We consider solutions of a refinement equation of the form
\[
ϕ
=
∑
γ
∈
Z
s
a
(
γ
)
ϕ
(
M
⋅
−
γ
)
,
\phi = \sum _{\gamma \in \mathbb {Z}^s} a(\gamma ) \phi ({M\cdot }-\gamma ),
\]
where
a
a
is a finitely supported sequence called the refinement mask. Associated with the mask
a
a
is a linear operator
Q
a
Q_a
defined on
L
p
(
R
s
)
L_p(\mathbb {R}^s)
by
Q
a
ψ
:=
∑
γ
∈
Z
s
a
(
γ
)
ψ
(
M
⋅
−
γ
)
Q_a \psi := \sum _{\gamma \in \mathbb {Z}^s} a(\gamma ) \psi ({M\cdot }-\gamma )
. This paper is concerned with the convergence of the cascade algorithm associated with
a
a
, i.e., the convergence of the sequence
(
Q
a
n
ψ
)
n
=
1
,
2
,
…
(Q_a^n\psi )_{n=1,2,\ldots }
in the
L
p
L_p
-norm. Our main result gives estimates for the convergence rate of the cascade algorithm. Let
ϕ
\phi
be the normalized solution of the above refinement equation with the dilation matrix
M
M
being isotropic. Suppose
ϕ
\phi
lies in the Lipschitz space
Lip
(
μ
,
L
p
(
R
s
)
)
\operatorname {Lip} (\mu ,L_p(\mathbb {R}^s))
, where
μ
>
0
\mu >0
and
1
≤
p
≤
∞
1 \le p \le \infty
. Under appropriate conditions on
ψ
\psi
, the following estimate will be established:
\[
‖
Q
a
n
ψ
−
ϕ
‖
p
≤
C
(
m
−
1
/
s
)
μ
n
∀
n
∈
N
,
\bigl \| Q_a^n\psi - \phi \bigr \|_p \le C (m^{-1/s})^{\mu n}\quad \forall \, n \in \mathbb {N},
\]
where
m
:=
|
det
M
|
m:=|\det M|
and
C
C
is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.