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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

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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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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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