Databases Reference
In-Depth Information
For example, one of the simplest scaling functions is the Haar scaling function:
10
t
<
1
φ(
t
)
=
(18)
0 otherwise
Then
f
(
t
)
can be any piecewise continuous function that is constant in the interval
[
k
,
k
+
1
)
for all
k
.
Let's define
φ
k
(
t
)
=
φ(
t
−
k
)
(19)
The set of all functions that can be obtained using a linear combination of the set
{
φ
k
(
t
)
}
,
represented by
f
(
t
)
=
a
k
φ
k
(
t
)
(20)
k
is called the
span
of the set
{
φ
k
(
t
)
}
, or Span
{
φ
k
(
t
)
}
. If we now add all functions that are limits
of sequence
s of function
s in Span
{
φ
k
(
t
)
}
, this is referred to as the closure of Span
{
φ
k
(
t
)
}
and
denoted by Span
. Let's call this set
V
0
.
If we want to generate functions at a higher resolution, say functions that are required to
be constant over only half a unit interval, we can use a dilated version of the “mother” scaling
function. In fact, we can obtain scaling functions at different resolutions in a manner similar
to the procedure used for wavelets:
{
φ
k
(
t
)
}
2
j
/
2
2
j
t
φ
j
,
k
(
t
)
=
φ(
−
k
)
(21)
The indexing scheme is the same as that used for wavelets, with the first index referring to
the resolution while the second index denotes the translation. For the Haar example,
√
20
1
2
t
<
φ
1
,
0
(
t
)
=
(22)
0
otherwise
We can use translates of
φ
1
,
0
(
t
)
to represent all functions that are constant over intervals
[
k
/
2
,(
k
+
1
)/
2
)
for all
k
. Notice that in general any function that can be represented by the
translates of
φ(
t
)
can also be represented by a linear combination of translates of
φ
1
,
0
(
t
)
.The
converse, however, is not true. Defining
V
1
=
Span
{
φ
1
,
k
(
t
)
}
(23)
we can see that
V
0
⊂
V
1
. Similarly, we can show that
V
1
⊂
V
2
and so on.
Example15.4.1:
Consider the function shown in Figure
15.7
. We can approximate this function using translates
of the Haar scaling function
φ(
t
)
. The approximation is shown in Figure
15.8
(a). If we call
φ
(
0
)
f
this approximation
(
t
)
, then
φ
(
0
)
f
(
t
)
=
c
0
,
k
φ
k
(
t
)
(24)
k
