The refinement equation
φ
(
t
)
=
∑
k
=
N
1
N
2
c
k
φ
(
2
t
−
k
)
\varphi (t) = \sum _{k=N_1}^{N_2} c_k \, \varphi (2t-k)
plays a key role in wavelet theory and in subdivision schemes in approximation theory. Viewed as an expression of linear dependence among the time-scale translates
|
a
|
1
/
2
φ
(
a
t
−
b
)
|a|^{1/2} \varphi (at-b)
of
φ
∈
L
2
(
R
)
\varphi \in L^2(\mathbf {R})
, it is natural to ask if there exist similar dependencies among the time-frequency translates
e
2
π
i
b
t
f
(
t
+
a
)
e^{2 \pi i b t} f(t+a)
of
f
∈
L
2
(
R
)
f \in L^2(\mathbf {R})
. In other words, what is the effect of replacing the group representation of
L
2
(
R
)
L^2(\mathbf {R})
induced by the affine group with the corresponding representation induced by the Heisenberg group? This paper proves that there are no nonzero solutions to lattice-type generalizations of the refinement equation to the Heisenberg group. Moreover, it is proved that for each arbitrary finite collection
{
(
a
k
,
b
k
)
}
k
=
1
N
\{(a_k,b_k)\}_{k=1}^N
, the set of all functions
f
∈
L
2
(
R
)
f \in L^2(\mathbf {R})
such that
{
e
2
π
i
b
k
t
f
(
t
+
a
k
)
}
k
=
1
N
\{e^{2 \pi i b_k t} f(t+a_k)\}_{k=1}^N
is independent is an open, dense subset of
L
2
(
R
)
L^2(\mathbf {R})
. It is conjectured that this set is all of
L
2
(
R
)
∖
{
0
}
L^2(\mathbf {R}) \setminus \{0\}
.