In applied mathematics, the transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an important role in wavelet theory and finite element theory.
For the mask
h
, which is a vector with component indexes from
a
to
b
, the transfer matrix of
h
, we call it
T
h
here, is defined as
(
T
h
)
j
,
k
=
h
2
⋅
j
−
k
.
More verbosely
T
h
=
(
h
a
h
a
+
2
h
a
+
1
h
a
h
a
+
4
h
a
+
3
h
a
+
2
h
a
+
1
h
a
⋱
⋱
⋱
⋱
⋱
⋱
h
b
h
b
−
1
h
b
−
2
h
b
−
3
h
b
−
4
h
b
h
b
−
1
h
b
−
2
h
b
)
.
The effect of
T
h
can be expressed in terms of the downsampling operator "
↓
":
T
h
⋅
x
=
(
h
∗
x
)
↓
2.