This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form
ϕ
(
x
)
=
g
(
x
)
+
∑
α
∈
Z
s
a
(
α
)
ϕ
(
M
x
−
α
)
,
x
∈
R
s
,
\begin{equation*} \phi (x) = g(x) + \sum _{\alpha \in \mathbb {Z}^s} a(\alpha ) \phi (Mx-\alpha ), \qquad x \in \mathbb {R}^s, \end{equation*}
where
ϕ
=
(
ϕ
1
,
…
,
ϕ
r
)
T
\phi = (\phi _1,\ldots ,\phi _r)^T
is the unknown,
g
=
(
g
1
,
…
,
g
r
)
T
g = (g_1,\ldots ,g_r)^T
is a given vector of functions on
R
s
\mathbb {R}^s
,
M
M
is an
s
×
s
s \times s
dilation matrix, and
a
a
is a finitely supported refinement mask such that each
a
(
α
)
a(\alpha )
is an
r
×
r
r \times r
(complex) matrix. Let
ϕ
0
\phi _0
be an initial vector in
(
L
2
(
R
s
)
)
r
(L_2(\mathbb {R}^s))^r
. The corresponding cascade algorithm is given by
ϕ
k
:=
g
+
∑
α
∈
Z
s
a
(
α
)
ϕ
k
−
1
(
M
⋅
−
α
)
,
k
=
1
,
2
,
…
.
\begin{equation*} \phi _k := g + \sum _{\alpha \in \mathbb {Z}^s} a(\alpha ) \phi _{k-1}({M \cdot } - \alpha ), \qquad k=1,2,\ldots . \end{equation*}
In this paper we give a complete characterization for the
L
2
L_2
-convergence of the cascade algorithm in terms of the refinement mask
a
a
, the nonhomogeneous term
g
g
, and the initial vector of functions
ϕ
0
\phi _0
.