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(1) If f is bounded and monotonic, then it is of bounded variation.
(2) The function f is bounded and of bounded variation if and only if it can be
expressed as the difference of two bounded increasing functions.
Proof.
See [Apos58].
Theorem 21.6.2(1) implies that the following increasing function is of bounded
variation:
The Heaviside unit step function H(x):
H (x) = 0,
x < 0,
= 1,
x ≥ 0.
Since the Heaviside function is of bounded variation, so is the box function B(x) since
it differs from
1
2
1
2
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
Hx
+
-
Hx
-
1
2
at the single point x =
.
Here is the theorem we were after.
21.6.3
Theorem.
Let f: R Æ R be an absolutely integrable function of bounded
variation, then
() + () =
+
-
(
)
fx
fx
(
)
Ú
Ú
()
2p i
ux t
-
fte
dtdu
.
(21.18)
2
-•
-•
If f is also continuous, then the left hand side of equation (21.18) equals f(x), that is,
f = FT -1 (FT (f)).
Proof.
See [Widd71] or [Apos58].
Theorem 21.6.3 is the fundamental theorem in the theory of Fourier transforms.
It says that if one thinks of the Fourier transform as a mapping of functions to func-
tions, then the inverse Fourier transform is basically the inverse of that mapping (if
one ignores the finite number of points of discontinuity that might exist). For example,
in Figure 21.4 the functions on the left are the inverse Fourier transform of the func-
tions on the right. There are different variants of Theorem 21.6.3. The problem is that
the hypotheses of the theorem are not satisfied by all the functions one might want
to deal with. For example, the sinc(x) function is not absolutely integrable. We have
defined what sometimes is called the “ordinary” Fourier transform. It is possible to
generalize the Fourier transform by defining it in terms of a convergent sequence of
integrals, but we shall not pursue this further here. At the end of the day, however, all
the functions that one wants to use are covered.
Using the different names “x” and “u” for the variables of a function and for its
Fourier transform, respectively, was a conscious decision. We shall do so throughout
the rest of this chapter. One considers functions f(x) as defined on a “spatial” domain
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