Graphics Reference
In-Depth Information
function as defined only on the image rectangle,
R
, or as being defined on the
whole plane (which we'll treat as
R
2
). In either case, we require that the integral
of the square of
f
is finite,
1
that is,
f
(
x
)
2
dx
< ∞
,
(18.22)
D
where
D
is the domain on which the function is defined. (Functions satisfying
this inequality are called
square integrable;
the interpretation, for many physi-
cally meaningful functions, is that they represent signals of finite total energy.)
The domain
D
might be the rectangle
R
, the whole plane
R
2
, the real line
R
,or
some interval
[
a
,
b
]
when we're discussing the one-dimensional situation. Func-
tions that are square integrable form a vector space called
L
2
, where we often
write something like
L
2
(
R
2
)
to indicate square-integrable functions on the plane.
We say “
f
is
L
2
” as shorthand for “
f
is square integrable.” The set of
L
2
functions
on any particular domain is generally a vector space. It takes a little work to show
that
L
2
is closed under addition, that is if
f
and
g
are
L
2
, then so is
f
+
g
;butwe'll
omit the proof, since it's not particularly instructive.
A function
x
→
f
(
x
)
in
L
2
(
R
)
must “fall off” as
x
→±∞
, because if
|
f
(
x
)
|
0, then
K
−
>
K
−
K
f
(
x
)
2
K
M
2
is always greater than some constant
M
>
dx
dx
=
2
KM
2
, which goes to infinity as
K
.
The next class of functions is the discrete analog of
L
2
: the set of all functions
f
:
Z
→∞
→
R
such that
f
(
i
)
2
< ∞
(18.23)
i
2
; these are called
square summable.
There are two ways in which
is denoted
2
functions arise. The first is through sampling
of
L
2
functions. Sampling is formally defined in the next section, but for now note
that if
f
is a
continuous L
2
function on
R
, then the samples of
f
are just the val-
ues
f
(
i
)
where
i
is an integer, so sampling in this case amounts to restricting the
domain from
R
to
Z
. The second way that
2
functions arise is as the Fourier trans-
form of functions in
L
2
([
a
,
b
])
for an interval
[
a
,
b
]
, which we'll describe presently.
Finally, both
2
and
L
2
have inner products. For
2
(
Z
)
we define
∞
a
,
b
=
a
(
i
)
b
(
i
)
,
(18.24)
i
=
−∞
w
=
i
=
1
v
i
w
i
.
For
L
2
(
D
)
, where
D
is either a finite interval or the real line, we define
which is analogous to the definition in
R
3
of
v
·
=
f
,
g
f
(
x
)
g
(
x
)
dx
.
(18.25)
D
This inner product on
L
2
has all the properties you might expect: It's linear in
each factor, and
=
0 if and only if
f
=
0, at least if we extend the notion of
f
=
0 to mean that
f
is zero “almost everywhere,” in the sense that if we picked a
f
,
f
1. Later we'll consider complex-valued functions rather than real-valued ones. When we
doso,wehavetoreplace
f
(
x
)
with
|
f
(
x
)
|
in the integral.
