Graphics Reference
In-Depth Information
1
2
The box function B(x):
B (x) = 0,
|x|>
,
1
2
= 1,
|x|£
.
The function sinc(x):
sinc x = 1,
x = 0,
=
sin p
p
x
,x π 0.
x
2
Ê
Á
-
x
ˆ
˜
()
=
The Gaussian function:
fx
exp
. (With this notation s
becomes the
2
s
standard deviation
and s
2
is the
variance
of f(x).
The Dirac delta “function”
d
(x):
This is not really a function in the mathematical
sense. It is usually described by the strange-looking conditions
()
=
d x
0
,
x
π
0
,
and
•
Ú
()
d xdx
=
1.
-•
As stated this last integral expression is really nonsense. A precise mathematical def-
inition (see [Frie63]) defines d(x) as a
generalized function,
which is a linear functional
on an appropriate space of functions. On the other hand, the actual definition is not
as important as its properties, the most important of which is
•
Ú
()
(
)
=
()
fx
d
x x dx fx
-
0
.
(21.17)
0
-•
An intuitive discussion of generalized functions can be found in [Brac86], where they
are explained in terms of limits of functions. The Dirac delta function is often called
the
impulse signal
or
impulse function
in Fourier analysis.
Now, we are after more than just the Fourier transform. We want what we called
the inverse Fourier transform to really be the inverse of the Fourier transform. In
order to state that theorem and its hypotheses, we need to extend the notion of a func-
tion being of bounded variation to functions defined on unbounded intervals.
Definition.
Let
I
be an unbounded interval of the form
R
, [a,•), or (-•,b]. A func-
tion f defined on
I
is said to be of
bounded variation
on
I
if it is of bounded variation
on every finite subinterval and there is a positive constant M so that V
f
(c,d) < M for
every finite subinterval [c,d] of
I
.
The next theorem states the most relevant extensions of Theorem 21.5.4.
21.6.2
Theorem.
Let f be a function defined on
R
.
