Characterization of Lp
c -solutions for the Dilation Equations on R2

Chun-Tai Liu1 and Guo-Tai Deng2
1Department of Mathematics and Physics, Wuhan Polytechnic University, Wuhan 430023, P. R. China
2College of Mathematics and Statistics, Huazhong Normal University, Wuhan 430079, P.R. China

Email: lct984@163.com, hilltower@163.com

Abstract In this paper, the author discussed the existence of compactly supported Lp-solutions for
the dilation equations on the plane. Furthermore, two examples are given to illustrate the general theory.

Keywords Dilation equation Compactly supportedLp-solutions Iteration function system

1. Introduction

A α-scale dilation equation is a functional equation of the form

f(x) =

N∑
n=0

cnf(αx− βn)

where f : R → R(or C), α > 1, β0 < β1 < · · · < βN are real constants, and cn are real
(complex) constants. The equation is called a lattice k-scale dilation equation if

f(x) =

N∑
n=0

cnf(kx− n)

for an integer k ≥ 2. A special case of the functional equation (k = 3, N = 4, and cn =
1, 2/3, 1/3, 1) was first studied by de Rham[1] as an example of a continuous nowhere dif-
ferentiable function. Recently this equation has attracted a lot of attention, especially for the
lattice case with k = 2. In wavelet theory, the study of multiresolution and the search of var-
ious orthogonal, compactly supported wavelets has lead to the investigation of the existence,
uniqueness, and smoothness of such continuous integrable solutions[2]. The equation also plays
an important role in the “subdivision schemes” and “interpolation schemes” of constructing
continuous spline curves, surfaces and fractal objects [3, 4] .

There are two major approaches to the equation: the Fourier method(the frequency domain
approaches) and the iteration method(the time-domain approaches). Using Fourier transforma-
tion, Daubechies and Lagarias[3] proved that the equation has a nonzero integrable solution. By
using the Fourier transform of f and the Paley-Wiener theorem, it was proved in [3] that f has
compact support in [0, βN/(α− 1)]. The Fourier method, however, does not give sharp criteria
for the existence of L1 − solutions in terms of the coefficients{cn}. Some partial results are
given in [5, 6].

The iteration method is restricted to the lattice case. It applies particularly well in the case
of compactly supported solutions. The basic idea is to identify a given function f supported by

ISSN 1078-6236 International Institute for General Systems Studies, Inc.

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 61-70



[0, N ] with the vector-valued function

f(x) = [f(x), · · · , f(x+ (N − 1))]T , x ∈ [0, 1],

and to use the right side of the dilation equation to construct two N × N matrices T0 and T1.
A constant vector v is used as the initial condition, followed by iteration with the matrices T0
and T1. The limit, if the sequence converges, will be the solution of the dilation equation.
Such an approach was used by Daubechies and Lagarias[4], and independently by Michelli and
Prautzsch[7]. It was also used by Collela and Heil[8] and [9] and Ka-sing Lau and Jianrong
Wang[10].

Similarly, on the plane the dilation equation is defined as the form

f(x) =
M∑
m=0

N∑
n=0

cmnf(Ax−Qmn) (1)

where f : R2 → R or C, A is an expand matrix, Qmn are vectors, cmn are real (or complex)
constants. The equation is called a lattice dilation equation if

f(x) =

M∑
m=0

N∑
n=0

cmnf(Ax−
(
m

n

)
) (2)

where A is an integer expand matrix.

In this paper we will study the existence of the compactly supported Lp-solutionof the

equation(2) on the plane with cmn ∈ R and A =

(
2 0
0 2

)
.

As usually the basic assumption on the coefficients is
M∑
m=0

N∑
n=0

cmn = | detA| = 4.

Let d1 =
(
0
0

)
, d2 =

(
0
1

)
, d3 =

(
1
0

)
, d4 =

(
1
1

)
, ϕk(x) = A−1(x + dk), k = 1, 2, 3, 4, there

exists an attractor T = [0, 1]× [0, 1] satisfing T =
4⋃

k=1

ϕk(T ) ,
4⋃

k=1

Tk.

At the same time there exist vectors {eis =
(
i
s

)
, 0 ≤ i ≤M − 1, 0 ≤ s ≤ N − 1} such that

suppf ⊂
M−1⋃
i=0

N−1⋃
s=0

(T + eis).

Let

Pi0 = (ci,2s−t)0≤s, t≤N−1, Pi1 = (ci,2s−t+1)0≤s, t≤N−1, i = 0, 1, · · · ,M.

For example,

P00 = (c0,2s−t)0≤s, t≤N−1 =


c0,0
c0,2 c0,1 c0,0
c0,4 c0,3 c0,2 c0,1 c0,0
· · · · · · · · · · · · · · · · · ·
0 0 0 0 0 · · · c0,N c0,N−1

 .

62 Liu:Characterization of Lp
c -solutions for the Dilation Equations on R2



And let

M1 = (P2i−j,0)0≤i,j≤M−1 =


P0,0

P2,0 P1,0 P0,0

P4,0 P3,0 P2,0 P1,0 P0,0

· · · · · · · · · · · · · · · · · ·
0 0 0 0 0 · · · PM,0 PM−1,0

 ,

M2 = (P2i−j,1)0≤i,j≤M−1 =


P0,1

P2,1 P1,1 P0,1

P4,1 P3,1 P2,1 P1,1 P0,1

· · · · · · · · · · · · · · · · · ·
0 0 0 0 0 · · · PM,1 PM−1,1

 ,

M3 = (P2i−j+1,0)0≤i,j≤M−1 =


P1,0 P0,0

P3,0 P2,0 P1,0 P0,0

P5,0 P4,0 P3,0 P2,0 P1,0 P0,0

· · · · · · · · · · · · · · · · · · · · ·
0 0 0 0 0 0 · · · 0 PM,0

 ,

M4 = (P2i−j+1,1)0≤i,j≤M−1 =


P1,1 P0,1

P3,1 P2,1 P1,1 P0,1

P5,1 P4,1 P3,1 P2,1 P1,1 P0,1

· · · · · · · · · · · · · · · · · · · · ·
0 0 0 0 0 0 · · · 0 PM,1


We define a vector function: F (x) = (f(x+e00), f(x+e01), f(x+e02), · · · , f(x+e0,N−1), f(x+
e10), · · · , f(x + e1,N−1), · · · , f(x + eM−1,N−1))

T for x ∈ T = [0, 1] × [0, 1], then equation
(2) will satisfy

F (x) =


M1F (ϕ−11 (x)) x ∈ T1 = [0, 1/2)× [0, 1/2);

M2F (ϕ−12 (x)) x ∈ T2 = [0, 1/2)× [1/2, 1);

M3F (ϕ−13 (x)) x ∈ T3 = [1/2, 1)× [0, 1/2);

M4F (ϕ−14 (x)) x ∈ T4 = [1/2, 1)× [1/2, 1);
0 x ∈ others.

(3)

Let v is 4-eigenvector of (M1 + M2 + M3 + M4),we have (M1 + M3 − 2I)v = −(M2 +
M4 − 2I)v.And let ṽ = (M1 + M3 − 2I)v, H(ṽ) be the subspace in RM×N spanned by
{Mσṽ : σ ∈ Σ∗}. Then the basic theorem is as follows.

Theorem 1.1. For 1 ≤ p ≤ ∞, the following are equivalent:

(1) equation (2) has a nonzero compactly supportedLp-solution;

(2) there exists a 4-eigenvector v of (M1 +M2 +M3 +M4) satisfying

lim
l→∞

1

4l

∑
|σ|=l

‖Mσṽ‖p = 0

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 63



(3) there exists a 4-eigenvector v of (M1+M2+M3+M4) such that there exists an integer
l ≥ 1 such that

1

4l

∑
|σ|=l

‖Mσu‖p < 1 for all u ∈ H(ṽ), ‖u‖ ≤ 1

2. Preliminaries

Lemma 2.1. If equation(2) exists compactly supportedLp-solutionf , then suppf ⊂ [0,M ]×
[0, N ].

Proof Let suppf ⊂ D, take x ∈ D with f(x) 6= 0 thenAx−
(
m
n

)
∈ D, i.e. x ∈ A−1(D+

(
m
n

)
).

Let E = {0, 1, 2 · · ·M} × {0, 1, 2 · · ·N}, then

D ⊂ A−1(D + E) = A−1D +A−1E

⊂ A−1(A−1D +A−1E) +A−1E = A−2D +A−2E +A−1E

⊂ · · ·
⊂ A−tD +A−tE +A−(t−1)E + · · ·+A−1E

let t→∞, then

D ⊂ {
∞∑
t=1

A−ty : y ∈ E} ⊂ [0,M ]× [0, N ]

for the closed set E.

Proposition 2.2. Let f be supported by [0,M ]× [0, N ], and let F be defined as above, then f
is an Lpc−solution of (2) if and only if F ∈ Lp and F = MF , i.e. F satisfies equation(3).

Proposition 2.3. If
M∑
m=0

N∑
n=0

cmn = 4, then 4 is an eigenvalue of (M1 +M2 +M3 +M4) with

left eigenvalue [1, 1, · · · , 1].

Proof Obviously, the sum of each column is equal to 4 in the matrix (M1 +M2 +M3 +M4).
�

It follows that the right 4-eigenvector of (M1 +M2 +M3 +M4) exists also; it will play a
central role in the existence of the solution of equation(2). Let f4 be the average of f over
4, i.e., f4 = 1

L(4)

∫
4 f .

Proposition 2.4. Let f be an compactly supported Lp-solutionof equation(2), v = [fT+e00 ,
fT+e01 · · · fT+eM−1,N−1

]T be the vector defined by the average of f on the M ×N subintervals
as indicated. Then v is 4-eigenvector of (M1 +M2 +M3 +M4).

Proof According to Proposition2.2, F = MF , i.e.,

F (x) =


M1F (ϕ−11 (x)) x ∈ T1 = [0, 1/2)× [0, 1/2)

M2F (ϕ−12 (x)) x ∈ T2 = [0, 1/2)× [1/2, 1)

M3F (ϕ−13 (x)) x ∈ T3 = [1/2, 1)× [0, 1/2)

M4F (ϕ−14 (x)) x ∈ T4 = [1/2, 1)× [1/2, 1)

(4)

64 Liu:Characterization of Lp
c -solutions for the Dilation Equations on R2



when we integrate the expression over T1, T2, T3 and T4 separately, we have

[f[0, 1
2
]×[0, 1

2
], · · · , f[M−1,N− 1

2
]×[M−1,N− 1

2
]]
T = M1v

[f[0, 1
2
]×[ 1

2
,1], · · · , f[M−1,N− 1

2
]×[M− 1

2
,N ]]

T = M2v

[f[ 1
2
,1]×[0, 1

2
], · · · , f[M− 1

2
,N ]×[M−1,N− 1

2
]]
T = M3v

[f[ 1
2
,1]×[ 1

2
,1], · · · , f[M− 1

2
,N ]×[M− 1

2
,N ]]

T = M4v

On the other hand, note that on each interval [i, i+ 1]× [i, i+ 1] the average satisfies

f[i,i+ 1
2
]×[i,i+ 1

2
] +f[i,i+ 1

2
]×[i+ 1

2
,i+1] +f[i+ 1

2
,i+1]×[i,i+ 1

2
] +f[i+ 1

2
,i+1]×[i+ 1

2
,i+1] = 4f[i,i+1]×[i,i+1],

hence we conclude that (M1 +M2 +M3 +M4)v = 4v.

Let Σ = {1, 2, 3, 4}, Σn = {(i1, i2, · · · , in) : ij ∈ Σ}, Σ0 = ∅, Σ∗ =
∞⋃
n=0

Σn, Σ∞ =

{(i1, i2, · · · ) : ij ∈ Σ}. For each σ = (i1, i2, · · · ) ∈ Σ∞, define σ|n = (i1, i2, · · · , in). Let
σ = (i1, i2, · · · , in) ∈ Σ∗, τ = (j1, j2, · · · , jm) ∈ Σ∗, define (σ, τ) := (i1, i2, · · · , in, j1, j2, · · · , jm).Tσ :=
4⋃
i=1

T(σ, i), and Mσ := Mi1Mi2 · · ·Min . So for any σ, τ ∈ Σ∗, we have T(σ, τ) ⊂ Tσ.

Lemma 2.5. Let F0(x) = v for x ∈ T , and Fk+1 = MFk for k ≥ 0. Then Fk = Mσv for
each x ∈ Tσ. Moreover, if f is an LPc −solution of equation(2) and v is the average vector of
f defined in Proposition 2.4, then

Fk = Mσv = [fTσ+e00 , fTσ+e01 · · · fTσ+eM−1,N−1
]T ,

where (Tσ + j) = {x+ j : x ∈ Tσ}. Also, Fk → F in Lp(T,RM×N ).

Proof We will use induction to show that Fk(x) = Mσv for x ∈ Tσ with |σ| = k. Suppose
that Fk(x) = Mσv for x ∈ Tσ. Let x ∈ T(1,σ) = ϕ1(Tσ); then ϕ−11 (x) ∈ Tσ and

Fk+1(x) = MFk(x) = M1Fk(ϕ
−1
1 (x)) = M1Mσv = M(1,σ)v.

Similarly, if x ∈ T(i,σ), then Fk+1(x) = M(i,σ)v, i = 2, 3, 4.

Moreover, F = MF and F (x) = MσF (ϕ−1σ (x)) for x ∈ Tσ. Integrating this over the
interval Tσ, we obtain [fTσ+e00 , fTσ+e01 , · · · , fTσ+eM−1,N−1

]T = Mσv.

Lemma 2.6. Let v be a 4-eigenvector of (M1 + M2 + M3 + M4), and let Fk be defined as
above;then for each k, ∫

T
Fk(x)dx = v. (5)

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 65



Proof This follows from the following induction argument:∫
T
Fk+1 dx =

∫
T1

M1Fk(ϕ
−1
1 (x)) dx+

∫
T2

M2Fk(ϕ
−1
2 (x)) dx

+

∫
T3

M3Fk(ϕ
−1
3 (x)) dx+

∫
T4

M4Fk(ϕ
−1
4 (x)) dx

=
1

4

(
M1

∫
T
Fk(x) dx+M2

∫
T
Fk(x) dx+M3

∫
T
Fk(x) dx+M4

∫
T
Fk(x) dx

)
=

1

4

(
M1 +M2 +M3 +M4

)∫
T
Fk(x) dx

=
1

4
(M1 +M2 +M3 +M4)v = v.

Theorem 2.7. For 1 ≤ p ≤ ∞, the following are equivalent:

(1) equation (2) has a nonzero compactly supportedLp-solution;

(2) there exists a 4-eigenvector v of (M1 +M2 +M3 +M4) satisfying

lim
l→∞

1

4l

∑
|σ|=l

‖Mσṽ‖p = 0

(3) there exists a 4-eigenvector v of (M1 +M2 +M3 +M4) such that there exists an integer
l ≥ 1 such that

1

4l

∑
|σ|=l

‖Mσu‖p < 1 for all u ∈ H(ṽ), ‖u‖ ≤ 1 (6)

Proof Let F0 = v and Fn+1 = MFn. By Lemma (2.5), for x ∈ Tσ and |σ| = n, Fn(x) =
Mσv. Let Gn = Fn+1 − Fn; then Fn+1 = F0 +G0 + · · ·+Gn, where

Gn(x) =

{
M(σ,1)v +M(σ,3)v − 2Mσv = Mσṽ if x ∈ T(σ,1) ∪ T(σ,3),
M(σ,2)v +M(σ,4)v − 2Mσv = −Mσṽ if x ∈ T(σ,2) ∪ T(σ,4),

and
‖Gn‖p =

1

4n

∑
|σ|=n

‖Mσṽ‖p.

Since (1) implies that ‖Gn‖ converges to zero, (2) follows immediately.

To prove that (2) implies (3), we note that H(ṽ) is finite dimensional and has a finite basis
of Mτ ṽ’s. Let u = Mτ ṽ with |τ | = k; then

1

4n

∑
|σ|=n

‖Mσu‖p =
1

4n

∑
|σ|=n

‖MσMτ ṽ‖p ≤ 4k
1

4n+k

∑
|σ|=n+k

‖Mσṽ‖p → 0

as n → ∞, and the convergence is uniform for all ‖u‖ ≤ 1. Hence (6) follows by taking
l = n for n sufficiently large.

66 Liu:Characterization of Lp
c -solutions for the Dilation Equations on R2



Now assume (3) holds. SinceH(ṽ) is finite dimensional, there is a constant 0 < c < 1 such
that for any u ∈ H(ṽ),

1

4l

∑
|σ|=l

‖Mσu‖p ≤ c‖u‖p.

For any |τ | = n, let u = Mτ ṽ ∈ H(ṽ); then

1

4l

∑
|σ|=l

‖MσMτ ṽ‖p ≤ c‖Mτ ṽ‖p.

Summing over all |τ | = n, we have

1

4l+n

∑
|σ|=l+n

‖Mσṽ‖p =
1

4l+n

∑
|σ|=l

∑
|τ |=n

‖MσMτ ṽ‖p <
c

4n

∑
|τ |=n

‖Mτ ṽ‖p.

It follows from the expression of ‖Gn‖ given above that

‖Gn+l‖p ≤ c‖Gn‖p.

For each fixed n, {‖Gn+kl‖}∞k=1 is dominated by a geometric series, hence Fn+1 = F0 +G0 +
· · ·+Gn converges in Lp. The limit F is nonzero by Lemma (2.6), and so by Proposition (5),
(1) follows.

Remark 2.8. we can also consider the equation

g(x) =
M∑
m=0

N∑
n=0

dmng(Bx−
(
m

n

)
)

with B =

(
1 1
1 −1

)
and

M∑
m=0

N∑
n=0

dmn = |detB| = 2, since iterating this equation again, we

obtain the equation (2).

Corollary 2.9. Under the same hypotheses of Theorem (2.7), assume that the solution f exists;
then v /∈ H(ṽ), and the dimension of H(ṽ) is at most MN − 1.

Proof By Theorem (2.7)(2),

1

4n

∑
|σ|=n

‖Mσu‖p → 0 for any u ∈ H(ṽ).

It follows that if v ∈ H(ṽ), then

‖v‖p =
1

4np
‖(M1 +M2 +M3 +M4)

nv‖p ≤ 1

4n

∑
|σ|=n

‖Mσv‖p → 0

as n→∞. This contradicts v 6= 0.

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 67



3. Some Examples

Example 1: We consider a dilation equation :

f(x) =
1∑

m=0

2∑
n=0

cmnf(Ax−
(
m

n

)
) (7)

where A =

(
2 0
0 2

)
,

1∑
m=0

2∑
n=0

cmn = 4.

Theorem 3.1. For 1 ≤ p <∞, equation(7) has a (nonzero)Lpc−solution if and only if either
c01 + c11 = 2 and

1

4
(|c00 + c10|p + |2− (c00 + c10)|p) < 1

or c00 + c10 = 2 and c02 + c12 = 2.

Proof Note that

M1 =

(
c00 0
c02 c01

)
,M2 =

(
c01 c00
0 c02

)
,M3 =

(
c10 0
c12 c11

)
,M4 =

(
c11 c10
0 c12

)
,

and M1 +M2 +M3 +M4 =

(
c00 + c10 + c01 + c11 c00 + c10
c02 + c12 c01 + c11 + c02 + c12

)
.

If (c00 + c10, c02 + c12) = (0, 0), then M1 + M2 + M3 + M4 = 4I . Any nonzero
vector v = [x, y]T will be a 4-eigenvector. It is a direct calculation that v ∈ H(ṽ) and, by
Corollary(2.9), no nonzero Lpc−solution exists.

We assume that (c00 + c10, c02 + c12) 6= (0, 0), then 4-eigenvector of M1 +M2 +M3 +M4

is v = [c00 + c10, c02 + c12]
T , so that

ṽ = (M1 +M3 − 2I)v =

(
(c00 + c10)(c00 + c10 − 2)

(c02 + c12)(2− (c02 + c12))

)
. (8)

For a Lpc−solution exists, H(ṽ) can only be {0} or one-dimensional(Corollary(2.9)).

In the first case, ṽ = 0, condition(6) is automatically satisfied. The only possible cases
are

(c00 + c10, c02 + c12) = {(2, 2), (0, 2), (2, 0)}.

In the second case, ṽ 6= 0. Since H(ṽ) is invariant under M1 +M3 and M2 +M4, (M1 +
M3)ṽ = cṽ for some c. Expression (8) yields the following cases:

(a) c00 + c10 = 0 or c02 + c12 = 0. In this case v ∈ H(ṽ) and Corollary(2.9) implies that
(7) has no Lpc−solution.

(b) c00 + c10 = 2 or c02 + c12 = 2. In this case a direct calculation shows that (M1 +
M3)ṽ, (M2 + M4)ṽ are independent. Since H(ṽ) is 2-dimensional and by Corollary(2.9) no
Lpc−solution exists.

68 Liu:Characterization of Lp
c -solutions for the Dilation Equations on R2



(c) c00 +c10 6= 0, 2 and c02 +c12 6= 0, 2. Let a = c00 +c10, b = c02 +c12. By equating (8)
and

(M1 +M3)ṽ =

(
a2(a− 2)

b[(2− b)(4− a− b) + (a− 2)a]

)
(9)

with (M1 +M3)ṽ = cṽ, we have c = a, so that by (8) and (9),

b[(2− b)(4− a− b) + (a− 2)a] = ab(2− b);

that is
(a+ b− 4)(a+ b− 2) = 0.

Hence, either (i) or (ii) below holds.

(i) a+ b = 4. In this case v = [a, 4− a]T and ṽ = (a− 2)v. Once again v ∈ H(ṽ) and no
Lpc−solution exists.

(ii) a+b = 2. In this case a direct calculation shows that (M1+M3)ṽ = aṽ, (M2+M4)ṽ =
bṽ. By Theorem(6), equation(7) has an Lpc−solution if and only if there exists an integer
l ≥ 1 such that

1

4l
(|a|p + |b|p)l‖ṽ‖p =

1

4l

∑
|σ|=l

‖Mσṽ‖p < ‖ṽ‖p.

This is equivalent to
1

4
(|a|p + |2− a|p) < 1,

i.e.,
1

4
(|c00 + c10|p + |2− (c00 + c10)|p) < 1.

The theorem follows by summarizing all the cases.

It follows directly from the theorem that if a+ b = 2 and if

(i) a ∈ (−1, 3), then an L1
c − solution exists;

(ii) a ∈ (0, 2), then an L2
c − solution exists;

(iii) a = 1, then an Lpc − solution exists for all 1 6 p <∞.
Example 2: Considering a dilation equation as follows:

g(x) =

1∑
m=0

1∑
n=0

dmng(Bx−
(
m

n

)
) (10)

where B =

(
1 1
1 −1

)
and

1∑
m=0

1∑
n=0

dmn = 2.. Iterating the equation once, we obtain the

suppg ⊂ [−1, 2]× [0, 3].

Let f(x) = g(x−
(
1
0

)
), A = B2 =

(
2 0
0 2

)
, and let

(cmn)0≤m,n≤3 =


0 d00d10 d01d10 0
d200 d00d01 + d210 d00d11 + d10d11 d01d11

d00d10 d00d11 + d00d01 d201 + d10d11 d211
0 d01d10 d01d11 0



Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 69



then (10) is rewritten as the form

f(x) =
3∑

m=0

3∑
n=0

cmnf(Ax−
(
m

n

)
) (11)

with suppf ⊂ [0, 3]× [0, 3], so the discussion of the equation (11) is similar with the Example
1.

References

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[2] Heil,C. Mathods of solving dilation equations, Proc.1991 Nato Adv.Sci.Ins.on Prob. and
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1993.

[3] Daubechies,I. & Lagarias J. Two-scale difference equation I.Existence and global regu-
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[4] Daubechies,I. & Lagarias J. Two-scale difference equation II. Local regulatity, infinite
products of matrices,and fractals, Siam J. Math.Anal., 23(1992) 1031-1079.

[5] Lawton, W. Necessary and sufficient conditions for constructing orthonormal wavelet
bases, J.Math.Phys., 32(1991) 57-64.

[6] Mallalt, S. Multiresolution approximation and wavelet orthonormal bases for L2(R),
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[7] Michelli, C.A. & Prautzsch,H. Uniform refinement of curves, Linear Algebra Appl.,
114/115(1989) 841-870.

[8] Colella, D. & Heil,C. The characterization of continuous,four-coefficient scaling functions
and wavelets, IEEE.Trans.Inform.Theory, 30(1992) 876-881.

[9] Colella, D. & Heil,C. Characterizations of scaling functions, I.Continuous solu-
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[10] Lau K.L. & Wang,J.R. Characterization of Lp−solutions for the two-scale dilation equa-
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70 Liu:Characterization of Lp
c -solutions for the Dilation Equations on R2