x,0<x<1, andf(x)=0,− 1 <x<0, thenf(± 1 )should be taken to
be 1, andf( 0 )should be^12.
Examples.
1.The square-wave functionf(x)={ 1 , 0 <x<1,
− 1 , − 1 <x< 0
is sectionally smooth; therefore the corresponding Fourier series con-
verges to
{ 1 , for 0<x<1,
− 1 , for− 1 <x<0,
0 , forx= 0 , 1 ,− 1
and is periodic with period 2.
2.For the functionf(x)=|x|^1 /^2 ,−π<x<π,f(x+ 2 π)=f(x),thepre-
ceding theorem does not guarantee convergence of the Fourier series at
any point, even though the function is continuous. Nevertheless, the se-
ries does converge at any pointx! This shows that the conditions in the
theorem are perhaps too strong. (But they are useful.) EXERCISES
- Foreachfunctiongiven,ifitisnotsectionallysmoothontheinterval,ex-
plain why not. Sketch.
a. f(x)=|x|−| 1 −x|, − 1 <x<2;
b. f(x)=
√
|x|, − 1 <x<1;
c. f(x)=ln(
2cos(x/ 2 ))
, −π<x<π;
d. f(x)=tan(x), 0 <x<π/2;
e. f(x)=tan(x), 0 <x<π.- Check each function described in what follows to see whether it is section-
ally smooth. If it is, state the value to which its Fourier series converges at
each pointxin the interval and at the endpoints. Sketch.