Page 1
Construction of Orthonormal Multi-Wavelets
with Additional Vanishing Moments
Charles K. Chuia,∗
aDepartment of Mathematics & Computer Science, University of Missouri–St.
Louis, St. Louis, MO 63121, USA, and
Department of Statistics, Stanford University, Stanford, CA 94305, USA
Jian-ao Lianb,†
bDepartment of Mathematics, Prairie View A&M University, Prairie View, TX
77446, USA
Abstract. An iterative scheme for constructing compactly supported orthonormal
(o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established.
Precisely, let φ = [φ1,···,φr]?be an r-dimensional o.n. scaling function vector with
polynomial preservation of order (p.p.o.) m, and ψ = [ψ1,···,ψr]?an o.n. multi-
wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a
new (r + 1)-dimensional o.n. scaling function vector φ?:= [φ?,φr+1]?and some
corresponding o.n. multi-wavelet ψ?are constructed in such a way that φ?has
p.p.o. = n > m and their two-scale symbols P?and Q?are lower and upper triangular
block matrices, respectively, without increasing the size of the supports. For instance,
for r = 1, if we consider the mthorder Daubechies o.n. scaling function φD
φ?:= [φD
for r = 2, if we use the symmetric o.n. scaling function vector φ in our earlier work
[3], then we obtain a new pair of scaling function vector φ?= [φ?,φ3]?and multi-
wavelet ψ?that not only increase the order of vanishing moments but also preserve
symmetry.
m, then
m,φ2]?is a scaling function vector with p.p.o. > m. As another example,
Keywords: scaling function vectors, multi-wavelets, two-scale symbols, two-scale
equations, orthonormality, compactly supported functions
Dedicated to Charles A. Micchelli in Honor of His Sixtieth Birthday
1. Introduction
A vector-valued function φ := [φ1,···,φr]?is called a refinable func-
tion vector of dimension r, or r-refinable function vector, if φ ∈ (L2)r,
L2:= L2(I R), and satisfies φ(x) =?M
j=0Pjφ(2x − j), P0, PM ?= Or,
for some finite matrix sequence {Pj} ⊂ I Rr×r, where Or denotes the
∗Supported in part by NSF grants CCR-9988289 and CCR-0098331 and Army
Research Office under grant DAAD 19-00-1-0512.
†Supported in part by Army Research Office under grant DAAD 19-01-1-0739.
c ? 2003 Kluwer Academic Publishers. Printed in the Netherlands.
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C. Chui & J.-A. Lian
zero square matrix of order r. The r-refinable function vector φ is
called an r-scaling function vector, if φ is L2-stable, meaning that
{φ?(·− k) : 1 ≤ ? ≤ r; k ∈ Z Z} is a Riesz basis of V0, where, for j ∈ Z Z,
Vj:= closL2 span2j/2φ?(2j·−k) : 1 ≤ ? ≤ r; k ∈ Z Z
??
,
which is also called a multiresolution analysis (MRA) of L2, provided
L2= closL2 ∪j∈Z ZVj. Corresponding to an r-scaling function vector
φ, a vector-valued function ψ(x) = [ψ1,···,ψr]?=
k), Q0, QN ?= Or, for some finite matrix sequence {Qj} ⊂ I Rr×r, is
called a (semi-orthogonal) multi-wavelet, if {ψ?(· − k) : 1 ≤ ? ≤ r;k ∈
Z Z} is a Riesz basis of the L2-orthogonalcomplementary subspace W0⊂
V1relativetoV0, i.e.,V1= V0⊕⊥W0. A function vector η = [η1,...,ηr]?
is said to be orthonormal (o.n.) if it satisfies ?η?(· − k),η??(· − k?)? =
δ?,?? δk,k?, ?, ??= 1,···,r; k, k?∈ Z Z. Hence, an o.n. multi-waveletψ cor-
responding to an o.n. r-scaling function vector φ generates an o.n. basis
{2−j/2ψ?(2j· −k) : ? = 1,...,r; j,k ∈ Z Z} of L2.
The above time-domain definitions have frequency-domain formula-
tions
?N
k=0Qkφ(2x −
?φ(ω) = P
?ψ(ω) = Q
?
?
e−iω/2??φ
e−iω/2??φ
?ω
?ω
2
?
?
,P(z) =1
2
M
?
N
?
j=0
Pjzj,
(1.1)
2
,Q(z) =1
2
j=0
Qjzj,
where the polynomial matrices P,Q ∈ πr×r, π := π(I R), are called
two-scale symbols of φ and ψ, respectively. To study the smoothness
of o.n. scaling function vectors, we consider the usual H¨ older spaces
Cγ:= Cγ(I R), where γ > 0 and 0 < γ − ?γ? < 1, defined by
Cγ(I R) =
f :
?
|f(?γ?)(·+ h) − f(?γ?)(·)| ≤ C |h|γ−?γ??
.
(1.2)
Finally, the supports of φ and ψ are defined by suppφ = ∪r
and suppψ = ∪r
The first main objective of this paper is to introduce an iterative
scheme for constructing compactly supported o.n. scaling function vec-
tors and multi-wavelets in terms of any given o.n. scaling function (or
scaling function vector) of lower dimension in order to achieve addi-
tional vanishing moments for the multi-wavelets. Precisely, let φ be
an o.n. scaling function (vector) with polynomial preservation of order
(p.p.o.) m, and ψ an o.n. (multi-)waveletcorresponding to φ, with two-
scale symbols P and Q, respectively. Then a new o.n. (r + 1)-scaling
?=1suppφ?
?=1suppψ?, respectively.
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Orthonormal Multi-Wavelets with Vanishing Moments
3
function vector φ?:= [φ?,φr+1]?and some corresponding o.n. multi-
wavelet ψ?can be constructed in such a way that φ?has p.p.o. = n > m
and that their two-scale symbols P?, Q?are lower and upper triangular
block matrices, without increasing the support. Furthermore, if φ is
symmetric and/or anti-symmetric, φ?and ψ?can be constructed to
preserve symmetry and/or anti-symmetry with only slight increase
in support. For instance, for r = 1, if we start with the mthorder
Daubechies o.n. scaling function φD
function vector with p.p.o. > m. As another example, for r = 2, if we
use the symmetric o.n. scaling function vectors φ = [φ1,φ2]?in our
earlier work [3], then we obtain new pairs of scaling function vectors
φ?= [φ?,φ3]?and multi-waveletsψ?that not only increase the order of
vanishing moments but also preserve symmetry. An important feature
of our method is that the corresponding multi-wavelets are easy to
construct.
The main results of this paper will be presented in §2 and proved in
§3. An interesting example of o.n. φ and ψ with support= [0,1] will be
considered in §4. This is a natural vector-valued extention of the Haar
wavelet. For r = 1 and with φ1= φD
multi-wavelets will be constructed in §5. The construction of o.n. sym-
metric scaling function vectors and multi-wavelets will constitute §6,
where r = 2 and φ1being scaling function vectors of those in [3] and
GHM [5].
m, then φ?:= [φD
m,φ2]?is a scaling
m, o.n. scaling function vectors and
2. Main Results
A scaling function vector φ has p.p.o. = m, if m is the largest integer for
which there is a set of row vectors {a0,?}m−1
that satisfy, for ? = 0,...,m− 1,
?
k
?
k
j∈Z Z
?=0⊂ I R1×r, with a0,0?= O1×r
?
?
k=0
(−1)k1
2k
?
?
?
a0,?−k?
j∈Z Z
(2j)kP2j =
1
2?a0,?; (2.1)
?
?
k=0
(−1)k1
2k
?
a0,?−k?
(2j + 1)kP2j+1 =
1
2?a0,?,
(2.2)
(see [8] for details). As shown in [11], if a scaling function vector φ has
p.p.o. = m, then det(P(z)) must be divisible by (1 + z)m. However,
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C. Chui & J.-A. Lian
this condition is not sufficient. As a simple counterexample, consider
which satisfies (1+z)3|det(P(z)). If P is the two-scale symbol of some
scaling function vector φ, then, with φ1being normalized as the sec-
ond order cardinal B-spline, the corresponding autocorrelation symbol
Φ(z) :=?
2(1 − β)
Hence, since Φ must be positive definite for |z| = 1, we have |β| < 1
(see, e.g., [4] and [6]), which implies that φ has p.p.o. = 2 when β ?= 1/4,
and p.p.o. = 3 when β = 1/4. In general, by applying (2.1)–(2.2), we
can establish the following.
P(z) =
?1 + z
2
α
?2
0
1 + z
2
β
, αβ ?= 0,
n∈Z Z(?φj(·),φk(· − n)?)1≤j,k≤2zn, is given by
Φ(z) =
1 + 4z + z2
6z
α
α
2(1− β)(1+ z)
2α2(2+ β)
3(1+ β)(1− β)2
?
1 +1
z
?
.
Theorem 2.1. Let φ be an r-scaling function vector with two-scale
symbol P and p.p.o. = m, and consider the polynomial matrix
?P(z) Or×1
where
a(z) =1
2
k∈Z Z
B(z) = [B1(z),...,Br(z)] =:1
P?=
B(z) a(z)
?
,
(2.3)
?
akzk; (2.4)
2
?
k∈Z Z
bkzk,
(2.5)
with row vectors bk, k ∈ Z Z. Then a necessary condition for P?to
be the two-scale symbol of some (r + 1)-scaling function vector φ?=
[φ?,φr+1]?that provides p.p.o. = n > m is that
?1 + z
s(1) = 1.
a(z) =
1
2m
2
?n−m
s(z);(2.6)
(2.7)
By applying Theorem 2.1 to any pair of o.n. scaling function vectors
and multi-wavelets,not only do we obtain a new scaling function vector
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Orthonormal Multi-Wavelets with Vanishing Moments
5
with higher p.p.o., but also some corresponding o.n. multi-wavelet can
be easily constructed. Precisely, we have the following.
Theorem 2.2. Let φ be an o.n. r-scaling function vector with two-
scale symbol P and p.p.o = m, and ψ be its corresponding o.n. multi-
wavelet with two-scale symbol Q(z). Then there is φr+1such that φ?=
[φ?,φr+1]?is an o.n. (r+1)-scaling function vector with p.p.o. ≥ m+1
and two-scale symbol P?given by (2.3)–(2.5) with
B(z) =
r
?
j=1
αjQj(z),
(2.8)
a(−1) = 0,a(1) =
1
2m,
(2.9)
|a(z)|2+ |a(−z)|2= 1 −
r
?
j=1
|αj|2=
1
22m,
|z| = 1,
(2.10)
for some constants α1,...,αr. In addition, ψ?is an o.n. multi-wavelet
corresponding to φ?, with two-scale symbol Q?being upper triangular
block matrix, given by
?
O1×r
Q?(z) =
AQ(z) 2ma(z)[ζ1,...,ζr]?
2mz2L−1a(−z)?
?
,
(2.11)
where A is a constant matrix, deg(a) = 2L − 1, and [ζ1,...,ζr]?is a
constant vector, such that
[α1,...,αr]A?+
1
2m[ζ1,...,ζr] = 0,
(2.12)
AA?+ [ζ1,...,ζr]?[ζ1,...,ζr] = Ir,
(2.13)
where Iris the identity matrix of order r.
One of the important features of the construction procedure de-
scribed here is that it can be applied repeatedly without increasing the
support (or filter length). However, if the support of φ?needs to be
increased, B in (2.8) can be chosen as
B(z) =
r
?
j=1
αj(z2)Qj(z),
(2.14)
for some Laurent polynomials α1,...,αr, while A in (2.11) is chosen
to be a Laurent polynomial matrix in z2. By doing so, the governing
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C. Chui & J.-A. Lian
equations (2.10), (2.12), and (2.13) of all parameters become
|a(z)|2+ |a(−z)|2= 1 −
r
?
j=1
|αj(z)|2=
1
2m[ζ1,...,ζr] = 0,
1
22m,
(2.15)
[α1(z),...,αr(z)]A(z)?+ (2.16)
A(z)A(z)?+ [ζ1,...,ζr]?[ζ1,...,ζr] = Ir,
|z| = 1.
(2.17)
We will demonstrate the procedure in §6 by constructing a symmetric
multi-wavelet with p.p.o. = 3 based on the first o.n. multi-wavelet
constructed in [3].
3. Proof of Main Results
In this section we prove the main results presented in previous section.
Proof of Theorem 2.1. First, by applying the p.p.o. = m of φ, there
are a0,?∈ I Rr,? = 0,...,m − 1, with a0,0?= O1×r, such that, from
(2.1)–(2.2),
j∈Z Z
j∈Z Z
?−1
?
a0,?
?
?
= −
P2j−1
2?Ir
= −
2?Ir
?−1
?
k=0
(−1)?−k
1
2?−k
?
?
k
?
a0,k?
j∈Z Z
(2j)?−kP2j,
a0,?
P2j+1−1
1
k=0
(−1)?−k
2?−k
?
?
k
?
a0,k?
j∈Z Z
(2j + 1)?−kP2j+1.
Secondly, by the Riesz property of φ?, for φ?to have p.p.o. = n > m, the
first m (r+1)-vectors in (2.1)–(2.2), denoted by ˜ a0,?, ? = 0,...,m−1,
must be of the form ˜ a0,?= [a0,?,0]. The additional n − m row vectors
are denoted by ˜ a0,m+?:= [a0,m+?,cm+?],
particular, ˜ a0,mmust satisfy cm?= 0. Indeed, if cm= 0, then ˜ a0,m=
[a0,m,0], so that φ itself would have p.p.o. > m, with last row r-vector
being a0,m. Hence, if we use the notation ˜ a0,?= [a?,c?], then from (2.1),
c?= 0 for ? = 0,...,m − 1, and since φ?has p.p.o. = n > m, we have,
on one hand,
???
? = 0,...,n − m − 1. In
[a0,?,c?]
j∈Z ZP2j
?
Or×1
j∈Z Za2j
j∈Z Zb2j
?
?
−1
2?Ir+1
?
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Orthonormal Multi-Wavelets with Vanishing Moments
7
= −
?−1
?
k=0
(−1)?−k
??
1
2?−k
?
?
k
?
[a0,k,ck]
×
j∈Z Z(2j)?−kP2j
?
Or×1
j∈Z Z(2j)?−kb2j
?
j∈Z Z(2j)?−ka2j
?
,
which is equivalent to
a0,?
?
j∈Z Z
P2j−1
2?Ir
+ c?
(2j)?−kP2j+ ck
?
j∈Z Z
b2j= −
?−1
?
k=0
(−1)?−k
1
2?−k
?
?
k
?
×
a0,k?
a2j−1
j∈Z Z
?
j∈Z Z
(2j)?−kb2j
,
(3.1)
c?
?
j∈Z Z
2?
(−1)?−k
= −
?−1
?
k=0
1
2?−k
?
?
k
?
ck
?
j∈Z Z
(2j)?−ka2j.
(3.2)
Due to the fact that c?= 0, ? = 0,···,m − 1, the identities in (3.2)
become
j∈Z Z
j∈Z Z
×
j∈Z Z
Since cm?= 0, these identities lead to
?
2m
2?− 1
k=0
? = 1,...,n− m − 1.
On the other hand, from (2.2), φ?having p.p.o. = n also implies,
cm
?
?
a2j−
1
2m
= 0,
1
2m+?
?
and
cm+?
a2j−
= −
(2j)?−ka2j,
?−1
?
k=0
(−1)?−k
1
2?−k
?
m + ?
? − k
?
cm+k
? = 1,...,n− m − 1.
j∈Z Z
a2j=
1
2m,
(3.3)
cm+?= −
?−1
?
(−1)?−k2k
?
m + ?
? − k
?
cm+k
?
j∈Z Z
(2j)?−ka2j,
(3.4)
?
j∈Z Z
a2j+1=
1
2m,
(3.5)
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C. Chui & J.-A. Lian
cm+?= −
2m
2?− 1
?−1
?
k=0
(−1)?−k2k
?
m + ?
? − k
?
cm+k
?
j∈Z Z
(2j + 1)?−ka2j+1,
? = 1,...,n − m − 1.
(3.6)
Again, due to the fact that cm?= 0, it follows from (3.3)–(3.6) that
2m?
?
which determines that a(z) must be of the form in (2.6)–(2.7). This
completes the proof of Theorem 2.1.
j∈Z Z
(2j)ka2j=
a2j= 2m?
j∈Z Z
?
a2j+1= 1,
j∈Z Zj∈Z Z
(2j + 1)ka2j+1,k = 1,...,n− m − 1,
As a consequence of Theorem 2.1, we having the following.
Remark 3.1. By applying c?= 0, ? = 0,···,m − 1, the identities in
(3.1) imply that
j∈Z Z
m+?−1
?
?−1
?
By changing the parity of indices in (3.7), we also have
j∈Z Z
m+?−1
?
?−1
?
The two identities (3.7) and (3.8) will be used in §5 to find a0,m+?.
Proof of Theorem 2.2. First, with lower triangular P?as in (2.3),
it is natural to set B in the form of (2.8) in order for φ?to be o.n.,
i.e., B is a linear combination of the rows of Q. Secondly, by applying
a0,m+?
?
= −
P2j−
1
2m+?Ir
+ cm+?
1
2m+?−k
?
m + ?
k
j∈Z Z
b2j
k=0
(−1)m+?−k
??
a0,k?
j∈Z Z
(2j)m+?−kP2j
−
k=0
(−1)?−k
1
2?−k
?
m + ?
? − k
?
cm+k
?
j∈Z Z
(2j)?−kb2j.
(3.7)
a0,m+?
?
P2j+1−
1
2m+?Ir
+ cm+?
?
?
?
?
j∈Z Z
b2j+1
= −
k=0
(−1)m+?−k
1
2m+?−k
?
? − k
m + ?
k
a0,k?
?
j∈Z Z
(2j + 1)m+?−kP2j+1
−
k=0
(−1)?−k
1
2?−k
m + ?
cm+k
j∈Z Z
(2j + 1)?−kb2j+1.
(3.8)
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Orthonormal Multi-Wavelets with Vanishing Moments
9
Theorem 2.1 with n ≥ m + 1 in (2.6)–(2.7), we have (2.9). Thirdly,
φ?being o.n. requires P?(z)P?(z)?+ P?(−z)P?(−z)?= Ir+1, |z| = 1,
which leads to
P(z)B(z)?+ P(−z)B(−z)?= 0,
B(z)B(z)?+ B(−z)B(−z)?+ |a(z)|2+ |a(−z)|2= 1, (3.10)
where the o.n. condition P(z)P(z)?+ P(−z)P(−z)?= Ir of φ has
been used. With Q(z) = [Q1(z)?,···,Qr(z)?]?, i.e., Qj(z) is the jth
row of Q(z), the orthogonality condition P(z)Q(z)?+P(−z)Q(−z)?=
Orbetween φ and ψ is equivalent to P(z)Q?(z)?+ P(−z)Q?(−z)?=
Or×1, ? = 1,...,r. Hence, with B in (2.8) as a linear combination
of Qj(z), the identity (3.9) is automatically satisfied. Similarly, the
o.n. condition Q(z)Q(z)?+ Q(−z)Q(−z)?= Ir of ψ is equivalent to
Qj(z)Qk(z)?+ Qj(−z)Qk(−z)?= δj,k, j,k = 1,...,r. Hence,
(3.9)
B(z)B(z)?+ B(−z)B(−z)?=
r
?
j=1
|αj|2,
which, together with (3.10), leads to (2.10). Finally, by writing Q?as
a upper triangular block matrix of the form (2.11) for some o.n. multi-
wavelet ψ?corresponding to φ?, it follows from the orthogonality be-
tween φ?and ψ?that
[α1,...,αr]A?+ 2m[ζ1,...,ζr]
?
|a(z)|2+ |a(−z)|2?
= 0,
(3.11)
where B(z)Q(z)?+B(−z)Q(−z)?= [α1,...,αr] has been used. Simi-
larly, the o.n. condition of ψ?implies
AA?+ 22m?
×[ζ1,...,ζr]?[ζ1,...,ζr] = 1,
22m?
so that |a(z)|2+ |a(−z)|2= 1/22min (2.10) follows from (3.13). Sub-
stituting (3.13) into (3.11)–(3.12) leads to (2.12)–(2.13). This complete
the proof of Theorem 2.2.
|a(z)|2+ |a(−z)|2?
|a(z)|2+ |a(−z)|2?
(3.12)
(3.13)= 1,
4. From Haar Wavelets to Multi-Wavelets: Support [0,1]
For r = 1 and φ1= χ[0,1), by applying Theorem 2.1 r − 1 times, we
obtain an o.n. r-scaling function vector φ = [φ1,···,φr]?that provides
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C. Chui & J.-A. Lian
p.p.o. = r, and suppφ?= [0,1], ? = 1,...,r. Let L?be the ?th-degree
Legendre polynomial on [−1,1], which can be formulated as
1
2??!
dx?
It can be shown that the r components of the o.n. r-scaling func-
tion vector φ are the dilation and translation of the first r Legendre
o.n. polynomials, up to sign changes. A corresponding multi-wavelet
could be derived by the Gram-Schmidt process ([1], [2], [10]).
First, we have the following.
L?(x) =
d?
?
x2− 1
??,? ∈ Z Z+.
Lemma 4.1. Define
φ?(x) :=
√2?− 1
√2? − 1L?−1(2x− 1)χ[0,1)(x)
1
(? − 1)!
Then φ = [φ1,···,φr]?is an o.n. r-scaling function vector that provides
p.p.o. = r.
(4.1)
=
d?−1
dx?−1[x(x− 1)]?−1χ[0,1)(x),? = 1,...,r.
Proof. First, the orthonormality of the family {φ?(· − k) : 1 ≤ ? ≤
r; k ∈ Z Z} follows immediately from the orthogonalityof {Lk}, namely,
?1
Secondly, observe that any polynomial of exact degree ? can be ex-
panded exactly as a linear combination of Legendre polynomials L0,
..., L?. Since L?((x− 1)/2) and L?((x + 1)/2) are polynomials of exact
degree ? when x ∈ [−1,1), there exist real numbers ξ?,0,...,ξ?,?and
η?,0,...,η?,?such that
−1L?−1(x)L??−1(x)dx =
2
2?− 1δ?,??, ?,??∈ Z Z+.
L?
?x − 1
2
?
=
?
?
k=0
ξ?,kLk(x),L?
?x + 1
2
?
=
?
?
k=0
η?,kLk(x).
for x ∈ [−1,1). Indeed, it clear that
2k + 1
ξ?,k =
2
?1
?1
−1L?
?x − 1
?x + 1
2
?
?
Lk(x)dx,
η?,k =
2k + 1
2
−1L?
2
Lk(x)dx,k = 0,...,?.
Hence,
L?(x) =
?
?
k=0
ξ?,kLk(2x + 1),x ∈ [−1,0),
(4.2)
ChuiLian.tex; 9/12/2003; 21:29; p.10
Page 11
Orthonormal Multi-Wavelets with Vanishing Moments
11
L?(x) =
?
?
k=0
η?,kLk(2x− 1),x ∈ [0,1).
(4.3)
However, (4.2) and (4.3) can be combined to be
L?(x) =
?
?
?
?
k=0
ξ?,kLk(2x + 1)χ[−1,0)(x)
+
k=0
η?,kLk(2x− 1)χ[0,1)(x),
(4.4)
which, in terms of φ?’s in (4.1), is exactly the two-scale relation of φ,
i.e.,
φ(x) = D−1/2
r
P?
0D1/2
r
φ(2x) + D−1/2
r
P?
1D1/2
r
φ(2x − 1),
(4.5)
where
Dr:= diag
?
1,1
3,...,
1
2r − 1
?
,
(4.6)
and P?
ηj,k’s, i.e., with
0and P?
1are the lower triangular matrices with entries ξj,k’s and
ξj,k= ηj,k= 0, j < k, j,k = 0,...,r − 1,
P?
0and P?
1are given by
P?
0:= (ξj,k)r−1
j,k=0,P?
1:= (ηj,k)r−1
j,k=0.
(4.7)
This completes the proof of Lemma 4.1.
For the exact values of P?
0and P?
1in (4.7), we establish the following.
Lemma 4.2. The o.n. scaling function vector φ in Lemma 4.1 satisfies
1.1) with P(z) = 1/2(P0+ P1z), where P0and P1are shown in (4.5),
i.e., Pj= D−1/2
r
P?
explicitly given by
jD1/2
r ,j = 0,1, with Drin (4.6), and P?
0and P?
1being
P?
0=
(−1)j+k2k + 1
1= SrP?
2k
j−k
?
p=0
(−1)p
2p
?
(j + k + p)!
p!(j − k − p)!(2k+ p + 1)!
1,−1,...,(−1)r−1?
r−1
j,k=0
,
P?
0Sr,Sr= diag
.
ChuiLian.tex; 9/12/2003; 21:29; p.11
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12
C. Chui & J.-A. Lian
Furthermore, the r row vectors a0,0,...,a0,r−1in (2.1)–(2.2) for the
p.p.o. = r of φ and the polynomial preservation identities are given by
a0,?= ˜ a0,?D1/2
?
x?=
j∈Z Z
where aj,?:= a0,?+??
r ,? = 0,...,r − 1,
[˜ a0,0]?,...,[˜ a0,r−1]???=
?
?
(j!)2(2k + 1)
(j + k + 1)!(j − k)!
? = 0,...,r − 1,
?r−1
j,k=0
;
aj,?φ(x − j),
p=1
??
p
?jpa0,?−p, j ∈ Z Z. In particular, when x ∈
Lr−1(2x − 1)
[0,1],
1
x
...
xr−1
=
˜ a0,0
˜ a0,1
...
˜ a0,r−1
L0(2x − 1)
L1(2x − 1)
...
.
The two-scale sequence {Q0,Q1} of some corresponding o.n. multi-
wavelet ψ is an upper triangular matrix and satisfies
P?
Q?
Q?
0Dr(Q?
0Dr(Q?
j= D1/2
Q?
0)?+ P?
0)?+ Q?
QjD−1/2
1Dr(Q?
1Dr(Q?
,
1)?= 0,
1)?= 2Dr,
j = 0,1,
rr
(4.8)
1= −SrQ?
0Sr,
and both Q?
0and Q?
1are upper triangular and have only integer entries.
Remark 4.3. It follows from Theorem 4.2 that
ηj,k= (−1)j+kξj,k,j ≥ k; j,k = 0,...,r − 1.
j,k=0in (4.8), then Q?
In addition, if Q?
Hence,
0= (˜ qj,k)r−1
1= −((−1)j+k˜ qj,k)r−1
j,k=0.
1
√2?− 1ψ?(x)
=
k=?−1
r−1
?
r−1
?
˜ q?−1,kLk(4x− 1)−
r−1
?
k=?−1
(−1)?−1+k˜ q?−1,kLk(4x− 3)
=
k=?−1
˜ q?−1,k(Lk(4x − 1) − (−1)?−1+kLk(4x− 3)).
ChuiLian.tex; 9/12/2003; 21:29; p.12
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Orthonormal Multi-Wavelets with Vanishing Moments
13
5. From Daubechies Wavelets to Multi-Wavelets
In this section , we will apply Theorem 2.1 to a natural extention of a
scaling function to a scaling function vector. The followingis immediate
from Theorems 2.1 and 2.2.
Proposition 5.1. Let P be the two-scale symbol of an o.n. scaling
function φ1 with p.p.o. = m, and Q be the two-scale symbol of an
o.n. wavelet corresponding to φ1. Let P?be the two-scale symbol of
the scaling function vector φ = [φ1,φ2]?constructed from Theorem 2.1
with p.p.o. = n > m. Then
?
where a satisfies (2.10) with r = 1 and α1= 2−m√22m− 1. Moreover,
the two-scale symbol Q?of some o.n. multi-wavelet ψ?corresponding
to φ?is upper triangular and given explicitly by
P?(z) =
P(z)0
2−m√22m− 1Q(z) a(z)
?
,
(5.1)
Q?(z) =
?
2−mQ(z) −√22m− 1a(z)
02mz2L−1a(−z)?
?
,
(5.2)
where we have assumed that deg(a) = 2L− 1.
If we start with the mthorder Daubechies o.n. scaling function φD
then φ?:= [φD
Observe that, to obtain φ?’s with high p.p.o., Theorem 2.2 can be
applied repeatedly. For convenience and simplicity, we will temporarily
fix a in (5.1)–(5.2) as a(z) = (1 + z)/2m+1. We have the following.
m,
m,φ2]?is a scaling function vector with p.p.o. > m.
Algorithm 5.2.
1. Denote by P[0](z) = 1/2?
1/2?
j∈Z Zpjzjthe two-scale symbol of φ[0]:=
φD
m, the mthorder Daubechies o.n. scaling function, and Q[0](z) =
j∈Z Zqjzja corresponding o.n. wavelet,respectively, and define
or calculate {a0,?
[0]}?=0,...,m−1by
a0,0
[0]= 1,
(5.3)
a0,?
[0]=
2?
2?− 1
?
?
k=1
(−1)k−11
2k
?
?
k
?
a0,?−k
[0]
?
j
(2j)kp2j,
? = 1,...,m− 1.
(5.4)
ChuiLian.tex; 9/12/2003; 21:29; p.13
Page 14
14
C. Chui & J.-A. Lian
2. For n ≥ 1, denote by P[n](z) and Q[n](z) the two-scale symbols
of the scaling function vector φ[n]and some corresponding multi-
wavelet ψ[n]at the nthstage in the application of Theorem 2.2,
with {a0,?
which has p.p.o. ≥ m + n. Then
B[n−1](z)
2m+n−1
where
[n]}?=0,...,m+n−1being the corresponding row vectors of φ[n],
P[n](z) =
P[n−1](z) On×1
1 1 + z
2
1 + z
2
1 − z
2
,
1
Q[n](z) =
A[n−1]Q[n−1](z)[ζ[n−1]
,...,ζ[n−1]
n
]?
On×n
,
B[n−1](z) =1
2
?
k
b[n−1]
k
zk=
n
?
j=1
α[n−1]
j
Qj
[n−1](z),
(5.5)
with Qj
[n−1](z), j = 1,...,n, being the n rows of Q[n−1](z), i.e.,
?
and the parameters α[n−1]
1
triangular matrix A[n−1], and the parameters ζ[n−1]
(5.6), satisfy (2.10) and (2.12)–(2.13), namely,
Q[n−1](z) =
Q1
[n−1](z)?,...,Qn
[n−1](z)???,
in (5.5), the constant upper
(5.6)
,...,α[n−1]
n
1
,...,ζ[n−1]
n
in
n
?
?
j=1
α[n−1]
1
???α[n−1]
j
???
2= 1−
1
22(m+n−1),
(5.7)
,...,α[n−1]
n
?
(A[n−1])?
?
+
1
2m+n−1
ζ[n−1]
1
,...,ζ[n−1]
n
?
= 0,
(5.8)
A[n−1](A[n−1])?
?
+
ζ[n−1]
1
,...,ζ[n−1]
n
???
ζ[n−1]
1
,...,ζ[n−1]
n
?
= In.
(5.9)
3. Define
a0,?
[n]=
?
a0,?
?
[n−1],0
a0,m+n−1
[n−1]
?
,? = 0,...,m+ n − 2,
,c[n]
.
(5.10)
a0,m+n−1
[n]
=
?
(5.11)
ChuiLian.tex; 9/12/2003; 21:29; p.14
Page 15
Orthonormal Multi-Wavelets with Vanishing Moments
15
Then, together with (5.7) and by applying (3.7)–(3.8), solve for
a0,m+n−1
[n]
j∈Z Z
m+n−2
?
×
j∈Z Z
j∈Z Z
m+n−2
?
×
j∈Z Z
in (5.11) from the linear system
a0,m+n−1
[n−1]
?
(−1)m+n−k
?
?
(−1)m+n−k
?
P[n−1]
2j
−
1
2m+n−1In
+ c[n−1]
m + n − 1
k
?
j∈Z Z
b[n−1]
2j
=
k=0
1
2m+n−1−k
??
a0,k
[n−1]
(2j)m+n−1−kP[n−1]
2j
,
(5.12)
a0,m+n−1
[n−1]
P[n−1]
2j+1−
1
2m+n−1In
+ c[n−1]
m + n − 1
k
?
j∈Z Z
b[n−1]
2j+1
=
k=0
1
2m+n−1−k
??
a0,k
[n−1]
(2j + 1)m+n−1−kP[n−1]
2j+1.
(5.13)
Example 5.3.
As a demonstrative example for Algorithm 5.2, we consider m = 2,
i.e., φ[0]= φD
from (2.6)–(2.7) and (5.1)–(5.2) that, with
?2?
?
z
[0]= (3 −√3)/2, the two-scale symbols P[1]and Q[1]
and the three row vectors a0,0
are given by
2, the 2ndorder Daubechies o.n. scaling function. It follows
P[0](z) =
?1 + z
2
1 +√3
2
?
+1−√3
2
?2?
z
?
1 −√3
2
,
Q[0](z) = −z3P[0]
−1
=
?1 − z
2
−1 +√3
2
z
?
,
and a0,0
[0]= 1,a0,1
[1],a0,1
[1],a0,2
[1]for φ[1], which has p.p.o. = 3,
P[1](z) =
P[0](z)
√15
4
1
4Q[0](z) −
0
0
Q[0](z)
1 + z
8
√15
8
1 − z
,
Q[1](z) =
(1 + z)
2
,
ChuiLian.tex; 9/12/2003; 21:29; p.15
Page 16
16
C. Chui & J.-A. Lian
a0,0
[1]= [1,0],a0,1
[1]=
?
?
3 −√3
2
3(2−√3)
2
,0
?
,
a0,2
[1]=
,−
√5
10
?
,
(5.14)
respectively. Furthermore, by applying the Matlab routines in [7] for
computing the H¨ older smoothness of scaling function vectors, see (1.2),
we have φ ∈ C0.5and φ?∈ C0.4251. See Fig. 1 for the graphs of φ?and
ψ?.
0 0.51 1.522.53
−0.5
0
0.5
1
1.5
0 0.511.52 2.53
−10
−5
0
5
10
0 0.51 1.52 2.53
−15
−10
−5
0
5
10
00.511.52 2.53
−10
−5
0
5
10
Figure 1. Graphs of φ (left) and ψ (right).
To further demonstrate Algorithm 5.2, we repeat the process again
to get
B[1](z)
23
where, from (5.7)–(5.9),
P[2](z) =
P[1](z)
A[1]Q[1](z)
O2×1
11 + z
2
1+ z
2
,
[ζ[1]
1− z
2
Q[2](z) =
1,ζ[1]
2]?
O1×2
,
???α[1]
α[1]
1
???
2+
???α[1]
2
(A[1])?+1
?
???
2=63
64,
(5.15)
?
A[1](A[1])?+
1,α[1]
2
?
8
?
???
ζ[1]
1,ζ[1]
2
?
= 0,
?
(5.16)
ζ[1]
1,ζ[1]
2
ζ[1]
1,ζ[1]
2
= I2.
(5.17)
The parameters will be determined in such a way that the p.p.o. of
the corresponding φ is 4. To this end, it follows from (5.12)–(5.13) that
α[1]
2?= 0 and a0,3
[2]in (5.11) is given by
?
28
a0,3
[2]=
189− 107√3
,−3√5(91− 10√3)
620
,
3√5
160α[1]
2
?
.
ChuiLian.tex; 9/12/2003; 21:29; p.16
Page 17
Orthonormal Multi-Wavelets with Vanishing Moments
17
By writing the upper triangular matrix A[1]as A[1]=
?
a[1]
0
11a[1]
12
a[1]
22
?
, the
identities in (5.15)–(5.17) become
???α[1]
11
???a[1]
α[1]
2
???
???
2=
20181
454601408
2
7103147
1
11215744
?
5121+ 1280√3
862891+ 201810√3
?
α[1]
2,
?
,
(5.18)
???a[1]
22
1= −8√5(2√3 − 1)
651√5
1401968
√5
5432626
ζ[1]
???
2=
??
,
(5.19)
2=862891− 201810√3
?
,
(5.20)
31
?
?
11.
a[1]
12= 1016√3 − 453
1927592√3 − 2073751
2a[1]
?
a[1]
11,
ζ[1]
1 =
?
α[1]
2a[1]
11.
2= −8α[1]
For simplicity, we choose positive roots in (5.18)–(5.20). This leads to
P[2](z) =
4α[1]
Q[2](z) =
In summary, we have P[2]and Q[2]in (5.21)–(5.22) and, by applying
(5.10) and (5.14), we also have
P[0](z)
√15
4
1
1Q[0](z) −
00
Q[0](z)
1 + z
8
α[1]
0
√15(1+ z)
8
1+1 − z
2
α[1]
2
1+ z
16
,
(5.21)
1
4a[1]
11Q[0](z) −
0
√15(1 + z)
8
a[1]
11+1 − z
1 − z
2
0
2
a[1]
12
1+ z
2
1+ z
2
1 − z
ζ[1]
1
ζ[1]
2
a[1]
22
0
2
. (5.22)
a0,0
[2]
a0,1
[2]
a0,2
[2]
a0,3
[2]
=
100
3 −√3
2
3(2−√3)
2
189− 107√3
28
0
√5
10
0
−
0
−3√5(91− 10√3)
620
3√5
160α[1]
2
,
ChuiLian.tex; 9/12/2003; 21:29; p.17
Page 18
18
C. Chui & J.-A. Lian
where α[1]
φ?∈ C0.3436([7]).
Example 5.4.
As another demonstrative example for Proposition 5.1, we consider
φ = φD
first that the two-scale symbols of φD
?1 + z
+1 −√10
2 36
2
is the positive root determined from (5.18). Furthermore,
3, i.e., the 3rdorder Daubechies o.n. scaling function. Observe
3and ψD
?3?
z +1+√10
3are given by
P(z) =
2
1 +√10
36
?
9 +
?
15 + 12√10
?
?
9 −
?
3has p.p.o. = 3 with,
15+ 12√10
?
z2
?
,
and Q(z) = −z5P (−1/z), respectively, and φD
from (5.3)–(5.4), a0,0= 1 and
2−1 +√10
15+√10
2
We apply (5.1)–(5.2) in Proposition 5.1 with m = 3, n = 5, i.e., the
new o.n. scaling function vector φ = [φD
ψ be a corresponding o.n. multi-wavelet. Then the two-scale symbols
P and Q of φ and ψ have the form
a0,1=
5
18
?
15+ 12√10,
−5(1+√10)
18
a0,2=
?
15 + 12√10.
3,φ2]?provides p.p.o. = 5. Let
P?(z) =
?P(z)0
B(z) a(z)
?
,Q?(z) =
1
8Q(z) −3√7a(z)
08z5a
?
−1
z
?
,
where B(z) = 3√7/8Q(z) and a(z) satisfies
a(z) =1
2
5
?
j=0
ajzj=1
8
?1 + z
1
64,
2
?2
s1(z),
(5.23)
|a(z)|2+ |a(−z)|2=
|z| = 1,
(5.24)
with s1being a polynomial of degree 3 satisfying s1(1) = 1. For φ to
have p.p.o. = 5, with
˜ a0,?= [a0,?,0],? = 0,1,2;˜ a0,3+?= [a0,?,c3+?],? = 0,1,
it follows from (3.7)–(3.8) that ˜ a0,3and ˜ a0,4are given by
?
4
˜ a0,3=25+15
√10 −172 + 145√10
126
?
15+ 12√10,−
√70
28
?
,
ChuiLian.tex; 9/12/2003; 21:29; p.18
Page 19
Orthonormal Multi-Wavelets with Vanishing Moments
√10−845+ 575√10
126
√7
28
19
˜ a0,4=
?
90+283
14
?
15 + 12√10,
−
?
10√10 −
?
15+ 12√10
??
.
012345
−0.5
0
0.5
1
1.5
012345
−1.5
−1
−0.5
0
0.5
1
1.5
012345
−2
−1
0
1
2
3
012345
−1.5
−1
−0.5
0
0.5
1
1.5
2
Figure 2. Graphs of the 2-scaling function vector φ (left) and multi-wavelet ψ (right).
From (3.4), we have c4= 32c3
?
j(2j)a2j, so that
?
j
(2j)a2j=
?
j
(2j + 1)a2j+1=
5
16−
√10
320
?
15+ 12√10,
which, together with s1(1) = 1, yields
s?
1(1) =3
2−
√10
40
?
15 + 12√10.
(5.25)
On the other hand, it follows from (5.23)–(5.24)that (cf., e.g., [9]) there
is a constant c0such that
j=0
j
|s1(z)|2=
1
?
?
1 + j
?
tj+ c0t2(1− 2t)
?????t=1
2
?
1−z−1+z
2
?.
(5.26)
Therefore, (5.25)–(5.26) leads to
s1(z) = 1 +
?
√10
80(3− 2
3
2−
√10
40
?
?
15 + 12√10
?
?
(z − 1)
+
?
11
64+
15+ 12√10)(z − 1)2+ d0(z − 1)3,
where d0has two solutions, namely,
√10
160(3 − 2
Finally, φ ∈ C1.0181while the two solutions for φ?are in C0.8796and
C0.8702([7]), respectively. See Fig. 2 and Fig. 3 for graphs of the two
solutions of φ?and ψ?.
d0= −21
128+
?
15 + 12√10) ±
√15
640
?
2049+ 424√10.
ChuiLian.tex; 9/12/2003; 21:29; p.19
Page 20
20
C. Chui & J.-A. Lian
012345
−0.5
0
0.5
1
1.5
012345
−1.5
−1
−0.5
0
0.5
1
1.5
012345
−2
−1
0
1
2
012345
−2
−1.5
−1
−0.5
0
0.5
1
1.5
Figure 3. Graphs of the 2-scaling function vector φ (left) and multi-wavelet ψ (right).
We end this section by indicating that, with φ1= φD
possible p.p.o. of an o.n. scaling function vector obtained from Proposi-
tion 5.1, with suppφ ⊂ [0,2m−1], is ≤ ?5m/3?. This can be seen from
the following three facts. (1) The linear system (3.7)–(3.8) always has
solution for [a0,m+?,cm+?], ? = 0,...,n − m − 1. (2) a in (2.6) satisfies
n − m − 1 linear conditions in (3.4). (3) a in (2.6) satisfying (2.10)
requires s to satisfy
?n−m−1
j=0
?
m, the highest
|s(z)|2=
?
?
n − m − 1 + j
j
(1 − 2t)2???????t=1
?
tj
+t2(1 − 2t)f
2
?
1−z−1+z
2
?,
where f is an arbitrary polynomial. Hence, we conclude, in general,
(n − m − 1)+ (n − m) ≤ deg(s) ≤ 3m − n − 1.
6. Symmetric Multi-Wavelets
For convenience, we say that a scaling function vector or multi-wavelet
is symmetric if all of its components are either symmetric or antisym-
metric. First, from [3], we have the following.
Lemma 6.1. Let φ and ψ be a pair of scaling function vector and multi-
wavelet with two-scale symbols P and Q, respectively. Then both φ and
ψ are symmetric if and only if
S1,rP(z)S1,r= D1,r(z2)P(z)D1,r(z−1),
S2,rQ(z)S1,r= D2,r(z2)Q(z)D1,r(z−1),
|z| = 1,
where
S1,r:= diag((−1)i1,...,(−1)ir),
D1,r(z) := diag(za1+b1,...,zar+br),
D2,r(z) := diag(zc1+d1,...,zcr+dr),
S2,r:= diag((−1)j1,...,(−1)jr),
ChuiLian.tex; 9/12/2003; 21:29; p.20
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Orthonormal Multi-Wavelets with Vanishing Moments
21
with suppφ?= [a?,b?], suppψ?= [c?,d?], ? = 1,...,r; i1,...,ir, j1, ...,
jrbeing either 0 or 1, depending on symmetry or antisymmetry of the
corresponding components of φ and ψ, respectively.
Certainly, for a new scaling function vector φ?constructed by apply-
ing Theorem 2.1 in §2 to be symmetric, we assume that the original φ
is symmetric itself. In this section, we are interested in a similar scheme
for constructing o.n. scaling function vectors and multi-wavelets with
symmetry based on constructed symmetric o.n. scaling function vectors
and multi-wavelets.In particular,we willstartwith symmetrico.n. scal-
ing function vectors φ and corresponding symmetric multi-wavelets ψ
in our earlier work [3]. Then the new pairs of scaling function vectors
and multi-wavelets φ?= [φ?,φ3]?and ψ?can be constructed so that
not only the order of vanishing moments increases but symmetry is
preserved also.
First, let us consider the o.n. scaling function vector and multi-
wavelet φ and ψ in [3] with support [0,2]. With
P(z) =
?1 + z
√7
8(1− z2) −1
?1 − z
1
8(1− z2)
2
?2
1
4(1− z2)
8(√7 − 2z +
−1
8(1 + 2√7z + z2)
−
√7z)
,
Q(z) =
−
2
?2
4(1− z2)
1
,
we construct a symmetric component φ3for a new o.n. scaling function
vector φ?with p.p.o. = 3 by setting a(z) in (2.3) to be (1 + z3)/8.
Under this condition, there are two solutions. The first solution for the
two-scale symbol P?of φ?is given by (2.3) with B in (2.14), while the
two-scale symbol Q?of ψ?is given by, similar to (2.11),
?A(z2)Q(z) −2ma(z)[ζ1,ζ2]?
Here, α1(z) and α2(z) satisfy |α1(z)|2+ |α2(z)|2= 15/16, A(z) and
[ζ1,ζ2] satisfy (2.16)–(2.17). Hence, by requiring symmetry on φ?and
the degree of α1(z) and α2(z) to be 1, we have
√15
8
?
Q?(z) =
O1×2
−22z3a(−z)?
?
.
(6.1)
α1(z) = (1 + z),
√15
64(1 − z)4,−
α2(z) =
√15
√15
8
(1 − z);
B(z) =
−
64(1 − z2)(1− 2√7z + z2)
?
;
ChuiLian.tex; 9/12/2003; 21:29; p.21
Page 22
22
C. Chui & J.-A. Lian
A(z) =
?−(1 − z)/2 −(1 + z)/2
(1+ z)/8 (1 − z)/8
?
,ζ1= 0, ζ2=
√15/4.
0 0.51 1.52
−0.5
0
0.5
1
1.5
0 0.51 1.52
−2
−1
0
1
2
0 0.51 1.52 2.53
−1.5
−1
−0.5
0
0.5
1
1.5
2
0 0.51 1.522.53
−2
−1
0
1
2
0 0.51 1.522.53
−2
−1.5
−1
−0.5
0
0.5
1
1.5
00.511.522.53
−2
−1
0
1
2
Figure 4. Graphs of φ?(top) and ψ?(bottom).
The three vectors in (2.1)–(2.2) are given by
a0,0= [1,0,0],a0,1=
√7 − 1
3
?
1,
1
√7 + 1,0
√15
90(4−
?
,
(6.2)
a0,2=
?
22 −√7
18
,,−
√7)
?
.
(6.3)
Moreover, φ ∈ C0.4963and φ?∈ C0.4084([7]). See Fig. 4 for graphs of
φ?and ψ?. Similarly, the second solution for P?and Q?has the same
formulation as (2.3) and (6.1) with the same ζ1and ζ2but with
√15
8
?
√15
64(1− z2)(3+ 2√7z + 3z2)
?−(1 − z)/2
The three vectors in (2.1)–(2.2) for polynomial preservation of this
solution for φ?and ψ?are also the same as those for the first solution,
namely, in (6.2)–(6.3). However, φ?∈ C0.4004([7]). The plots for the
second solution of φ?and ψ?are omitted here.
α1(z) = (1+ z),
√15
64(1 − z)2(3+ 2z + 3z2),
α2(z) = −
√15
8
(1− z);
B(z) =
−
−
?
;
A(z) =
(1 + z)/2
−(1− z)/8(1+ z)/8
?
.
ChuiLian.tex; 9/12/2003; 21:29; p.22
Page 23
Orthonormal Multi-Wavelets with Vanishing Moments
23
Next, let us consider the o.n. scaling function vector and multi-
wavelet φ and ψ in [3] with support= [0,3]. Precisely, with
P(z) =
?P1,1(z) P1,2(z)
P2,1(z) P2,2(z)
?
,Q(z) =
?Q1,1(z) Q1,2(z)
Q2,1(z) Q2,2(z)
?
,
where
P1,1(z) =
10 − 3√10
80
5√6 − 2√15
80
5√6 − 3√15
80
5 − 3√10
1040
5√6 − 2√15
80
(1 + z)(1 + (38+ 12√10)z + z2),
P1,2(z) = (1 − z)(1 + z)2,
P2,1(z) = (1 − z)(1 − 10(3+
(1+ z)(13− (10+ 6√10)z + 13z2),
√10)z + z2),
P2,2(z) =
Q1,1(z) =
Q1,2(z) = −10 − 3√10
Q2,1(z) = −5 − 3√10
5√6 − 3√15
(1 − z)2(1+ z),
(1− z)(1− (38 + 12√10)z + z2),
(1− z)(13+ (10 + 6√10)z + 13z2),
(1 + z)(1 + 10(3+√10)z + z2),
80
1040
Q2,2(z) =
80
it follows from Theorem 2.2 that a new pair of o.n. scaling function
vector φ?with p.p.o. = 4 and multi-wavelet ψ?can be obtained from φ
and ψ. The two-scale symbols P?and Q?of φ?and ψ?are given by
P?(z) =
P(z)
O2×1
1 + z3
16
3√7
8
0
Q2(z)
,
Q?(z) =
1 0
O1×2
1
8
Q(z)
1 + z3
2
1 − z3
2
0
−3√7
8
,
where Q2in P?is the second row of Q. It can be verified by applying
Lemma 6.1 that both φ3and ψ?
ψ?
Q?to be (1 + z3)/16. In addition, the four vectors in (2.1)–(2.2) for
2are antisymmetric while both ψ?
1and
3are symmetric. This is also the reason for choosing a(z) in P?and
ChuiLian.tex; 9/12/2003; 21:29; p.23
Page 24
24
C. Chui & J.-A. Lian
polynomial preservation are given by
a0,0= [1,0,0],a0,1=
?
3
2,
√15−√6
6
?
,0
?
,
a0,2=
?
?
17−√10
6
3(8−√10)
4
,
√15 −√6
2
√3(1280√5− 1373)
996
,0
,
a0,3=
,,
√7(9√10− 35)
581
?
.
Finally, φ ∈ C0.8797and φ?∈ C0.7907([7]). See Fig. 5 for graphs of φ
and ψ.
0 0.51 1.52 2.53
−0.5
0
0.5
1
1.5
0 0.511.52 2.53
−1.5
−1
−0.5
0
0.5
1
1.5
0 0.511.522.53
−1.5
−1
−0.5
0
0.5
1
1.5
0 0.511.522.53
−1.5
−1
−0.5
0
0.5
1
1.5
00.5 1 1.522.53
−1.5
−1
−0.5
0
0.5
1
1.5
00.511.52 2.53
−1.5
−1
−0.5
0
0.5
1
1.5
Figure 5. Graphs of φ?(top) and ψ?(bottom).
We end this section by pointing out that the GHM [5] scaling func-
tion vector φF= [φF
be expanded to give a new pair of o.n. scaling function vector φ?
[φF
does not exist a component φ3that is symmetric or antisymmetric. To
be more explicit, with
1,φF
2]?and multi-wavelet ψF= [ψF
1,ψF
2]?can also
F=
1,φF
2,φ3]?, with p.p.o. = 3, and multi-wavelet ψ?
F. However, there
PF(z) =
3
10(1+ z)
2√2
5
−
√2
40(1+ z)(1− 10z + z2) −1
√2
40(1 + z)(1 − 10z + z2) −1
1
20(1 − z)(1 − 8z + z2)
20(3− 10z + 3z2)
,
QF(z) =
−
20(1 + 3z)(3+ z)
3√2
20(1 − z)(1+ z)
,
ChuiLian.tex; 9/12/2003; 21:29; p.24
Page 25
Orthonormal Multi-Wavelets with Vanishing Moments
25
it follows from Theorem 2.2 that the two-scale symbols P?
φ?, with p.p.o. = 3, and multi-wavelet ψ?are given by
−
4
O1×2
Fand Q?
Fof
P?
F(z) =
PF(z)
√15
O2×1
1 + z
8
Q1(z)
,
Q?
F(z) =
1
4
0 1
0
QF(z)
1+ z
2
1− z
2
−
√15
4
0
,
respectively, where Q1in P?
three vectors satisfying (2.1)–(2.2) are given by
Fis the first row of QF. In addition, the
a0,0= [√2,1,0],a0,1=
?√2
2,1,0
?
,a0,2=
?
2√2
7
,13
14,−
√15
70
?
.
Although all the entries of P?
component φ3of φ?is neither symmetric nor antisymmetric. This can
also be verified by applying Lemma 6.1.
Fare reciprocal polynomials, the third
Acknowledgment
The authors are indebted to the referee for pointing out references [4],
[6] and helpful comments.
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