Graphics Reference
In-Depth Information
Now that we have seen the usefulness of Fourier series in the context of a specific
application, we look at the general question of their existence and uniqueness. Fourier
series do not converge to any arbitrary function.
Definition.
Let [a,b] be a finite interval. A function f defined on [a,b] is said to be
of
bounded variation
on [a,b] if there is a positive constant M so that
n
Â
()
-
(
)
£
fx
fx
M
i
i
-
1
i
=
1
for all partitions a = x
0
< x
1
< ...< x
n
= b of [a,b]. If f is of bounded variation, then
the
total variation
V
f
(a,b) is defined by
n
Ì
Ó
˝
˛
Â
(
)
=
()
-
(
)
Vab
,
sup
fx
fx
,
f
i
i
-
1
i
=
1
where the supremum is taken over the sums associated to all possible partitions of
[a,b].
Here are some basic facts related to functions of bounded variation on finite
intervals.
21.5.4
Theorem.
Let f be a function defined on a finite interval [a,b].
(1) If f is monotonic, then the set of points where it is discontinuous is countable.
(2) If f is monotonic, then it is of bounded variation.
(3) If f is continuous and f¢ exists and is bounded on (a,b), then f is of bounded
variation. It follows that if f has a continuous derivative on [a,b], then f is of bounded
variation.
(4) The function f is of bounded variation if and only if it can be expressed as the
difference of two increasing functions. Furthermore, if f is continuous, then f is of
bounded variation if and only if it can be expressed as the difference of two increas-
ing continuous functions.
Proof.
See [Apos58]
21.5.5
Example.
The function
()
=
()
fx
x
cos
1
x
,
0
<£
x
1
,
()
=
f
00
is not of bounded variation on [0,1]. See Figure 21.5. We start with the graph of the
function cos(1/x), which wiggles “infinitely often” near 0. This function is not contin-
uous at 0. Multiplying by x decreases the magnitude of the wiggles (but not their
number) and turns it into a continuous function because
()
Æ
f x
0
as
x
Æ
0.
