Bessel multiwavelet sequences and dual multiframelets in

Adv Comput Math (2013) 38:491–529DOI 10.1007/s10444-011-9246-8

Bessel multiwavelet sequences and dualmultiframelets in Sobolev spaces

Youfa Li · Shouzhi Yang · Dehui Yuan

Received: 8 August 2010 / Accepted: 10 October 2011 /Published online: 27 October 2011© Springer Science+Business Media, LLC 2011

Abstract The dual 2Id-framelets in (Hs(Rd), H−s(Rd)), s > 0, were introducedby Han and Shen (Constr Approx 29(3):369–406, 2009). In this paper, wesystematically study the Bessel property of multiwavelet sequences in Sobolevspaces. The conditions for Bessel multiwavelet sequence in H−s(Rd) take greatdifference from those for Bessel wavelet sequence in this space. Precisely,the Bessel property of multiwavelet sequence in H−s(Rd) is not only relatedto multiwavelets themselves but also to the corresponding refinable functionvector. We construct a class of Bessel M-refinable function vectors withM being an isotropic dilation matrix, which have high Sobolev smoothness,and of which the mask symbols have high sum rules. Based on the con-structed Bessel refinable function vector, an explicit algorithm is given for dualM-multiframelets in (Hs(Rd), H−s(Rd)) with the multiframelets in H−s(Rd)

Communicated by Qiyu Sun.

Y. Li (B)College of Mathematics and Information Science, Guangxi University,Nanning 530004, People’s Republic of Chinae-mail: [email protected]

Y. LiDepartment of Mathematics, Faculty of Science and Technology,University of Macau, Taipa, Macao, People’s Republic of China

S. YangDepartment of Mathematics, Shantou University,Shantou 515063, People’s Republic of Chinae-mail: [email protected]

D. YuanDepartment of Mathematics, Hanshan Normal University,Chaozhou 521041, People’s Republic of Chinae-mail: [email protected]

492 Y. Li et al.

having high vanishing moments. On the other hand, based on the dual mul-tiframelets, an algorithm for dual M-multiframelets with symmetry is given. InSection 6, we give an example to illustrate the constructing procedures of dualmultiframelets.

Keywords Bessel property · Dual multiframelet · Sobolev space ·Isotropic dilation matrix · Symmetry

Mathematics Subject Classifications (2010) 42C15 · 94A12

1 Introduction

Framelets and multiframelets in L2(Rd) have been studied extensively in the

literature, see [1–9, 13, 14, 16, 18, 19, 24, 25, 29, 30, 33] and the referencestherein for a few examples. As a redundant system, framelets and multi-framelets have great design freedom and are of interest in signal denoising,image compression, and numerical algorithm [18]. As well known to all,framelets and multiframelets in L2(R

d) must have at least one vanishing mo-ment so that the frame series converges unconditionally (i.e., Bessel property).The Bessel property of framelets or multiframelets is of great importance andinterest for researchers [13, 23].

On the other hand, some applications imply that it is desirable to have awavelet (or multiwavelet) system in Sobolev space Hs(Rd) with its dual systemin H−s(Rd) [20]. Recently, Han and Shen in [20] first introduced pairs of dual2Id-framelets in (Hs(Rd), H−s(Rd)). The most significant difference betweenwavelet sequences in Hs(Rd), s > 0, and those in L2(R

d) is that the frameletsin Hs(Rd) do not necessarily have vanishing moment for the convergenceof the frame series. Therefore, it makes the construction of framelets inSobolev spaces much easier. Except some introductory discussion in [26, 27],to our best knowledge, dual multiframelets in (Hs(Rd), H−s(Rd)) have notbeen systematically addressed so far in the literature. In this paper, we shallstudy multiframelets in (Hs(Rd), H−s(Rd)). Nevertheless, it is not the trivialgeneralization of [20]. For example, conditions for the Bessel property ofmultiwavelet sequence in H−s(Rd) take great differences from those for theone of wavelet sequence in this space. Such conditions are not only relatedto multiwavelets themselves but also to the corresponding refinable functionvector, which will be seen in Theorem 2.2 or Comparison 2.1. Moreover, theconstruction of dual multiframelets in (Hs(Rd), H−s(Rd)) is based on a newcanonical factorization form for mask symbols of refinable function vectors[16, Section 5]. On the other hand, motivated by [24, 31, 32], we shall give analgorithm for dual multiframelets with symmetry in (Hs(Rd), H−s(Rd)).

Before proceeding further, let us introduce some notations and definitions.For a matrix A, we denote by A∗ its transpose conjugate. Let N0 denotethe set of all nonnegative integers. A d × d integer matrix M is regarded asa dilation matrix if limn→∞ M−n = 0. That is, all the eigenvalues of M are

Multiframelets in Sobolev spaces 493

strictly greater than one in modulus. Throughout this paper, we assume thatM is isotropic, i.e., M is similar to a diagonal matrix diag(λ1, λ2, · · · , λd) suchthat |λ1| = |λ2| = · · · = |λd| = | det M|1/d. Moreover, denote m = | det M|1/d forconvenient narration. In applications such as image processing and computeraided geometric design (CAGD), the following isotropic dilation matrices areusually used:

qId,

[1 11 −1

],

⎡⎣ 0 2 1

−1 −1 01 1 1

⎤⎦

with q (≥ 2) being an integer. The second matrix above is regarded as aquincunx dilation matrix. Readers are referred to [1, 6, 12, 15, 17] for someinvestigation and applications of wavelets or multiwavelets with isotropicdilation matrices. Denote by �MT the complete set of representatives ofdistinctive cosets of the quotient group [(MT)−1

Zd]/Z

d. Let x · ξ be the innerproduct in R

d with x = (x1, · · · , xd)T ∈ R

d and ξ = (ξ1, · · · , ξd)T ∈ R

d, i.e.,x · ξ = ∑d

j=1 x jξ j. The Euclidean norm || · ||2 on Rd is defined by ||x||2 = (x ·

x)1/2 = (∑d

j=1 x2j)

1/2. On the norm of MT x, x ∈ Rd, we have the following

result, which is necessary for the proof of Theorem 2.1.

Lemma 1.1 Let M be a d × d isotropic dilation matrix. Then there exists a norm|| · || on R

d such that ||MT x|| = m||x||. Moreover, there exist positive constants�1 and �2 such that �2||x|| ≤ ||x||2 ≤ �1||x||.

Proof It is not difficult to check that MT is isotropic and similar todiag(λ1, λ2, · · · , λd) as well with |λ1| = |λ2| = . . . = |λd| = m. Maybe some ofthe eigenvalues λ1, . . . , λd are complex. Without losing generality, we assumethat λ1 and λ2 are a pair of simple complex conjugate eigenvalues while λ j isreal, j = 3, . . . , d. Moreover, Mζ1 = λ1ζ1, Mζ2 = λ2ζ2. Since M is real, Mζ1 =Mζ1 = λ1ζ1 = λ2ζ1, which implies that ζ1 is the eigenvector corresponding toλ2. Recall that λ1 and λ2 are both simple. Then we can find a set of linearlyindependent eigenvectors ζ1, ζ2, . . . , ζd such that MTζ j = λ jζ j, j = 1, 2, . . . , d,

and ζ1 = ζ2. It is straightforward to see that (ζ1 + ζ2)/2, (ζ1 − ζ2)/2i, ζ3, . . . , ζd

are linearly independent as well and are all real, where i is the imaginary unit.Therefore, (ζ1 + ζ2)/2, (ζ1 − ζ2)/2i, ζ3, . . . , ζd constitute a system of basis in R

d,which implies that there exists a unique real sequence (r1, r2, . . . , rd) such that∀x ∈ R

d can be represented as x = r1(ζ1 + ζ2)/2 + r2(ζ1 − ζ2)/2i + ∑dj=3 r jζ j.

Equivalently, there exists an unique complex sequence (c1, c2, . . . , cd) such thatx = ∑d

j=1 c jζ j. Now, define a functional || · || on Rd by ||x|| = ∑d

j=1 |c j|. It is notdifficult to check that || · || is a norm. Moreover,

∣∣∣∣MT x∣∣∣∣ =

∣∣∣∣∣∣

∣∣∣∣∣∣d∑

j=1

λ jc jζ j

∣∣∣∣∣∣

∣∣∣∣∣∣ =d∑

j=1

∣∣λ jc j∣∣ = m

d∑j=1

∣∣c j∣∣ = m||x||.

494 Y. Li et al.

The existences of �1 and �2 are guaranteed by the equivalence of norms in afinite dimensional space.

For s ∈ R, the Sobolev space Hs(Rd) is defined by

Hs (R

d) ={

f :∫

Rd

∣∣ f (ξ)∣∣2 (1 + ||ξ ||22

)sdξ < ∞

}

with the Fourier transform of g ∈ L1(Rd) being defined to be

g(ξ) =∫

Rdg(x)e−ix·ξ dx,

which can be naturally extended to square integrable functions and tempereddistributions. It is straightforward to see that Hs1(Rd) ⊇ Hs2(Rd) iff s1 ≤ s2,and L2(R

d) = H0(Rd). The quantity ν2( f ) = sup{s : f ∈ Hs(Rd)} is regardedas the Sobolev exponent of f . Readers are referred to [12, 20] for methods ofcomputing ν2( f ).

The inner product 〈·, ·〉Hs(Rd) is defined to be

〈 f, g〉Hs(Rd) = 1

(2π)d

∫Rd

f (ξ )g(ξ)(1 + ||ξ ||22

)sdξ, ∀ f, g ∈ Hs (

Rd) .

Moreover, for any f ∈ Hs(Rd) and g ∈ H−s(Rd), the functional

〈 f, g〉 = 1

(2π)d

∫Rd

f (ξ )g(ξ)dξ

is well defined. For f, g : Rd �−→ C, define the so called bracket product

[f , g

]γ

(ξ) =∑k∈Zd

f (ξ + 2kπ) g (ξ + 2kπ)(

1 + ||ξ + 2kπ ||22)γ

, γ ∈ R.

When f is compactly supported, μ2( f ) = ν2( f ) with μ2( f ) = sup{γ : [ f , f ]γ ∈L∞(Rd)} [20, Proposition 2.6].

Let φ = (φ1, . . . , φr)T ∈ (Hs(Rd))r be an M-refinable function vector satis-

fying the refinement equation

φ = md∑k∈Zd

akφ (M · −k) (1.1)

with {ak}k∈Zd being a sequence of r × r matrices. If φ ∈ (L2(Rd))r is com-

pactly supported, we get by [13, Theorem 3.2] that [φ�, φ�]γ (ξ) ∈ L∞(Rd) iffγ ≤ ν2(φ�), � = 1, 2, . . . , r. We get from (1.1) that φ(MTξ) = a(ξ)φ(ξ) witha(ξ) = ∑

k∈Zd ake−ik·ξ being called the mask symbol of φ. Suppose {ψ� =(ψ�

1 , . . . , ψ�r )T}L

�=1 is a set of MRA function vectors derived from

ψ�

(MTξ

) = b �(ξ)φ(ξ)


for r × r matrix b �(ξ) of 2πZd-periodic trigonometric polynomials. Define a

multiwavelet system

Xs (φ; ψ1, . . . , ψL) := {φn;0,k : n = 1, . . . , r; k ∈ Z

d}

∪{ψ

�,sn; j,k : n = 1, . . . , r; k ∈ Z

d; j ∈ N0; � = 1, . . . , L}

,

where φn;0,k = φn(· − k) and ψ�,sn; j,k = m j(d/2−s)ψ�

n(M j · −k). If there exist twopositive constants C1 and C2 such that

C1|| f ||2Hs(Rd)≤

r∑n=1

∑k∈Zd

∣∣∣⟨ f, φn;0,k⟩Hs(Rd)

∣∣∣2 +r∑

n=1

L∑�=1

∑j∈N0

∑k∈Zd

∣∣∣∣⟨

f, ψ�,sn; j,k

⟩Hs(Rd)

∣∣∣∣2

≤ C2|| f ||2Hs(Rd), ∀ f ∈ Hs (

Rd) , (1.2)

then we say that Xs(φ; ψ1, . . . , ψL) is an M-multiframelet in Hs(Rd). If thesecond inequality of (1.2) holds, i.e.,

r∑n=1

∑k∈Zd

∣∣∣⟨ f, φn;0,k⟩Hs(Rd)

∣∣∣2 +r∑

n=1

L∑�=1

∑j∈N0

∑k∈Zd

∣∣∣∣⟨

f, ψ�,sn; j,k

⟩Hs(Rd)

∣∣∣∣2

≤ C2|| f ||2Hs(Rd),

we say that Xs(φ; ψ1, . . . , ψL) is a Bessel M-multiwavelet sequence in Hs(Rd).Furthermore, if there exists another M-multiframelet X−s(φ; ψ1, . . . , ψL),which is related to an M-refinable function vector φ ∈ (H−s(Rd))r and a setof MRA function vectors {ψ� = (ψ�

1 , . . . , ψ�r )T}L

�=1 satisfying

φ (MTξ

) = a(ξ )φ(ξ), ψ�

(MTξ

) = b �(ξ )φ(ξ),

such that for any f ∈ Hs(Rd) and g ∈ H−s(Rd),

〈 f, g〉 =r∑

n=1

∑k∈Zd

⟨φn;0,k, g

⟩ ⟨f, φn;0,k

⟩ +r∑

n=1

L∑�=1

∑j∈N0

∑k∈Zd

⟨ψ

�,sn; j,k, g

⟩ ⟨f, ψ�,−s

n; j,k

⟩,

(1.3)

then we say that Xs(φ; ψ1, . . . , ψL) and X−s(φ; ψ1, . . . , ψL) are a pair of dualM-multiframelets in (Hs(Rd), H−s(Rd)). It follows directly from (1.3) that

f =r∑

n=1

∑k∈Zd

⟨f, φn;0,k

⟩φn;0,k +

r∑n=1

L∑�=1

∑j∈N0

∑k∈Zd

⟨f, ψ�,−s

n; j,k

⟩ψ

�,sn; j,k

and

g =r∑

n=1

∑k∈Zd

⟨g, φn;0,k

⟩φn;0,k +

r∑n=1

L∑�=1

∑j∈N0

∑k∈Zd

⟨g, ψ

�,sn; j,k

⟩ψ

�,−sn; j,k.

Here we should point out that Han and Shen in [20, Proposition 2.1]provided an isomorphic map θs : Hs(R) �−→ H−s(R) defined by

θs( f )(ξ) = f (ξ)(1 + ||ξ ||22

)s, ∀ f ∈ Hs (

Rd) .

496 Y. Li et al.

Moreover, they proved that for any h ∈ Hs(Rd), 〈 f, h〉Hs(Rd) = 〈θs( f ), h〉, whichleads to that (1.2) is equivalent to

C1||g||2H−s(Rd)≤

r∑n=1

∑k∈Zd

∣∣⟨g, φn;0,k⟩∣∣2 +

r∑n=1

L∑�=1

∑j∈N0

∑k∈Zd

∣∣∣⟨g, ψ�,sn; j,k

⟩∣∣∣2

≤ C2||g||2H−s(Rd), ∀g ∈ H−s (

Rd) .

We shall see that for s > 0, the M-multiframelet Xs(φ; ψ1, . . . , ψL) in (1.4)does not necessarily imply that ψ� has vanishing moment, � = 1, . . . , L. This isthe greatest difference between multiframelets in Sobolev spaces and those inL2(R

d).The structure of this paper is as follows. In Section 2, we systematically study

the Bessel property of multiwavelet sequences in Sobolev spaces. In Section 3,a class of Bessel refinable function vectors are constructed. In Section 4.1, wegive Mixed Extension Principle (MEP) for dual M-multiframelets. Based onthe Bessel refinable function vectors constructed in Section 3, we constructdual M-multiframelets in (Hs(Rd), H−s(Rd)) in Section 4.2. The multiframeletsin H−s(Rd) have high vanishing moments. In Section 5, we give an algorithmfor dual M-multiframelets with symmetry in (Hs(Rd), H−s(Rd)).

2 A study of Bessel property of multiframelets in Sobolev spaces

Since it plays a crucial role in the convergence of a framelet series, Besselproperty is one of the most important properties of a wavelet (or multiwavelet)system ([13, Section 3], [23]). In this section, we shall study the Bessel propertyof multiframelets in Sobolev spaces.

Let us give some necessary notations first. For x = (x1, . . . , xd)T ∈ R

d andα = (α1, . . . , αd)

T ∈ Nd0 , denote xα = ∏d

j=1 xα j

j and |x| = ∑dj=1 |x j|. Let ∂ j be the

partial differential operator with respect to the j-th variable x j, j = 1, . . . , d.Define ∂α = ∂

α11 . . . ∂

αdd with ∂ = (∂1, . . . , ∂d)

T and α = (α1, . . . , αd)T ∈ N

d0 . The

equality f (ξ) = g(ξ) + O(||ξ ||β+12 ), ξ ∈ R

d and β ∈ N0, implies that ∂α f (0) −∂αg(0) = 0 for any α ∈ N

d0 with |α| ≤ β. We say that a function f : R

d �−→ C

has κ + 1 vanishing moments, κ ∈ N0, if ∂α f (0) = 0, ∀α ∈ Nd0 with |α| ≤ κ. The

following lemma is necessary for Theorem 2.1.

Lemma 2.1 For η > ζ > 0, def ine

Bζ,η(ξ) =∑j∈N0

m−2 jζ (1 + �21||ξ ||2)ζ (1 + m−2 j−2�2

2||ξ ||2)−η,

where �1, �2, and || · || are as in Lemma 1.1. Then there exists a positive constantC such that Bζ,η(ξ) < C, ∀ξ ∈ R

d.


Proof The proof is in Section Appendix.

Theorem 2.1 Let φ = (φ1, . . . , φr)T ∈ (Hη(Rd))r, η > 0, be a compactly sup-

ported M-ref inable function vector satisfying (1.1). For any r × r matrix b(ξ) =(b i, j(ξ))r

i, j=1 of 2πZd-periodic trigonometric polynomials, def ine a function

vector ψ = (ψ1, . . . , ψr)T via ψ(MTξ) = b(ξ)φ(ξ). Then Xζ (φ; ψ) is a Bessel

M-multiwavelet sequence in Hζ (Rd), ∀ζ ∈ (0, η).

Proof For any � ∈ {1, 2, . . . , r} and any g ∈ H−ζ (Rd), by Plancherel’s Theoremand Parseval’s identity, it is not difficult to verify that

1

(2π)d

∫[−π,π ]d

∣∣[g, φ�

]0 (ξ)

∣∣2 dξ = 1

(2π)2d

∑k∈Zd

∣∣∣∣∫

[−π,π ]d

[g, φ�

]0 (ξ)e−ik·ξ dξ

∣∣∣∣2

= 1

(2π)2d

∑k∈Zd

∣∣∣∣∣∣∫

[−π,π ]d

∑j∈Zd

g (ξ + 2 jπ) φ� (ξ + 2 jπ)e−ik·ξ dξ

∣∣∣∣∣∣2

= 1

(2π)2d

∑k∈Zd

∣∣∣∣∫

Rdg(ξ)φ�(ξ)e−ik·ξ dξ

∣∣∣∣2

= 1

(2π)2d× (2π)2d

∑k∈Zd

|〈g(x), φ�(x − k)〉|2

=∑k∈Zd

|〈g(x), φ�(x − k)〉|2, (2.1)

from which we get

r∑�=1

∑k∈Zd

|〈g(x), φ�(x − k)〉|2 = 1

(2π)d

r∑�=1

∫[−π,π ]d

∣∣[g, φ�

]0 (ξ)

∣∣2 dξ

≤ 1

(2π)d

r∑�=1

∫[−π,π ]d

[g, g

]−ζ

[φ�, φ�

]ζ

dξ

≤ 1

(2π)d

r∑�=1

∣∣∣∣∣∣[φ�, φ�

]ζ

∣∣∣∣∣∣L∞(Rd)

∫[−π,π ]d

[g, g

]−ζ

dξ

=r∑

�=1

∣∣∣∣∣∣[φ�, φ�

]ζ

∣∣∣∣∣∣

L∞(Rd)||g||2H−ζ (Rd)

< ∞. (2.2)

498 Y. Li et al.

The first inequality in (2.2) is obtained by |[g, φ�]0(ξ)|2 ≤ [g, g]−ζ [φ�, φ�]ζ ,which can be proved by Cauchy–Schwarz inequality.

Denote ψ�(x) = ∑rn=1 ψn

� (x) with

ψn�

(MTξ

) = b �,n(ξ)φn(ξ) and ψn,ζ

�; j,k(x) = m j(d/2−ζ )ψn�

(M jx − k

).

Then

∑k∈Zd

∣∣∣⟨g, ψn,ζ

�; j,k

⟩∣∣∣2 = m jdm−2 jζ∑k∈Zd

∣∣∣⟨g ((MT)− j

x)

, ψn� (x − k)

⟩∣∣∣2

= m jdm−2 jζ

(2π)d

∫[−π,π ]d

∣∣∣[g((

MT)− j ·)

, ψn�

]0(ξ)

∣∣∣2 dξ

= m jdm−2 jζ

(2π)d

∫[−π,π ]d

∣∣∣∣∣∣∑

γ∈�MT

b �,n

((MT)−1

ξ + 2πγ)

×[g((

MT) j+1 ·)

, φn

]0

((MT)−1

ξ + 2πγ)∣∣∣2 dξ

≤ m jdm−2 jζ md

(2π)d

∑γ∈�MT

∫[−π,π ]d

∣∣∣b �,n

((MT)−1

ξ + 2πγ)

×[g((

MT) j+1 ·)

, φn

]0

((MT)−1

ξ + 2πγ)∣∣∣2 dξ, (2.3)

where the first equality is obtained by integral transform while the secondequality by (2.1), and the last inequality is guaranteed by the fact that

∣∣∣∣∣∣∑

γ∈�MT

b �,n

((MT)−1

ξ + 2πγ) [

g((

MT) j+1 ·)

, φn

]0

((MT)−1

ξ + 2πγ)∣∣∣∣∣∣

2

≤ md∑

γ∈�MT

∣∣∣b �,n

((MT)−1

ξ + 2πγ)

×[g((

MT) j+1 ·)

, φn

]0

((MT)−1

ξ + 2πγ)∣∣∣2 .


Moreover, we estimate the last inequality of (2.3) as follows

m jdm−2 jζ md

(2π)d

∑γ∈�MT

∫[−π,π ]d

∣∣∣b �,n

((MT)−1

ξ + 2πγ)

×[g((

MT) j+1 ·)

, φn

]0

((MT)−1

ξ + 2πγ)∣∣∣2 dξ

≤ m jdm−2 jζ m2d

(2π)d

∫[−π,π ]d

∣∣∣b �,n(ξ)[g((

MT) j+1 ·)

, φn

]0(ξ)

∣∣∣2 dξ


(2π)d

∫[−π,π ]d

∣∣b �,n(ξ)∣∣2

×[g((

MT) j+1 ·)

, g((

MT) j+1 ·)]

−η(ξ)

[φn, φn

]η(ξ)dξ


(2π)d

∣∣∣∣∣∣[φn, φn]η(ξ)

∣∣∣∣∣∣L∞(Rd)

∫[−π,π ]d

∣∣b �,n(ξ)∣∣2

×[g((

MT) j+1 ·)

, g((

MT) j+1 ·)]

−η(ξ)dξ

= m jdm−2 jζ m2d

(2π)d

∣∣∣∣∣∣[φn, φn

]η(ξ)

∣∣∣∣∣∣

L∞(Rd)

∫Rd

∣∣b �,n(ξ)∣∣2

×∣∣∣g ((

MT) j+1ξ)∣∣∣2 (1 + ||ξ ||22

)−ηdξ

= m−2 jζ md

(2π)d

∣∣∣∣∣∣[φn, φn]η(ξ)

∣∣∣∣∣∣L∞(Rd)

∫Rd

∣∣∣b �,n

((MT)− j−1

ξ)∣∣∣2

× |g(ξ)|2(

1 +∣∣∣∣∣∣(MT)− j−1

ξ

∣∣∣∣∣∣22

)−η

dξ. (2.4)

By Lemma 1.1, we estimate the last equality of (2.4) as follows

m−2 jζ md

(2π)d

∣∣∣∣∣∣[φn, φn]η(ξ)

∣∣∣∣∣∣L∞(Rd)

×∫

Rd

∣∣∣b �,n

((MT)− j−1

ξ)∣∣∣2 |g(ξ)|2

(1 + ∣∣∣∣(MT)− j−1ξ

∣∣∣∣22

)−η

dξ

≤ m−2 jζ md

(2π)d

∣∣∣∣∣∣[φn, φn

]η(ξ)

∣∣∣∣∣∣

L∞(Rd)

×∫

Rd

∣∣∣b �,n

((MT)− j−1

ξ)∣∣∣2 |g(ξ)|2 (1 + �2

2m−2 j−2||ξ ||2)−ηdξ

500 Y. Li et al.

= md

(2π)d

∣∣∣∣∣∣[φn, φn]η(ξ)

∣∣∣∣∣∣L∞(Rd)

∫Rd

∣∣∣b �,n

((MT)− j−1

ξ)∣∣∣2 |g(ξ)|2 (1 + ||ξ ||22

)−ζ

× m−2 jζ (1 + ||ξ ||22)ζ (

1 + �22m−2 j−2||ξ ||2)−η

dξ

≤ md

(2π)d

∣∣∣∣∣∣[φn, φn]η(ξ)

∣∣∣∣∣∣L∞(Rd)

∫Rd

∣∣∣b �,n

((MT)− j−1

ξ)∣∣∣2 |g(ξ)|2 (1 + ||ξ ||22

)−ζ

× m−2 jζ (1 + �21||ξ ||2)ζ (1 + �2

2m−2 j−2||ξ ||2)−ηdξ. (2.5)

According to Lemma 2.1, there exists a positive constant C′ such that∑j∈N0

m−2 jζ (1 + �21||ξ ||2)ζ (1 + �2

2m−2 j−2||ξ ||2)−η< C′.

Then we get from (2.3), (2.4) and (2.5) that

∑j∈N0

∑k∈Zd

∣∣∣⟨g, ψn,ζ

�; j,k

⟩∣∣∣2 ≤ md

(2π)dC′ max

n

{∣∣∣∣∣∣[φn, φn

]η(ξ)

∣∣∣∣∣∣

L∞(Rd)

}

× ∣∣∣∣b(ξ)∣∣∣∣2


,

which leads to

∑j∈N0

∑k∈Zd

∣∣∣⟨g, ψn�; j,k

⟩∣∣∣2 ≤ r2mdC′

(2π)dmax

n

{∣∣∣∣∣∣[φn, φn]η(ξ)

∣∣∣∣∣∣L∞(Rd)

}

× ∣∣∣∣b(ξ)∣∣∣∣2


. (2.6)

Consequently, (2.2) and (2.6) lead to that

r∑�=1

∑k∈Zd

|〈g(x), φ�(x − k)〉|2 +r∑

n=1

∑j∈N0

∑k∈Zd

∣∣∣⟨g, ψn�; j,k

⟩∣∣∣2 < C||g||2H−ζ (Rd),

where

C=r∑

�=1

∣∣∣∣∣∣[φ�,φ�

]ζ

∣∣∣∣∣∣L∞(Rd)

+ r3mdC′

(2π)dmax

n

{∣∣∣∣∣∣[φn,φn]η(ξ)

∣∣∣∣∣∣L∞(Rd)

}∣∣∣∣b(ξ)∣∣∣∣2

L∞(Rd)<∞.

In other words, Xζ (φ; ψ) is a Bessel M-multiwavelet sequence in Hζ (Rd).

Remark 2.1 Theorem 2.1 implies that Xζ (φ; ψ) being a Bessel M-multiwavelet sequence in Hζ (Rd) with ζ > 0 does not necessarily mean thatψ has vanishing moment.

We pass to the Bessel M-multiwavelet sequences in H−s(Rd), s > 0. Thefollowing two lemmas are necessary.


Lemma 2.2 Let b(ξ) be a 2πZd-periodic trigonometric polynomial. Moreover,

there exists β ∈ N0 such that b(ξ) = O(||ξ ||β+12 ), ξ → 0. Denote

B−s,t(ξ) =∑j∈N0

∣∣b (m− j−1ξ

)∣∣2 m2 js (1 + �22||ξ ||2)−s (

1 + �21m−2( j+1)||ξ ||2)t

with 0 < t < s < β + 1, where �1, �2, and || · || are as in Lemma 1.1. Then thereexists a positive constant C such that B−s,t(ξ) ≤ C, ∀ξ ∈ R

d.

Proof The proof is in Section Appendix.

Lemma 2.3 For s > 0, let φ ∈ (H−s(Rd))r be an M-ref inable function vector

satisfying φ(MTξ) = a(ξ )φ(ξ). Moreover, assume that φ(0) = (α1, α2, . . . , αr)T.

Def ine ψ via ψ(MTξ) = b(ξ )φ(ξ) with b(ξ) being an r × r matrix of 2πZd-

trigonometric polynomials. Construct φ� via

φ�(ξ) = P(ξ )φ(ξ), (2.7)

where P(ξ) is an r × r strongly invertible matrix of 2πZd-periodic trigonometric

polynomials, i.e., P(ξ)−1 is also an r × r matrix of 2πZd-periodic trigonometric

polynomials. Then it is not dif f icult to check that

(i) there exists a strongly invertible P(ξ) such that φ�(0) = (1, 0, . . . , 0)T. Forexample, if α1 �= 0, one can select

P(ξ) =

⎡⎢⎢⎢⎢⎢⎣

1α1

0 . . . 0

− α2α1

1 . . . 0...

.... . .

...

− αrα1

0 . . . 1

⎤⎥⎥⎥⎥⎥⎦

;

(ii) φ� and ψ satisfy φ�(MTξ) = P(MTξ )a(ξ)P(ξ)−1 φ�(ξ) and ψ(MTξ) =b(ξ)P(ξ)−1 φ�(ξ), respectively;

(iii) for f ixed β ∈ N0, ∂α φ�(2kπ) = 0 if and only if ∂αφ(2kπ) = 0, ∀k ∈ Zd\{0}

and ∀α ∈ Nd0 with |α| ≤ β;

(iv) X−s(φ; ψ) is a Bessel M-multiwavelet sequence if and only if X−s(φ�; ψ)

is also a Bessel M-multiwavelet sequence.

Note 2.1 In Lemma 2.3 (i), the construction of φ� such that φ�(0) =(1, 0, . . . , 0)T can facilitate studying the Bessel property of X−s(φ; ψ). One cansee the details in Theorem 2.2.

Theorem 2.2 For t > 0, let φ = (φ1, . . . , φr)T ∈ (H−t(Rd))r be a compactly sup-

ported M-ref inable function vector satisfying φ(MTξ) = a(ξ )φ(ξ). Assume that

502 Y. Li et al.

there exists an r × r strongly invertible matrix P(ξ) of 2πZd-periodic trigono-

metric polynomials such that φ� = (φ�

1, . . . , φ�r )

T def ined by (2.7) satisf ies thefollowing items (i)–(iii)

(i) φ�(0) = (1, 0, . . . , 0)T;

(ii) [ φ�

�,φ

�

�]−t(ξ) ≤ C0||ξ ||2β+22 , where C0 is a positive constant and β ∈ N0 with

β + 1 > t, � = 2, . . . , r.

Moreover, def ine ψ = (ψ1, . . . , ψr)T by ψ(MTξ) = b(ξ )φ(ξ), where b(ξ) is an

r × r matrix of 2πZd-periodic trigonometric polynomials such that

(iii) ∂αψ(0) = 0, ∀α ∈ Nd0 with |α| ≤ β.

Then X−s(φ�; ψ) is a Bessel M-multiwavelet sequence with t < s < β + 1, orequivalently, X−s(φ; ψ) is a Bessel M-multiwavelet sequence.

Proof According to Lemma 2.3 (ii), we have

φ�(MTξ) = P(MTξ )a(ξ)P(ξ)−1 φ�(ξ)

and

ψ(MTξ) = b(ξ)P(ξ)−1 φ�(ξ).

Denote

a�(ξ) = P(MTξ )a(ξ)P(ξ)−1 and b �(ξ) = (b �

i, j(ξ))ri, j=1 = b(ξ)P(ξ)−1.

By Lemma 2.3 (iv), we can equivalently prove that X−s(φ�; ψ) is a Bessel M-multiwavelet sequence. Denote

ψ�(ξ) =r∑

n=1

ψn� (ξ) and ψ

n,−s�; j,k (x) = m j(d/2+s)ψn

� (M jx − k)

with

ψn� (MTξ) = b �

�,n(ξ)φ

�n(ξ),

for � = 1, . . . , r. For any g ∈ Hs(Rd), we obtain through the same process as(2.1), (2.2), and (2.4) that

r∑n=1

∑k∈Zd

∣∣⟨g(x), φ�n(x − k)

⟩∣∣2 ≤r∑

n=1

∣∣∣∣∣∣[ φ�n,

φ

�n

]−s

∣∣∣∣∣∣L∞(Rd)

||g||2Hs(Rd)< ∞ (2.8)


and

∑k∈Zd

∣∣∣⟨g, ψ−s�; j,k

⟩∣∣∣2 =∑k∈Zd

∣∣∣∣∣⟨

g,

r∑n=1

ψn,−s�; j,k

⟩∣∣∣∣∣2

≤ rm jdm2 jsm2d

(2π)d

r∑n=1

∫[−π,π ]d

∣∣∣ b ��,n(ξ)

∣∣∣2

×[g((

MT) j+1 ·)

, g((

MT) j+1 ·)]

t(ξ)

[ φ

�n,

φ

�n

]−t

(ξ)dξ

≤ rm jdm2 jsm2d

(2π)d

∣∣∣∣∣∣[ φ�

1,φ

�

1

]−t

∣∣∣∣∣∣L∞(Rd)

∫[−π,π ]d

∣∣∣ b ��,1(ξ)

∣∣∣2

×[g((

MT) j+1·)

, g((MT) j+1·)]

t(ξ)dξ+ rm jdm2 jsm2d

(2π)d

r∑n=2

∣∣∣∣∣∣ b ��,n(ξ)

∣∣∣∣∣∣2L∞(Rd)

×∫

[−π,π ]d

[g((

MT) j+1 ·)

, g(

ł(MT) j+1 ·

)]t(ξ)

[ φ

�n,

φ

�n

]−t

(ξ)dξ. (2.9)

Since ∂αψ(0) = 0 for any α ∈ Nd0 with |α| ≤ β, it follows directly from condition

(i) and (ii) that

b ��,1(ξ) = O

(||ξ ||β+1

2

), ξ → 0,

which together with condition (ii) leads to that there exists a positive constantC such that

∣∣∣ b ��,1(ξ)

∣∣∣ ≤ C||ξ ||β+12 , j = 1, . . . , r,

[φ�n,

φ�n

]−t

(ξ) ≤ C||ξ ||2β+22 , n = 2, . . . , r.

Therefore, we estimate the first part of the last inequality of (2.9) as follows.

rm jdm2 jsm2d

(2π)d

∣∣∣∣∣∣[ φ�

1,φ

�

1

]−t

∣∣∣∣∣∣L∞(Rd)

∫[−π,π ]d

∣∣∣ b ��,1(ξ)

∣∣∣2

×[g((

MT) j+1 ·)

, g((

MT) j+1 ·)]

t(ξ)dξ

= rm jdm2 jsm2d

(2π)d

∣∣∣∣∣∣[ φ�

1,φ

�

1

]−t

∣∣∣∣∣∣L∞(Rd)

∫Rd

∣∣∣ b ��,1(ξ)

∣∣∣2

×∣∣∣g ((

MT) j+1ξ)∣∣∣2 (1 + ||ξ ||22

)tdξ

504 Y. Li et al.

= rm2 jsmd

(2π)d

∣∣∣∣∣∣[ φ�

1,φ

�

1

]−t

∣∣∣∣∣∣L∞(Rd)

×∫

Rd

∣∣∣ b ��,1

((MT)− j−1

ξ)∣∣∣2 |g(ξ)|2

(1 +

∣∣∣∣∣∣(MT)− j−1ξ

∣∣∣∣∣∣22

)t

dξ

≤ rC2m2 jsmd

(2π)d

∣∣∣∣∣∣[ φ�

1,φ

�

1

]−t

∣∣∣∣∣∣

L∞(Rd)

×∫

Rd

∣∣∣∣∣∣(MT)− j−1ξ

∣∣∣∣∣∣2β+2

2|g(ξ)|2 (1 + �2

1m−2( j+1)||ξ ||2)tdξ

≤ rC2md

(2π)d

∣∣∣∣∣∣[ φ�

1,φ

�

1

]−t

∣∣∣∣∣∣

L∞(Rd)

∫Rd

|g(ξ)|2 (1 + ||ξ ||22)s

× �2β+21

∣∣∣∣m− j−1ξ∣∣∣∣2β+2

m2 js (1 + ||ξ ||22)−s (

1 + �21m−2( j+1)||ξ ||2)t

dξ

≤ rC2md

(2π)d

∣∣∣∣∣∣[ φ�

1,φ

�

1

]−t

∣∣∣∣∣∣L∞(Rd)

∫Rd

|g(ξ)|2 (1 + ||ξ ||22)s

�2β+21

∣∣∣∣m− j−1ξ∣∣∣∣2β+2

× m2 js (1 + �22||ξ ||2)−s (

1 + �21m−2( j+1)||ξ ||2)t

dξ, (2.10)

where the last three inequalities are guaranteed by Lemma 1.1. According toLemma 2.2, there exists a positive constant C such that

∑j∈N0

∣∣∣∣m− j−1ξ∣∣∣∣2β+2

m2 js (1 + �22||ξ ||2)−s (

1 + �21m−2( j+1)||ξ ||2)t

< C, (2.11)

which together with (2.10) leads to that

rm jdm2 jsm2d

(2π)d

∣∣∣∣∣∣[ φ�

1,φ

�

1

]−t

∣∣∣∣∣∣L∞(Rd)

∑j∈N0

∫[−π,π ]d

∣∣∣ b ��,1(ξ)

∣∣∣2

×[g((

MT) j+1 ·)

, g((

MT) j+1 ·)]

t(ξ)dξ

≤ �2β+21 rCC2md

∣∣∣∣∣∣[ φ�

1,φ

�

1

]−t

∣∣∣∣∣∣L∞(Rd)

||g||2Hs(Rd). (2.12)

We get via the same way as (2.10) that

rm jdm2 jsm2d

(2π)d

r∑n=2

∣∣∣∣∣∣ b ��,n(ξ)

∣∣∣∣∣∣2L∞(Rd)

×∫

[−π,π ]d

[g((

MT) j+1 ·)

, g((

MT) j+1 ·)]

t(ξ)

[ φ

�n,

φ

�n

]−t

(ξ)dξ


≤ rm jdm2 jsm2d

(2π)d

r∑n=2

∣∣∣∣b �,n(ξ)∣∣∣∣2

L∞(Rd)

×∫

Rd

∣∣∣g ((MT) j+1

ξ)∣∣∣2 (1 + ||ξ ||22

)t[ φ

�n,

φ

�n

]−t

(ξ)dξ

≤ rm2 jsmd

(2π)d

r∑n=2

∣∣∣∣∣∣ b ��,n(ξ)

∣∣∣∣∣∣2L∞(Rd)

×∫

Rd|g(ξ)|2

(1 +

∣∣∣∣∣∣(MT)− j−1ξ

∣∣∣∣∣∣22

)t [ φ

�n,

φ

�n

]−t

((MT)− j−1

ξ)

dξ

≤ rCm2 jsmd

(2π)d

r∑n=2

∣∣∣∣∣∣ b ��,n(ξ)

∣∣∣∣∣∣2L∞(Rd)

×∫

Rd|g(ξ)|2

(1 +

∣∣∣∣∣∣(MT)− j−1ξ

∣∣∣∣∣∣22

)t ∣∣∣∣∣∣(MT)− j−1ξ

∣∣∣∣∣∣2β+2

2dξ

≤ rCmd

(2π)d

r∑n=2

∣∣∣∣∣∣ b ��,n(ξ)

∣∣∣∣∣∣2L∞(Rd)

∫Rd

|g(ξ)|2 (1 + ||ξ ||22)s

�2β+21

× ∣∣∣∣m− j−1ξ∣∣∣∣2β+2

m2 js (1 + �22||ξ ||2)−s (

1 + �21||m− j−1ξ ||2)t

dξ. (2.13)

It follows from (2.11) and (2.13) that

rm jdm2 jsm2d

(2π)d

r∑n=2

∑j∈N0

∣∣∣∣∣∣ b ��,n(ξ)

∣∣∣∣∣∣2L∞(Rd)

×∫

[−π,π ]d

[g((

MT) j+1 ·)

, g((

MT) j+1 ·)]

t(ξ)

[ φ

�n,

φ

�n

]−t

(ξ)dξ

≤ �2β+21 rCC0md

r∑n=2

∣∣∣∣∣∣ b �

�,n(ξ)

∣∣∣∣∣∣2

L∞(Rd)||g||2Hs(Rd)

. (2.14)

Now, it is not difficult to see from (2.8), (2.12) and (2.14) that X−s(φ�; ψ) is aBessel M-multiwavelet sequence.

Comparison 2.1 For the case of r = 1, to make sure that X−s(φ�; ψ) referredin Theorem 2.2 is a Bessel M-wavelet sequence, we just require the followingcondition ([20, Theorem 2.3]):

∂αψ(0) = 0, ∀α ∈ Nd0 with |α| ≤ β and t < s < β + 1.

But for r > 1, we can see from (2.10) that the Bessel property of X−s(φ�; ψ) cannot be guaranteed by the conditions above. Or put it in another way, the Besselproperty of multiwavelet sequence is not only related to multiwavelets themselvesbut also to the ref inable function vector φ. In next section, we shall construct aclass of such appropriate ref inable function vectors in H−s(Rd).

506 Y. Li et al.

3 A class of Bessel refinable function vectors in H−s(Rd)

In this section, we shall construct a class of M-refinable function vectors inH−s(Rd), s > 0, such that Theorem 2.2 (i) and (ii) hold simultaneously. We callthem Bessel refinable function vectors. The following lemma gives a sufficientcondition for Theorem 2.2(ii).

Lemma 3.1 Let F(x) ∈ Ht(Rd) be compactly supported, t > 0. For f ixed β ∈N0, if ∂α F(2kπ) = 0 for any k ∈ Z

d and any α ∈ Nd0 with |α| ≤ β, then there

exists a positive constant C such that

[F, F]−ζ (ξ) ≤ C||ξ ||2β+22 , ∀ζ ≥ 0, a.e. ξ ∈ [−π, π ]d.

Proof Clearly, F(x) ∈ L2(Rd). Since [F, F]−ζ (ξ) ≤ [F, F]0(ξ), ζ ≥ 0, it suffices

to prove [F, F]0(ξ) ≤ C||ξ ||2β+22 for some positive constant C. Since F(x) ∈

L2(Rd) is compactly supported, by [12, Theorem 3.6], there exists a compactly

supported function h(x) ∈ L2(Rd) and a 2πZ

d-periodic trigonometric polyno-mial L (ξ) such that

F(ξ) = L (ξ )h(ξ)

and

∂αL (0) = 0, ∀α ∈ Nd0 with |α| ≤ β. (3.1)

Note that (3.1) implies there exists a positive constant g such that

L (ξ) ≤ g||ξ ||β+12 .

Then [F, F

]0 (ξ) =

∑k∈Zd

F (ξ + 2kπ) F (ξ + 2kπ)

= L (ξ)L (ξ)∑k∈Zd

h (ξ + 2kπ) h (ξ + 2kπ)

≤ C||ξ ||2β+22 , a.e. ξ ∈ [−π, π ]d,

where C = g2||[h, h]0(ξ)||L∞[−π,π ]d < ∞.

For an r × r matrix a(ξ) of 2πZd-periodic trigonometric polynomials, we say

that it has κ + 1 sum rules, κ ∈ N0, with respect to dilation matrix M if thereexists a 1 × r vector Y(ξ) = (Y1(ξ), . . . , Yr(ξ)) of 2πZ

d-periodic trigonometricpolynomials such that

Y(0) �= 0, Y(MTξ

)a (ξ + 2πγ ) = δγ Y(ξ) + O

(||ξ ||κ+12

), ∀γ ∈ �MT . (3.2)

From now on, denote the greatest sum rule a(ξ) has by sum(a). Readers arereferred to [12, 16] for some methods of computing sum rule. We say that a(0)

satisfies Condition E if and only if 1 is its simple eigenvalue while the othereigenvalues are strictly smaller than 1 in modulus.


Theorem 3.1 Let φo ∈ L2(Rd) be a compactly supported M-ref inable function,

i.e., φo(MTξ) = h(ξ) φo(ξ) for some 2πZd-periodic trigonometric polynomialh(ξ). Construct an r × r matrix a(ξ) via a(ξ) = h(ξ)H(ξ), where H(ξ) is an

r × r matrix of 2πZd-periodic trigonometric polynomials with nonnegative

coef f icients and H(0) satisf ies Condition E with H(0)(1, . . . , 1)T = (1, . . . , 1)T.Def ine φ = (φ1, . . . , φr)

T via

φ (MTξ

) = a(ξ )φ(ξ). (3.3)

Then

(i) ν2(φ) ≥ ν2(φo) > 0, sum(a) ≥ sum(h);

(ii) ∂αφ(2kπ) = 0, ∀k ∈ Zd \ {0}, ∀α ∈ N

d0 with |α| ≤ sum(h) − 1.

Proof Recall that h(0) = 1. Then it is straightforward to see that a(0) aswell as H(0) satisfies Condition E, and a(0)(1, . . . , 1)T = (1, . . . , 1)T . By [10,Section 1] or [22], φ is a compactly supported distribution vector defined by itsFourier transform

φ(ξ) =∞∏

�=1

(a((

MT)−�ξ)

(1, . . . , 1)T)

= limn→∞

(n∏

�=1

a ((MT)−�

ξ)

(1, . . . , 1)T

)

= limn→∞

(n∏

�=1

h ((MT)−�

ξ) H ((

MT)−�ξ)

(1, . . . , 1)T

)(3.4)

with limn→∞∏n

�=1h((MT)−�ξ) and limn→∞

(∏n�=1

H((MT)−�ξ)(1, . . . , 1)T)

converging uniformly on compact sets.Denote H(ξ) = (Hi, j(ξ))r

i, j=1,

Hn(ξ) =

(Hn

i, j(ξ))r

i, j=1=

n∏�=1

H ((MT)−�

ξ)

,

(Vn

1(ξ), . . . , Vn

r (ξ))T = Hn

(ξ) (1, . . . , 1)T

with n ∈ N, and

(V∞1 (ξ), . . . , V∞

r (ξ))T = lim

n→∞

((Vn

1(ξ), . . . , Vn

r (ξ))T

)

= limn→∞

(n∏

�=1

H ((MT)−�

ξ)

(1, . . . , 1)T

).

508 Y. Li et al.

We are to prove by induction that φ ∈ L2(Rd) with ν2(φ) ≥ ν2(φ

o) > 0. In fact,since the coefficients of H j,�(ξ) are all nonnegative and H(0)(1, . . . , 1)T =(1, . . . , 1)T ,∣∣∣∣V1

j(ξ)

∣∣∣∣ ≤r∑

�=1

∣∣∣H j,�

((MT)−1

ξ)∣∣∣ ≤

r∑�=1

H j,�(0) = 1, j = 1, . . . , r, ∀ξ ∈ Rd.

Assume that |Vn

j (ξ)| ≤ 1 for n ≥ 2 and j = 1, . . . , r, then∣∣∣∣Vn+1

j (ξ)

∣∣∣∣ =r∑

�=1

∣∣∣H j,�

((MT)−(n+1)

ξ) Vn

� (ξ)

∣∣∣

≤r∑

�=1

∣∣∣H j,�

((MT)−(n+1)

ξ)∣∣∣ ≤

r∑�=1

H j,�(0) = 1. (3.5)

That is, |Vn

j (ξ)| ≤ 1, ∀n ∈ N and ∀ξ ∈ Rd. On the other hand, (3.4) leads to that

φ(ξ) =⎛⎝ ∞∏

j=1

h ((MT)− j

ξ)⎞⎠ ×

⎛⎝ ∞∏

j=1

H ((MT)− j

ξ)

× (1, . . . , 1)T

⎞⎠

=⎛⎝ ∞∏

j=1

h ((MT)− j

ξ)⎞⎠ × lim

n→∞

(Vn

1(ξ), . . . , Vn

r (ξ))T

. (3.6)

Therefore, we derive from (3.5) and (3.6) that∣∣∣φ�(ξ)

∣∣∣ ≤∣∣∣ φo(ξ)

∣∣∣ , ∀ξ ∈ Rd, � = 1, . . . , r,

which leads to ν2(φ�) ≥ ν2(φo) and consequently ν2(φ) ≥ ν2(φ

o), where ν2(φ) =min{ν2(φ�) : � = 1, . . . , r}. Again by [13, Theorem 3.2], we get ν2(φ

o) > 0. Thus,ν2(φ) ≥ ν2(φ

o) > 0.Next, we address the sum rules of H(ξ). By the definition of sum(h), we

have

h (ξ + 2πγ ) = δγ + O

(||ξ ||sum

(h)

2

), ξ → 0, ∀γ ∈ �MT .

Then we have

(σ1, . . . , σr)a (ξ + 2πγ ) = (σ1, . . . , σr)h (ξ + 2πγ ) H (ξ + 2πγ )

= δγ (σ1, . . . , σr) + O

(||ξ ||sum

(h)

2

)

with (σ1, . . . , σr) being the left eigenvector of H(0) corresponding to theeigenvalue 1, which implies that sum(a) ≥ sum(h).

Now we prove item (ii). For any k ∈ Zd\{0}, it is not difficult to see that

there exists j0 ∈ N such that (MT)− j0 k ∈ {γ1, . . . , γmd−1}, where {γ0, γ1, . . . ,


γmd−1} = �MT with γ0 = 0. Assume (MT)− j0 k = γ� + k0 with k0 ∈ Zd and � ∈

{1, . . . , md − 1}. That is, k = (MT) j0(γ� + k0). On the other hand, we deducefrom (3.3) that

φ(ξ) = a ((MT)−1

ξ)φ ((

MT)−1ξ)

=⎛⎝ 2 j0−1∏

j= j0+1

h ((MT) j0− j

ξ) H ((

MT) j0− jξ)⎞⎠

× h ((MT)− j0

ξ) H ((

MT)− j0ξ)φ ((

MT)− j0ξ)

. (3.7)

By the sum rule and period ofh(ξ), ∂αh(2πγ� + 2nπ) = 0, where ∀n ∈ Zd, ∀α ∈

Nd0 and |α| ≤ sum(h) − 1. Then

∂α[h((

MT)− j0 ·)]

(2kπ) = ∂α[h((

MT)− j0 ·)] (

2π(MT) j0

(γ� + k0))

= 0.

That is, ∂α [h((MT)− j0 ·)](2kπ) = 0. Now by (3.7), we get ∂αφ(2kπ) = 0 for allk ∈ Z

d \ {0} and all α ∈ Nd0 with |α| ≤ sum(h) − 1.

Lemma 3.2 [16, Theorem 5.1] Let a compactly supported M-ref inable functionvector φ = (φ1, . . . , φr)

T ∈ (L2(Rd))r satisfy

φ(MTξ) = a(ξ)φ(ξ).

Assume a(ξ) has κ + 1 sum rules, κ ∈ N0. Then there exists a strongly invertibler × r matrix P(ξ) of 2πZ

d-periodic trigonometric polynomials such that themask symbol

a�(ξ) = P(MTξ

)a(ξ)P(ξ)−1 =

[a�

11(ξ) a�12(ξ)

a�21(ξ) a�

22(ξ)

](3.8)

takes the form⎧⎨⎩

a�11 (ξ + 2πγ ) = δγ + O

(||ξ ||κ+12

),

a�12 (ξ + 2πγ ) = O

(||ξ ||κ+12

),

a�21(ξ) = O

(||ξ ||κ+12

),

∀γ ∈ �MT , ξ → 0, (3.9)

where a�11(ξ), a�

12(ξ) and a�21(ξ) are some 1 × 1, 1 × (r − 1) and (r − 1) ×

1 matrices of 2πZd-periodic trigonometric polynomials, respectively. Conse-

quently, the M-ref inable function vector φ� = (φ�

1, . . . , φ�r )

T def ined by φ�(ξ) =P(ξ)φ(ξ) satisf ies

φ�

1(ξ) = 1 + O(||ξ ||κ+1

2

), φ

�

�(ξ) = O(||ξ ||κ+1

2

), ξ → 0, � = 2, . . . , r. (3.10)

Theorem 3.2 Let φ = (φ1, . . . , φr)T be as in Theorem 3.1. Then there exists an

r × r strongly invertible matrix P(ξ) such that φ�, which is def ined by (2.7), is aBessel M-ref inable function vector in H−s(Rd), ∀s ∈ R

+. Precisely,

510 Y. Li et al.

(i) φ�(0) = (1, 0, . . . , 0)T;

(ii) [ φ�

�,φ

�

�]−t(ξ) ≤ C||ξ ||2sum(a)2 , where t > 0, a(ξ) is the mask symbol of φ,

and C is some positive constant, � = 2, . . . , r.

Proof According to Lemma 3.2, there exists an r × r strongly invertible matrixP(ξ) such that φ� satisfies the property of (3.10). That is,

φ�

1(ξ) = 1 + O(||ξ ||sum(a)

2

), φ

�

�(ξ) = O(||ξ ||sum(a)

2

), ξ → 0, � = 2, . . . , r.

(3.11)

Now, (3.11) together with Theorem 3.1(ii) and Lemma 2.3 (iii), leads to

∂α φ�

�(2kπ) = 0 for any k ∈ Zd and any α ∈ N

d0 with |α| ≤ sum(a) − 1, � =

2, . . . , r. By Lemma 3.1, there exists a positive constant C such that[φ

�

�,φ

�

�

]−t

(ξ) ≤ C||ξ ||2sum(a)2 ,

where t > 0, � = 2, . . . , r, and C is some positive constant. Now, it is easy to seethat φ� is a Bessel M-refinable function vector in H−s(Rd), ∀s ∈ R

+.

4 Construction of real-valued dual M-multiframelets in Sobolev spaces

In this section, we shall construct a pair of dual M-multiframelets in (Hs(Rd),

H−s(Rd)) with the multiframelets in H−s(Rd) having high vanishing moments.Moreover, the dual multiframelets are all real-valued. We first give the mixedextension principle (MEP) for dual multiframelets in Sobolev spaces.

4.1 Mixed extension principle for dual multiframelets in Sobolev spaces

Theorem 4.1 For f ixed t > 0, let compactly supported M-ref inable functionvectors φ = (φ1, . . . , φr)

T ∈ (Ht(Rd))r and φ = (φ1, . . . , φr)T ∈ (H−t(Rd))r are

given by

φ(MTξ

) = a(ξ)φ(ξ), φ (MTξ

) = a(ξ )φ(ξ),

respectively. Assume that∑r

�=1[φ�, φ�]s∗(ξ) ∈ L∞(Rd) with s∗ > t. Select astrongly invertible r × r matrix P(ξ) of 2πZ

d-periodic trigonometric polyno-mials and construct φ� = (φ

�

1, . . . , φ�r )

T via φ�(ξ) = P(ξ)φ(ξ) such that φ�(0) =(1, 0, . . . , 0)T. Assume there exists another strongly invertible r × r matrixP(ξ) of 2πZ

d-periodic trigonometric polynomials such that φ� = (φ�

1, . . . , φ�r )

T

def ined by (2.7) satisf ies Theorem 2.2 (i) and (ii). Furthermore, suppose

that there exist r × r matrices b 1(ξ), . . . , b L(ξ) and b 1(ξ), . . . ,b L(ξ) of 2πZ

d-periodic trigonometric polynomials such that

∂α[ b � φ�

](0) = 0, ∀α ∈ N

d0 with |α| ≤ β, β ∈ N0, � = 1, . . . , L, (4.1)


and

⎡⎢⎢⎢⎣

b 1 (ξ + 2πγ0)∗ b 2 (ξ + 2πγ0)

∗ . . . b L (ξ + 2πγ0)∗

b 1 (ξ + 2πγ1)∗ b 2 (ξ + 2πγ1)

∗ . . . b L (ξ + 2πγ1)∗

......

. . ....

b 1(ξ + 2πγmd−1

)∗b 2

(ξ + 2πγmd−1

)∗. . . b L

(ξ + 2πγmd−1

)∗

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

b 1(ξ)b 2(ξ)...b L(ξ)

⎤⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎣

Ir − a� (ξ + 2πγ0)∗ a�(ξ)

−a� (ξ + 2πγ1)∗ a�(ξ)

...

−a�(ξ + 2πγmd−1

)∗ a�(ξ)

⎤⎥⎥⎥⎦ (4.2)

with a�(ξ)= P(MTξ )a(ξ)P(ξ)−1, a�(ξ)= P(MTξ )a(ξ)P(ξ)−1,{γ0, γ1,. . . ,γmd−1}=�MT , andγ0 =0.Def ine{ψ� =(ψ�

1 , . . . , ψ�r )T}L

�=1and{ψ� =(ψ�1 , . . . , ψ�

r )T}L�=1 via

ψ�

(MTξ

) = b �(ξ)φ�(ξ), ψ�

(MTξ

) = b �(ξ)φ�(ξ), � = 1, . . . , L. (4.3)

Then Xs(φ�; ψ1, . . . , ψL) and X−s(φ�; ψ1, . . . , ψL) are a pair of dual M-multiframelets in (Hs(Rd), H−s(Rd)) with t < s < min{s∗, β + 1}.

Proof From Lemma 2.3, we know

φ�(MTξ

) = a�(ξ)φ�(ξ) and φ�(MTξ

) = a�(ξ)φ�(ξ).

By Theorem 2.1, Xs(φ�; ψ1, . . . , ψL) is a Bessel M-multiwavelet sequence inHs(Rd). It is straightforward to derive from (4.1) and (4.3) that ∂αψ�(0) =0, � = 1, . . . , L, ∀α ∈ N

d0 with |α| ≤ β. Moreover, since there exists a strongly

invertible matrix P(ξ) such that φ� satisfies Theorem 2.2 (i) and (ii),X−s(φ�; ψ1, . . . , ψL) is a Bessel M-multiwavelet sequence in H−s(Rd).

Let B(Rd) = { f : f (ξ) is compactly supported, ξ ∈ Rd}. It is not difficult

to check that B(Rd) ⊆ Hν(Rd) for any ν ∈ R. By a similar argument as [18,Theorem 3.1], we get from (4.2) that

r∑n=1

L∑�=1

∑k∈Zd

⟨f, ψ�

n; j,k

⟩ ⟨ψ�

n; j,k, g⟩=

r∑n=1

∑k∈Zd

⟨f, φ�

n; j+1,k

⟩ ⟨φ

�

n; j+1,k, g⟩

−r∑

n=1

∑k∈Zd

⟨f, φ�

n; j,k

⟩ ⟨φ

�

n; j,k, g⟩, ∀ f, g ∈ B

(R

d) ,

(4.4)

where ψ�n; j,k = m jd/2ψ�

n

(M j · −k

)and φ

�

n; j,k = m jd/2φ�n(M j · −k

). Since

⟨f, ψ�,−s

n; j,k

⟩ ⟨ψ

�,sn; j,k, g

⟩=

⟨f, ψ�

n; j,k

⟩ ⟨ψ�

n; j,k, g⟩,

512 Y. Li et al.

(4.4) is equivalent to

r∑n=1

L∑�=1

∑k∈Zd

⟨f, ψ�,−s

n; j,k

⟩ ⟨ψ

�,sn; j,k, g

⟩=

r∑n=1

∑k∈Zd

⟨f, φ�

n; j+1,k

⟩ ⟨φ

�

n; j+1,k, g⟩

−r∑

n=1

∑k∈Zd

⟨f, φ�

n; j,k

⟩ ⟨φ

�

n; j,k, g⟩,

which leads to that

N∑j=0

r∑n=1

L∑�=1

∑k∈Zd

⟨f, ψ�,−s

n; j,k

⟩ ⟨ψ

�,sn; j,k, g

⟩=

r∑n=1

∑k∈Zd

⟨f, φ�

n;N+1,k

⟩ ⟨φ

�

n;N+1,k, g⟩

−r∑

n=1

∑k∈Zd

⟨f, φ�

n;0,k

⟩ ⟨φ

�

n;0,k, g⟩. (4.5)

Through the same way as [20, Theorem 2.4], one can prove that

limN→∞

∑k∈Zd

⟨f, φ�

n;N+1,k

⟩ ⟨φ

�

n;N+1,k, g⟩

= 1

(2π)d

∫Rd

f (ξ )g(ξ) limN→∞

φ�n

((MT)−N

ξ)

φ�n

((MT

)−Nξ)

dξ (4.6)

with any n ∈ {1, 2, . . . , r}. Then (4.5) and (4.6) together with φ�(0)T φ�(0) = 1lead to that

〈 f, g〉 =r∑

n=1

∑k∈Zd

⟨f, φ�

n;0,k

⟩ ⟨φ

�

n;0,k, g⟩

+∑j∈N0

r∑n=1

L∑�=1

∑k∈Zd

⟨f, ψ�,−s

n; j,k

⟩ ⟨ψ

�,sn; j,k, g

⟩, ∀ f, g ∈ B

(R

d) . (4.7)

Since B(Rd) is dense in B(Rd) ⊆ Hν(Rd) for any ν ∈ R, (4.7) holds for any f ∈Hs(Rd) and g ∈ H−s(Rd). Moreover, as [20, Theorem 2.4], we can prove thatXs(φ�; ψ1, . . . , ψL) and X−s(φ�; ψ1, . . . , ψL) are M-multiframelets in Hs(Rd)

and H−s(Rd), respectively. Consequently, Xs(φ�; ψ1, . . . , ψL) and X−s(φ�; ψ1,

. . . , ψL) are a pair of dual M-multiframelets in (Hs(Rd), H−s(Rd)).

4.2 Construction of real-valued dual M-multiframelets in (Hs(Rd), H−s(Rd))

We can see from Theorems 2.1 and 4.1 that no restriction is imposed on thevanishing moment of ψ�, � = 1, . . . , L. Naturally, an unsolved problem relatedto Theorem 4.1 is how to construct the mask symbols b 1(ξ), . . . , b L(ξ) andb 1(ξ), . . . ,

b L(ξ) such that ψ�, � = 1, . . . , L, has high vanishing moments. Thefollowing theorem will answer this question.


Theorem 4.2 Let φ = (φ1, . . . , φr)T be a real-valued M-ref inable function

vector given by (1.1). Its mask symbol a(ξ) has κ + 1 sum rules, and∑r�=1[φ�, φ�]s∗(ξ) ∈ L∞(Rd), κ ∈ N0, s∗ > 0. Suppose that φ is the real-valued

M-ref inable function vector constructed in Theorem 3.2. Its mask symbol a(ξ)

has κ + 1 sum rules, κ ∈ N0. Then there exist two strongly invertible r × rmatrices P(ξ), P(ξ), and r × r matrices b 1(ξ), . . . , b md

(ξ), b 1(ξ), . . . ,b md

(ξ)

of 2πZd-periodic trigonometric polynomials with real-valued coef f icients such

that Xs(φ�; ψ�

1, . . . , ψ�

md) and X−s(φ�; ψ�

1, . . . , ψ�

md) are a pair of dual M-

multiframelets in (Hs(Rd), H−s(Rd)), where φ�, φ�, ψ�

� , ψ�

� are def ined via

φ�(ξ) = P(ξ)φ(ξ), φ�(ξ) = P(ξ )φ(ξ),

ψ�

�(MTξ) = b �(ξ)φ�(ξ),ψ

�

�(MTξ) = b �(ξ)φ�(ξ),

� = 1, . . . , md. Here 0 < s < min{β + 1, s∗} and β = min{κ, κ}. Moreover, ψ�

�

has β + 1 vanishing moments, � = 1, . . . , md.

Proof Since a(ξ) has κ + 1 sum rules, according to Lemma 3.2, there existsa strongly invertible r × r matrix P(ξ) such that a�(ξ) = P(MTξ )a(ξ)P(ξ)−1

has the property of (3.9). Define φ� via φ�(ξ) = P(ξ)φ(ξ). Then φ� satisfiesφ�(MTξ) = a�(ξ)φ�(ξ), and has the property of (3.10). Similarly, according toTheorem 3.2, there exists a strongly invertible r × r matrix P(ξ) such thata�(ξ) = P(MTξ )a(ξ)P(ξ)−1 has the property of (3.9) with κ being replaced byκ . Moreover, φ� is a Bessel refinable function vector in H−s(Rd), ∀s ∈ R

+.Next we shall construct the r × r matrices b 1(ξ), . . . , b md

(ξ) and b 1(ξ), . . . ,b md(ξ). Suppose p(ξ) and p(ξ) are the mask symbols of a pair of real-valued

biorthogonal M-refinable functions, and the mask symbols of their biorthogo-

nal M-wavelets are q1(ξ), . . . , qmd−1(ξ) and q1(ξ), . . . , qmd−1(ξ). That is,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

∑γ∈�MT

p (ξ + 2πγ )p (ξ + 2πγ ) = 1,

∑γ∈�MT

q� (ξ + 2πγ ) qk (ξ + 2πγ ) = δ�−k,

∑γ∈�MT

p (ξ + 2πγ ) qk (ξ + 2πγ ) = 0,

∑γ∈�MT

qk (ξ + 2πγ )p (ξ + 2πγ ) = 0,

(4.8)

where �, k = 1, . . . , md − 1. Define

b1(ξ) = p(ξ)Ir, b �(ξ) = q�−1(ξ)Ir, c1(ξ) = p(ξ)Ir, c�(ξ) = q�−1(ξ)Ir, (4.9)

514 Y. Li et al.

for � = 2, . . . , md. Then (4.8) and (4.9) lead to that

Imdr =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

b1 (ξ + 2πγ0)∗ b2 (ξ + 2πγ0)

∗ . . . b md(ξ + 2πγ0)

∗

b1 (ξ + 2πγ1)∗ b2 (ξ + 2πγ1)

∗ . . . b md(ξ + 2πγ1)

∗

......

. . ....

b1(ξ + 2πγmd−1

)∗b2

(ξ + 2πγmd−1

)∗. . . b md

(ξ + 2πγmd−1

)∗

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

c1 (ξ + 2πγ0) c1 (ξ + 2πγ1) . . . c1(ξ + 2πγmd−1

)c2 (ξ + 2πγ0) c2 (ξ + 2πγ1) . . . c2

(ξ + 2πγmd−1

)...

.... . .

...

cmd(ξ + 2πγ0) cmd

(ξ + 2πγ1) . . . cmd(ξ + 2πγmd−1

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (4.10)

Construct⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

c1(ξ)

c2(ξ)

...

cmd(ξ)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

c1 (ξ + 2πγ0) c1 (ξ + 2πγ1) . . . c1(ξ + 2πγmd−1

)c2 (ξ + 2πγ0) c2 (ξ + 2πγ1) . . . c2

(ξ + 2πγmd−1

)...

.... . .

...


(ξ + 2πγ1) . . . cmd(ξ + 2πγmd−1

)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Irδγ0 − a� (ξ + 2πγ0)∗ a�(ξ)

Irδγ1 − a� (ξ + 2πγ1)∗ a�(ξ)

...

Irδγmd − a�(ξ + 2πγmd−1

)∗ a�(ξ)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (4.11)

Next we shall prove that (4.2) holds with b �(ξ) being replaced by c�(ξ), � =1, . . . , md. In fact,⎡⎢⎢⎢⎢⎢⎢⎣

b1 (ξ + 2πγ0)∗ b2 (ξ + 2πγ0)

∗ . . . b md(ξ + 2πγ0)

∗

b1 (ξ + 2πγ1)∗ b2 (ξ + 2πγ1)

∗ . . . b md(ξ + 2πγ1)

∗

......

. . ....

b1(ξ + 2πγmd−1

)∗b2

(ξ + 2πγmd−1

)∗. . . b md

(ξ + 2πγmd−1

)∗

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

c1(ξ)

c2(ξ)

...

cmd(ξ)

⎤⎥⎥⎥⎥⎥⎥⎦


=

⎡⎢⎢⎢⎢⎢⎢⎣

b1 (ξ + 2πγ0)∗ b2 (ξ + 2πγ0)

∗ . . . b md(ξ + 2πγ0)

∗

b1 (ξ + 2πγ1)∗ b2 (ξ + 2πγ1)

∗ . . . b md(ξ + 2πγ1)

∗

......

. . ....

b1(ξ + 2πγmd−1

)∗b2

(ξ + 2πγmd−1

)∗. . . b md

(ξ + 2πγmd−1

)∗

⎤⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎢⎢⎣

c1 (ξ + 2πγ0) c1 (ξ + 2πγ1) . . . c1(ξ + 2πγmd−1

)c2 (ξ + 2πγ0) c2 (ξ + 2πγ1) . . . c2

(ξ + 2πγmd−1

)...

.... . .

...


(ξ + 2πγ1) . . . cmd(ξ + 2πγmd−1

)

⎤⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎢⎣

Irδγ0 − a� (ξ + 2πγ0)∗ a�(ξ)

Irδγ1 − a� (ξ + 2πγ1)∗ a�(ξ)

...


)∗ a�(ξ)

⎤⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎣

Irδγ0 − a� (ξ + 2πγ0)∗ a�(ξ)

Irδγ1 − a� (ξ + 2πγ1)∗ a�(ξ)

...


)∗ a�(ξ)

⎤⎥⎥⎥⎥⎥⎦

, (4.12)

where the lase equality is guaranteed by (4.10). Construct b �(ξ) = [c�(ξ) +c�(−ξ)]/2. Now we prove that (4.2) holds. In the words of components, (4.12)is represented as

md∑�=1

b � (ξ + 2πγ )∗ c�(ξ) = Irδγ − a� (ξ + 2πγ )∗ a�(ξ), ∀γ ∈ �MT , (4.13)

which is equivalent to

md∑�=1

b � (ξ − 2πγ )∗ c�(−ξ) = Irδγ − a� (ξ − 2πγ )∗ a�(ξ), (4.14)

where we use the fact that the matrix coefficients of b �(ξ), a�(ξ) and a�(ξ) are

all real-valued, i.e., b �(ξ) = b �(−ξ). Note that (4.14) is equivalent to

md∑�=1

b � (ξ + 2πγ )∗ c�(−ξ) = Irδγ − a� (ξ + 2πγ )∗ a�(ξ). (4.15)

516 Y. Li et al.

Therefore, we deduce from (4.13) and (4.15) that

md∑�=1

b �(ξ + 2πγ )∗[c�(ξ) + c�(−ξ)]/2 = Irδγ − a�(ξ + 2πγ )∗a�(ξ), ∀γ ∈ �MT .

That is,

md∑�=1

b �(ξ + 2πγ )∗ b �(ξ) = Irδγ − a�(ξ + 2πγ )∗a�(ξ), ∀γ ∈ �MT .

Thus, (4.2) holds. Next we check the vanishing moment conditions on ψ�

1, . . . ,

ψ�

md . By (4.11), compute

c�(ξ) [1, 0, . . . , 0]T = c�(ξ)(

Ir − a�(ξ)∗a�(ξ))

[1, 0, . . . , 0]T

−md−1∑

j=1

c�(ξ + 2πγ j

)a�

(ξ + 2πγ j

)∗ a�(ξ) [1, 0, . . . , 0]T

= c�(ξ)(

[1, 0, . . . , 0]T − a�(ξ)∗a�(ξ) [1, 0, . . . , 0]T)

−md−1∑

j=1

c�(ξ + 2πγ j)a�(ξ + 2πγ j

)∗ a�(ξ) [1, 0, . . . , 0]T

= c�(ξ)[1 −

(a�

11(ξ)a�11(ξ) + a�

21(ξ)∗a�21(ξ)

),

−(

a�12(ξ)∗a�

11(ξ) + a�22(ξ)∗a�

21(ξ))T

]T

−md−1∑

j=1

c�(ξ + 2πγ j

) [ a�11(ξ + 2πγ j

)a�11(ξ) + a�

21(ξ + 2πγ j

)∗ a�21(ξ)

a�12(ξ + 2πγ j

)∗ a�11(ξ) + a�

22(ξ + 2πγ j

)∗ a�21(ξ)

].

Again by (3.9), we have

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 −(

a�11(ξ)a�

11(ξ) + a�21(ξ)∗a�

21(ξ))

= O(||ξ ||κ+1+κ+1

2

),

a�12(ξ)∗a�

11(ξ) + a�22(ξ)∗a�

21(ξ) = O(||ξ ||β+1

2

),

a�11(ξ + 2πγ j

)a�11(ξ) + a�

21(ξ + 2πγ j

)∗ a�21(ξ) = O

(||ξ ||β+1

2

),

a�12(ξ + 2πγ j

)∗ a�11(ξ) + a�

22(ξ + 2πγ j

)∗ a�21(ξ) = O

(||ξ ||β+1

2

),

ξ → 0.


Therefore, c�(ξ)(1, 0, . . . , 0)T = O(||ξ ||β+12 ), ξ → 0. Obviously,

b �(ξ)(1, 0, . . . , 0)T = O(||ξ ||β+1

2

), ξ → 0,

which together with

φ

�

�(ξ) = O(||ξ ||κ+1

2

), ξ → 0

leads to that ∂αψ�

�(0) = 0 for any α ∈ Nd0 with |α| ≤ β. That is, {ψ�

1, . . . , ψ�

md}have β + 1 vanishing moments. According to Theorems 3.2 and 2.2,X−s(φ�; ψ

�

1, . . . , ψ�

md) is a Bessel M-multiwavelet sequence in H−s(Rd). ByTheorem 4.1, Xs(φ�; ψ

�

1, . . . , ψ�


1, . . . , ψ�

md) are a pair ofdual M-multiframelets in (Hs(Rd), H−s(Rd)) with t < s < min{s∗, β + 1}. Onthe other hand, φ ∈ (Ht(Rd))r and φ� ∈ (H−t(Rd))r for any t ∈ (0, s∗). Conse-quently, Xs(φ�; ψ

�

1, . . . , ψ�


1, . . . , ψ�

md) are a pair of dual M-multiframelets in (Hs(Rd), H−s(Rd)) with 0 < s < min{s∗, β + 1}.

Next we give an algorithm for summarizing the process in Theorem 4.2for constructing dual M-multiframelets. In the algorithm, based on the proofof [16, Theorem 5.1], we shall present the steps for constructing the stronglyinvertible matrices P(ξ) and P(ξ) in Theorem 4.2.

Algorithm 4.1 Let φ = (φ1, . . . , φr)T and φ = (φ1, . . . , φr)

T be as in Theorem4.2. Assume the mask symbol a(ξ) of φ has κ + 1 sum rules, κ ∈ N0, and satisf ies(3.2). Since Y(0) �= 0, where Y(ξ) is referred in (3.2), without loosing generality,we suppose Y1(0) �= 0. Otherwise, we can rearrange the components of φ. Thematrix P(ξ) in Theorem 4.2 can be constructed through Step 1 to Step 6 asfollows:

Step 1: Construct a 1 × r vector c(ξ) = (c1(ξ), . . . , cr(ξ)) of 2πZd-periodic

trigonometric polynomials such that

c1(ξ)=1/Y1(ξ)+O(||ξ ||κ+12 ) and c�(ξ)= Y�(ξ)/Y1(ξ)+O(||ξ ||κ+1

2 ),ξ→0

for �=2, . . . , r.Step 2: Def ine two strongly invertible matrices P1(ξ) and P2(ξ) of 2πZ

d-periodic trigonometric polynomials via

P1(ξ) =[

1 − f (ξ)

O Ir−1

],

P2(ξ) =⎡⎣ c1(ξ) −(1 − c1(ξ)/c1(0))κ+1 O

(1 − c1(ξ)/c1(0))κ+1 Qκ+1(ξ) OO O Ir−2

⎤⎦ ,

518 Y. Li et al.

where

f (ξ) = (c2(ξ), . . . , cr(ξ)) , Qκ+1(ξ)

=2κ+1∑�=0

(−1)�(2κ+2�+1

)[c1(ξ)]� / [c1(0)]�+1 .

Step 3: Def ine P0(ξ) = [P1(ξ)P2(ξ)]−1 and a0(ξ) = P0(MTξ )a(ξ)P0(ξ)−1.

Construct an M-ref inable function vector φ0 = (φ0,1, . . . , φ0,r)T by

φ0(MTξ) = P0(ξ)φ(ξ). Consequently, φ0(MTξ) = a0(ξ)φ0(ξ).Step 4: Construct a 1 × r vector d(ξ) = (d1(ξ), . . . , dr(ξ)) of 2πZ

d-periodictrigonometric polynomials such that

d1(ξ)φ0,1(ξ) = 1 + O(||ξ ||κ+12 ), φ0,�(ξ) − d�(ξ) = O(||ξ ||κ+1

2 ), ξ → 0

where � = 2, . . . , r.Step 5: Def ine two strongly invertible matrices T1(ξ) and T2(ξ) of 2πZ

d-periodic trigonometric polynomials via

T1(ξ) =[

1 −g(ξ)

O Ir−1

],

T2(ξ) =⎡⎢⎣

d1(ξ) − (1 − d1(ξ)/d1(0)

)κ+1O(

1 − d1(ξ)/d1(0))κ+1

Rκ+1(ξ) OO O Ir−2

⎤⎥⎦

where

g(ξ) = (d2(ξ), . . . , dr(ξ)

), Rκ+1(ξ)

=2κ+1∑�=0

(−1)�(2κ+2

�+1

) [d1(ξ)

]�/[d1(0)

]�+1.

Step 6: Def ine P(ξ) = [T1(ξ)T2(ξ)]T P0(ξ) and a�(ξ) = P(MTξ )a(ξ)P(ξ)−1.Step 7: Construct an M-ref inable function vector φ with desired sum rules and

Sobolev exponent via Theorem 3.1. Through the same procedures in

Step 1 to Step 6, construct a strongly invertible matrix P(ξ) such thata�(ξ) = P(MTξ )a(ξ)P(ξ)−1 takes the form of (3.9).

Step 8: For mask symbols { p(ξ), q1(ξ), . . . , qmd−1(ξ)} and {p(ξ), q1(ξ), . . . ,

qmd−1(ξ)} of a pair of real-valued biorthogonal M-wavelets, def ineb �(ξ) and c�(ξ) via (4.9), � = 1, . . . , md. Construct c�(ξ) via (4.11) andb �(ξ) = [c�(ξ) + c�(−ξ)]/2.

Step 9: Def ine {ψ�

�}md

�=1 and {ψ�

�}md

�=1 via ψ�

�(MTξ)= b �(ξ)φ�(ξ)and ψ

�

�(MTξ) =b �(ξ)φ�(ξ), � = 1, . . . , md.


5 A scheme for dual M-multiframelets with symmetry in Sobolev spaces

Symmetry is of great importance for multiframelets or framelets [11, 25, 28].Motivated by [24, 31, 32], we shall give a scheme for dual M-multiframeletswith symmetry in Sobolev spaces.

Theorem 5.1 Let Xs(φ�; ψ�

1, . . . , ψ�


1, . . . , ψ�

md) be as in Theo-rem 4.2. Construct

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

φnew(x) = (φnew

1 (x), φnew2 (x), . . . , φnew

2r (x))T

=√

22

(φ�(x)T + φ�(−x)T , φ�(x)T − φ�(−x)T

)T,

φnew(x) = (φnew

1 (x), φnew2 (x), . . . , φnew

2r (x))T

=√

22

(φ�(x)T + φ�(−x)T , φ�(x)T − φ�(−x)T

)T,

ψnew� (x) = (

ψnew�,1 (x), ψnew

�,2 (x), . . . , ψnew�,2r (x)

)T

=√

22

(ψ

�

�(x)T + ψ�

�(−x)T , ψ�

�(x)T − ψ�

�(−x)T)T

,

ψnew� (x) = (

ψnew�,1 (x), ψnew

�,2 (x), . . . , ψnew�,2r (x)

)T

=√

22

(ψ

�

�(x)T + ψ�

�(−x)T , ψ�

�(x)T − ψ�

�(−x)T)T

,

� = 1, . . . , md.

(5.1)

Then the function vectors in (5.1) are of symmetry and satisfy the following M-ref inable equations

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

φnew(MTξ

) = anew(ξ)φnew(ξ)

=⎡⎢⎣(

a�(ξ) + a�(−ξ))

/√

2(

a�(ξ) − a�(−ξ))

/√

2(

a�(ξ) − a�(−ξ))

/√

2(

a�(ξ) + a�(−ξ))

/√

2

⎤⎥⎦ φnew(ξ),

φnew(MTξ) = anew(ξ) φnew(ξ)

=⎡⎢⎣(a�(ξ) + a�(−ξ)

)/√

2(a�(ξ) − a�(−ξ)

)/√

2(a�(ξ) − a�(−ξ)

)/√

2(a�(ξ) + a�(−ξ)

)/√

2


(5.2)

520 Y. Li et al.

and⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ψnew�

(MTξ

) = b new� (ξ)φnew(ξ)

=⎡⎢⎣(

b �(ξ) + b �(−ξ))

/√

2(

b �(ξ) − b �(−ξ))

/√

2(

b �(ξ) − b �(−ξ))

/√

2(

b �(ξ) + b �(−ξ))

/√

2


ψnew�

(MTξ

) = b new� (ξ) φnew(ξ)

=⎡⎢⎣(b �(ξ) + b �(−ξ)

)/√

2( b �(ξ) − b �(−ξ)

)/√

2( b �(ξ) − b �(−ξ)

)/√

2( b �(ξ) + b �(−ξ)

)/√

2

⎤⎥⎦ φnew(ξ).

(5.3)

Moreover, Xs(φnew; ψnew1 , . . . , ψnew

md ) and X−s(φnew; ψnew1 , . . . , ψnew

md ) are a pairof dual M-multiframelets.

Proof It is easy to check that function vectors defined in (5.1) are of symmetry.From φ�(MTξ) = a�(ξ)φ�(ξ), we get[

φ�(MTξ

)φ�

(−MTξ)]

=[

a�(ξ) Or×r

Or×r a�(−ξ)

] [φ�(ξ)

φ�(−ξ)

].

Then

φnew(ξ) = 1√2

[φ�(ξ) + φ�(−ξ)

φ�(ξ) − φ�(−ξ)

]=

[1/

√2 1/

√2

1/√

2 −1/√

2

] [φ�(ξ)

φ�(−ξ)

],

from which we deduce

φnew(MTξ) =[

1/√

2 1/√

21/

√2 −1/

√2

] [a�(ξ) Or×r

Or×r a�(−ξ)

] [1/

√2 1/

√2

1/√

2 −1/√

2

]−1

φnew(ξ).

Now, it is easy to check that (5.2) and (5.3) hold. On the other hand,

2r∑n=1

∑k∈Zd

⟨φnew

n;0,k, g⟩ ⟨

f, φnewn;0,k

⟩

=r∑

n=1

∑k∈Zd

[⟨φnew

2n−1;0,k, g⟩ ⟨

f, φnew2n−1;0,k

⟩ + ⟨φnew

2n;0,k, g⟩ ⟨

f, φnew2n;0,k

⟩]

= 1

2

r∑n=1

∑k∈Zd

[⟨φ�

n(x − k) + φ�n(−x + k), g

⟩ ⟨f, φ�

n(x − k) + φ�n(−x + k)

⟩

+ ⟨φ�

n(x − k) − φ�n(−x + k), g

⟩ ⟨f, φ�

n(x − k) − φ�n(−x + k)

⟩]

=r∑

n=1

∑k∈Zd

⟨φ

�

n;0,k, g⟩ ⟨

f, φ�

n;0,k

⟩.


Similarly,

2r∑n=1

md∑�=1

∑j∈N0

∑k∈Zd

⟨ψ

new,�,sn; j,k , g

⟩ ⟨f, ψnew,�,−s

n; j,k

⟩=

r∑n=1

md∑�=1

∑j∈N0

∑k∈Zd

⟨ψ

�,sn; j,k, g

⟩ ⟨f, ψ�,−s

n; j,k

⟩,

where ψnew,�,sn; j,k = m j(d/2−s)ψnew

�,n (M j · −k). Therefore, Xs(φnew; ψnew1 , . . . , ψnew

md )

and X−s(φnew; ψnew1 , . . . , ψnew

md ) are a pair of dual M-multiframelets in (Hs(Rd),

H−s(Rd)).

6 An example

In this section, we shall give an example to illustrate the procedures ofconstructing real-valued dual multiframelets in Sobolev spaces.

In [21, Example 2.2], Han and Zhuang constructed an M-refinable inter-polating vector φ = (φ1, φ2)

T ∈ (L2(R2))2, where the isotropic dilation matrix

M =[

1 11 −1

]. It satisfies the refinement equation φ(Mξ) = a(ξ)φ(ξ), where

a(ξ) =[

1/2(eiξ1 + ei(ξ1+ξ2) + eiξ2 + 1

)/4

e−iξ1/2 0

], ξ = (ξ1, ξ2)

T ∈ R2.

Moreover, a(ξ) has 2 sum rules and ν2(φ) = 1.5. It is easy to check that �M ={(0, 0)T , (1/2, 1/2)T}.

We calculate Y(ξ) = (Y1(ξ), Y2(ξ)) first such that (3.2) holds with κ = 1.That is,

{∂α

[Y(M·)a(·)] (0) − ∂αY(0) = 0,

∂α[Y(M·)a(·)] ((π, π)T

) = 0,(6.1)

where Y(0) �= 0, ∀α ∈ {(0, 0)T , (0, 1)T , (1, 0)T}. For α = (0, 0)T , it is easy tocheck that (6.1) is equivalent to Y1(0) = Y2(0) �= 0. By replacing variable andLeibniz differential formula, we have

{∂(1,0)

[Y(M·)] (ξ) = ∂(1,0)Y(Mξ) + ∂(0,1)Y(Mξ),

∂(0,1)[Y(M·)] (ξ) = ∂(1,0)Y(Mξ) − ∂(0,1)Y(Mξ).

(6.2)

Compute

Y(Mξ)

[1/2

(eiξ1 + ei(ξ1+ξ2) + eiξ2 + 1

)/4

e−iξ1/2 0

]

= [Y1(Mξ)/2 + Y2(Mξ)e−iξ1/2, Y1(Mξ)

(1 + eiξ1 + ei(ξ1+ξ1) + eiξ2

)/4

].

522 Y. Li et al.

We get from (6.2) that⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∂(1,0)(Y1(Mξ)/2 + Y2(Mξ)e−iξ1/2

) = [∂(1,0)Y1

](Mξ)/2

+ [∂(0,1)Y1

](Mξ)/2 − (sin ξ1 + i cos ξ1) Y2(Mξ)/2

+ ([∂(1,0)Y2

](Mξ)/2 + [

∂(0,1)Y2](Mξ)/2

)e−iξ1 ,

∂(1,0)(Y1(Mξ)

(1 + eiξ1 + ei(ξ1+ξ1) + eiξ2

)/4

)= ([

∂(1,0)Y1](Mξ) + [

∂(0,1)Y1](Mξ)

) (1 + eiξ1 + ei(ξ1+ξ1) + eiξ2

)/4

+Y1(Mξ) (− sin ξ1 + i cos ξ1 − sin (ξ1 + ξ2) + i cos (ξ1 + ξ2)) /4,

∂(0,1)(Y1(Mξ)/2 + Y2(Mξ)e−iξ1/2

) = [∂(1,0)Y1

](Mξ)/2

− [∂(0,1)Y1

](Mξ)/2 + ([

∂(1,0)Y2](Mξ)/2 − [

∂(0,1)Y2](Mξ)/2

)e−iξ1 ,

∂(0,1)(Y1(Mξ)

(1 + eiξ1 + ei(ξ1+ξ1) + eiξ2

)/4

)= ([

∂(1,0)Y1](Mξ) − [

∂(0,1)Y1](Mξ)

) (1 + eiξ1 + ei(ξ1+ξ1) + eiξ2

)/4

+Y1(Mξ) (− sin ξ2 + i cos ξ2 − sin (ξ1 + ξ2) + i cos (ξ1 + ξ2)) /4.

(6.3)

For ξ = 0 and ξ = (π, π)T , we derive from (6.1) and (6.3) that⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−∂(1,0)Y1(0) + ∂(0,1)Y1(0) + ∂(1,0)Y2(0) + ∂(0,1)Y2(0) = i,

∂(1,0)Y1(0) + ∂(0,1)Y1(0) − ∂(1,0)Y2(0) = −i/2,

∂(1,0)Y1(0) − 3∂(0,1)Y1(0) + ∂(1,0)Y2(0) − ∂(0,1)Y2(0) = 0,

∂(1,0)Y1(0) − ∂(0,1)Y1(0) − ∂(0,1)Y2(0) = −i/2,

∂(1,0)Y1(0) + ∂(0,1)Y1(0) − ∂(1,0)Y2(0) − ∂(0,1)Y2(0) = −i,

∂(1,0)Y1(0) − ∂(0,1)Y1(0) − ∂(1,0)Y2(0) + ∂(0,1)Y2(0) = 0.

(6.4)

The solution of (6.4) is ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

∂(1,0)Y1(0) = 0,

∂(0,1)Y1(0) = 0,

∂(0,1)Y2(0) = i/2,

∂(1,0)Y2(0) = i/2.

Select

Y1(ξ) = 2ei(ξ1+ξ2) − ei2(ξ1+ξ2), Y2(ξ) = 3

2ei(ξ1+ξ2) − 1

2ei2(ξ1+ξ2).

Next, we calculate P(ξ) in Algorithm 4.1 such that

a�(ξ) = P(MTξ

)a(ξ)P(ξ)−1

takes the form of (3.9).

Step 1: Select c1(ξ) = Y1(ξ) and c2(ξ) = Y2(ξ) such that

c1(ξ)Y1(ξ)=1+O(||ξ ||22

)and c2(ξ)Y1(ξ)= Y2(ξ)+O

(||ξ ||22), ξ → 0.


Step 2: Define two strongly invertible matrices P1(ξ) and P2(ξ)

P1(ξ) =[

1 −c2(ξ)

0 1

]=

[1 − 3

2 ei(ξ1+ξ2) + 12 ei2(ξ1+ξ2)

0 1

]

and

P2(ξ) =[

2ei(ξ1+ξ2) − ei2(ξ1+ξ2) − (2ei(ξ1+ξ2) − ei2(ξ1+ξ2) − 1

)2(2ei(ξ1+ξ2) − ei2(ξ1+ξ2) − 1

)2Q2(ξ)

]

where

Q2(ξ) = 4 + ei6(ξ1+ξ2) − 6ei5(ξ1+ξ2) + 16ei4(ξ1+ξ2)

− 24ei3(ξ1+ξ2) + 22ei2(ξ1+ξ2) − 12ei(ξ1+ξ2).

Step 3: Define

P0(ξ) = [P1(ξ)P2(ξ)

]−1 =[

P0,11(ξ) P0,12(ξ)

P0,21(ξ) P0,22(ξ)

],

where⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

P0,11(ξ) = 4 + ei6(ξ1+ξ2) − 6ei5(ξ1+ξ2) + 16ei4(ξ1+ξ2) − 24ei3(ξ1+ξ2)

+ 22ei2(ξ1+ξ2) − 12ei(ξ1+ξ2),

P0,12(ξ) = −14ei2(ξ1+ξ2) + 35ei3(ξ1+ξ2) + 2ei(ξ1+ξ2) − 46ei4(ξ1+ξ2)

+1 + 9ei7(ξ1+ξ2)/2 − 17ei6(ξ1+ξ2) + 36ei5(ξ1+ξ2) − ei8(ξ1+ξ2)/2

P0,21(ξ) = − (−2ei(ξ1+ξ2) + ei2(ξ1+ξ2) + 1)2

,

P0,22(ξ) = ei(ξ1+ξ2)/2 + 11ei2(ξ1+ξ2)/2 − 11ei3(ξ1+ξ2) + 9ei4(ξ1+ξ2)

− 7ei5(ξ1+ξ2)/2 + ei6(ξ1+ξ2)/2.

Construct a0(ξ) = P0(MTξ )a(ξ)P0(ξ)−1 and φ0 = (φ0,1, . . . , φ0,r)T by

φ0(ξ) = P0(ξ)φ(ξ). (6.5)

It is easy to calculate from φ(Mξ) = a(ξ)φ(ξ) that{∂(1,0)φ(0) + ∂(0,1)φ(0) = ∂(1,0)a(0)φ(0) + a(0)∂(1,0)φ(0),

∂(1,0)φ(0) − ∂(0,1)φ(0) = ∂(0,1)a(0)φ(0) + a(0)∂(0,1)φ(0).(6.6)

Considering φ(0) = (2, 1)T , we get from (6.5) and (6.6) that

φ0(0)=(

31

), ∂(1,0)φ0(0)=

(−11.0000i−0.0714i

), ∂(0,1)φ0(0) =

(−11.5000i−0.1429i

).

Step 4: Construct

d1(ξ) = −0.5 + 0.3889ei(2ξ1+ξ2) + 0.4445ei(ξ1+2ξ2),

d2(ξ) = 1.0714 − 0.0715ei(ξ1+2ξ2),

such that d1(ξ)φ0,1(ξ) = 1 + O(||ξ ||22) and φ0,2(ξ) − d2(ξ) =O(||ξ ||22), ξ → 0.

524 Y. Li et al.

Step 5: Define two strongly invertible matrices T1(ξ) and T2(ξ) via

T1(ξ) =(

1 −d2(ξ)

0 1

), T2(ξ) =

(d1(ξ) − (

1 − d1(ξ))2

(1 − d1(ξ)

)2R2(ξ)

),

where R2(ξ) = 3∑3

�=0(−1)�( 4

�+1

)[3d1(ξ)]�.

Step 6: Define P(ξ) = [T1(ξ)T2(ξ)]T P0(ξ). Then a�(ξ) = P(MTξ )a(ξ)P(ξ)−1

takes the form of (3.9) and φ� defined via φ�(ξ) = P(ξ)φ(ξ) satisfies(3.10) with κ = 1.

Step 7: In this step, we shall construct a Bessel M-refinable function vectorφ� by Theorems 3.1 and 3.2. In [17, Example 4.1], Han and Jia con-structed an M-refinable function φo given by φo(MTξ) = h(ξ) φo(ξ),

where

h(ξ) = −ei(2ξ1+ξ2)/32 − ei(2ξ1−ξ2)/32 + eiξ1/4 + eiξ2/16 + 1/2

+ e−iξ2/16 + e−iξ1/4 − ei(−2ξ1+ξ2)/32 − ei(−2ξ1−ξ2)/32.

Moreover, h(ξ) has 2 sum rules and ν2(φo) = 2.44077. Construct φ =

(φ1, φ2)T via (3.3) with

a(ξ) = h(ξ)

[ei(ξ1+ξ2)/3 2e−i(ξ1+ξ2)/3

2ei(ξ1+ξ2)/3 e−i(ξ1+ξ2)/3

].

According to Theorem 3.1, a(ξ) has 2 sum rules, ν2(φ) ≥ 2.44077, and∂α φ�(2kπ) = 0 for all k ∈ Z

2\{0} and all α ∈ Z2 with |α| ≤ 1, � = 1, 2.

Select

P(ξ) =[

1 0(1 − e−i(ξ1+ξ2)

)2 − 1 1

]

and define φ� = (φ�

1, φ�

2)T via φ�(ξ) = P(ξ )φ(ξ). Then φ� satisfies item

(i) and (ii) of Theorem 2.2 with β = 1.Step 8: In [11, Example 3.6], Han gave an orthogonal N-refinable function,

of which the mask symbol is

p(ξ) =√

3 − 1

16e−i(ξ1+ξ2) + 3 + √

3

16e−i2ξ1 + 3 − √

3

16e−i(2ξ1+ξ2)

+ 3 − √3

16e−i(2ξ1+2ξ2) + 3 + √

3

16e−i3ξ1 + 3 + √

3

16e−i(3ξ1+ξ2)

+ 3 − √3

16e−i(3ξ1+2ξ2)+−1−√

3

16e−i(4ξ1+ξ2)

with N =[

1 −11 1

].


Considering �NT = �MT , construct q(ξ) = e−iξ1 p(ξ1 + π, ξ2 + π). It iseasy to check that

p (ξ1, ξ2) q (ξ1, ξ2) + p (ξ1 + π, ξ2 + π) q (ξ1 + π, ξ2 + π) = 0.

According to (4.13) and (4.14), define⎧⎪⎪⎪⎨⎪⎪⎪⎩

b 1(ξ)= c1(ξ)= p(ξ)I2,

b 2(ξ)= c2(ξ)= q(ξ)I2,b 1(ξ)= c1(ξ)(I2−a(ξ)∗a(ξ))−c1 (ξ1+π, ξ2+π) a (ξ1+π, ξ2+π)∗a(ξ),b 2(ξ)= c2(ξ)(I2−a(ξ)∗a(ξ)

)−c2 (ξ1+π, ξ2+π) a (ξ1+π, ξ2+π)∗a(ξ).

Step 9: Construct ψ�

� and ψ�

� via

ψ�

�(Mξ) = b �(ξ)φ�(ξ),ψ

�

�(Mξ) = b �(ξ)φ�(ξ), � = 1, 2.

By Theorem 4.2, Xs(φ�; ψ�

1, ψ�

2) and X−s(φ�; ψ�

1, ψ�

2) are a pair ofdual M-multiframelets in (Hs(R2), H−s(R2)), where s ∈ (0, 1.5). ByTheorem 5.1, we can construct dual M-multiframelets with symmetryin (Hs(R2), H−s(R2)).

Acknowledgements We appreciate the anonymous reviewer for his/her suggestions that im-proved the paper a lot. Especially, suggested by the reviewer, we add Lemma 1.1, revise theproof of Theorem 2.1 for being better understandable, and reprove Lemma 3.1 by Prof. Han Bin’smethod in [14, Lemma 3.2]. The work is supported by the National Natural Science Foundation ofChina (Grant No.11071152), the Natural Science Foundation of Guangdong Province (Grant No.S2011010004511). The research of the first author is by Natural Scientific Project of Guangxi Uni-versity (Grant No. XBZ110572), and his Postdoctoral research at Macau University is supportedby Macao Science and Technology Fund FDCT/056/2010/A3.

Appendix

Proof of Lemma 2.1 If �1||ξ || ≤ 1, then∑j∈N0

m−2 jζ (1 + �21||ξ ||2)ζ (1 + m−2 j−2�2

2||ξ ||2)−η

≤∑

j∈N0m−2 jζ 2ζ

= 2ζ

1 − m−2ζ.

For �1||ξ || > 1, there exists J ∈ N0 such that mJ ≤ �1||ξ || < mJ+1. Denote

Bζ,η(ξ) = B1ζ,η(ξ) + B2

ζ,η(ξ),

where

B1ζ,η(ξ) =

J∑j=0

m−2 jζ (1 + �21||ξ ||2)ζ (1 + m−2 j−2�2

2||ξ ||2)−η

526 Y. Li et al.

and

B2ζ,η(ξ) =

∞∑j=J+1

m−2 jζ (1 + �21||ξ ||2)ζ (1 + m−2 j−2�2

2||ξ ||2)−η.

Then

B1ζ,η(ξ) ≤

J∑j=0

m−2 jζ (1 + m2J+2)ζ (

1 + �22

�21

m−2 j−2m2J)−η

≤J∑

j=0

m−2 jζ 2ζ m2Jζ+2ζ m2 jη+2ηm−2Jη

(�1

�2

)2η

≤ m2Jζ+2ζ m2η2ζ m−2Jη

(�1

�2

)2η J∑j=0

m−2 jζ m2 jη

= m2Jζ+2ζ m2η2ζ m−2Jη

(�1

�2

)2η 1 − mJ(2η−2ζ )

1 − m2η−2ζ

≤m2ζ m2η2ζ

(�1

�2

)2η

m2η−2ζ − 1.

Moreover,

B2ζ,η(ξ) ≤

∞∑j=J+1

m−2 jζ (1 + m2J+2)ζ (

1 + �22

�21

m−2 j−2m2J)−η

≤∞∑

j=J+1

m−2 jζ (1 + m2J+2)ζ

=∞∑

j=J+1

m−2 jζ m2Jζ+2ζ(1 + m−2J−2

)ζ

≤ 2ζ m2ζ

∞∑j=J+1

m−2 jζ m2Jζ

= 2ζ m2ζ

∞∑j=1

m−2 jζ = 2ζ

1 − m−2ζ.

Above all, Bζ,η(ξ) ≤ C for any ξ ∈ Rd, where

C = m2ζ m2η2ζ (�1

�2)2η

m2η−2ζ − 1+ 2ζ

1 − m−2ζ< ∞.


Proof of Lemma 2.2 Recall that b(ξ) is 2πZd-periodic and

b(ξ) = O(||ξ ||β+1

2

), ξ → 0.

By Lemma 1.1, there exists a positive constant D such that

∣∣b(ξ)∣∣ ≤ D||ξ ||β+1, ∀ξ ∈ R

d.

When �1||ξ || < 1,

B−s,t(ξ) =∑j∈N0

∣∣b (m− j−1ξ

)∣∣2 m2 js (1 + �22||ξ ||2)−s (

1 + �21m−2( j+1)||ξ ||2)t

≤ D2∑j∈N0

m−2 j(β+1)−2(β+1)||ξ ||2(β+1)m2 js2t

≤ D22t(1 − m−(2(β+1)−2s)

)�

2β+21

.

When �1||ξ || ≥ 1, there exists J ∈ N0 such that mJ ≤ �1||ξ || ≤ mJ+1. DenoteB−s,t(ξ) = B−s,t,1(ξ) + B−s,t,2(ξ) with

B−s,t,1(ξ) =J∑

j=0

∣∣b (m− j−1ξ

)∣∣2 m2 js (1 + �22||ξ ||2)−s (

1 + �21m−2 j−2||ξ ||2)t

,

B−s,t,2(ξ) =∞∑

j=J+1

∣∣b (m− j−1ξ

)∣∣2 m2 js (1 + �22||ξ ||2)−s (

1 + �21m−2 j−2||ξ ||2)t

.

Estimate

B−s,t,1(ξ) ≤ ∣∣∣∣b(ξ)∣∣∣∣2

L∞(R)

J∑j=0

m2 js(

1 + �22

�21

m2J)−s (

1 + �21m−2 j−2m2J+2

)t

≤ ∣∣∣∣b(ξ)∣∣∣∣2

L∞(R)

�2s1

�2s2

J∑j=0

m2 jsm−2Js (1 + �21m2(J− j))t

≤ ∣∣∣∣b(ξ)∣∣∣∣2

L∞(R)

�2s1

�2s2

J∑j=0

m2 jsm−2Js ((1 + �21

)m2(J− j))t

≤∣∣∣∣b(ξ)

∣∣∣∣2L∞(R)

�2s1

�2s2

(1 + �2

1

)t

m−2t+2s − 1

528 Y. Li et al.

and

B−s,t,2(ξ) ≤ D2�−2s2

∞∑j=J+1

m−2(β+1)( j+1)m2 js

× m−2Jsm2(β+1)(J+1)(1 + �2

1m−2( j−J))t

≤ D2(1 + �2

1

)t�−2s

2

∞∑j=J+1

m−2(β+1)( j−J)m2 jsm−2Js

≤ D2(1 + �2

1

)t�−2s

2 m−2Jsm2J(β+1)

∞∑j=J+1

m−2(β+1) jm2 js

= D2(1 + �2

1

)t�−2s

2 m−2((β+1)−s)∞∑

k=0

m−2((β+1)−s)k

= D2(1 + �2

1

)t�−2s

2

1 − m−2((β+1)−s).

Therefore, B−s,t(ξ) ≤ C, where C =||b(ξ)||2L∞(R)

�2s1

�2s2

(1+�21)

t

m−2t+2s−1 + D22t

1−m−2((β+1)−s) .

References

1. Belogay, E., Wang, Y.: Arbitrary smooth orthogonal nonseparable wavelets in Rs. SIAM J.

Math. Anal. 30(3), 678–697 (1999)2. Bownik, M.: A characterization of affine dual frames in L2(Rn). Appl. Comput. Harmon.

Anal. 8, 203–221 (2000)3. Cohen, A., Daubechies, I.: Nonseparable bidimensional wavelet bases. Rev. Mat. Iberoam. 9,

51–137 (1993)4. Daubechies, I., Han, B.: Pairs of dual wavelet frames from any two refinable functions. Constr.

Approx. 20, 325–352 (2004)5. Daubechies, I., Han, B., Ron, A., Shen, Z.: Framelets: MRA-based constructions of wavelet

frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)6. Ehler, M.: On multivariate compactly supported bi-frames. J. Fourier Anal. Appl. 13, 511–532

(2007)7. Ehler, M.: The Construction of Nonseparable Wavelet Bi-Frames and Associated Approxi-

mation Schemes. Logos Verlag, Berlin (2007)8. Ehler, M., Han, B.: Wavelet bi-frames with few generators from multivariate refinable func-

tions. Appl. Comput. Harmon. Anal. 25, 407-414 (2008)9. Han, B.: On dual wavelet tight frames. Appl. Comput. Harmon. Annal. 4, 380–413 (1997)

10. Han, B.: Approximation properties and construction of Hermite interpolants and biorthogonalmultiwavelets. J. Approx. Theory 110, 18–53 (2001)

11. Han, B.: Symmetry property and construction of wavelets with a general dilation matrix.Linear Algebra Appl. 353, 207–225 (2002)

12. Han, B.: Vector cascade algorithms and refinable function vectors in Sobolev spaces. J.Approx. Theory 124, 44–88 (2003)

13. Han, B.: Construction of wavelets and framelets by the projection method. Int. J. Appl. Math.Appl. 1, 1–40 (2008)

14. Han, B.: Dual multiwavelet frames with high balancing order and compact fast frame trans-form. Appl. Comput. Harmon. Anal. 26, 14–42 (2009)


15. Han, B.: Bivariate (Two-dimensional) wavelets. In: Meyers, R.A. (ed.) Encyclopedia ofComplexity and System Science, vol. 9, pp. 589–599 (2009)

16. Han, B.: The structure of balanced multivariate biorthogonal multiwavelets and dual multi-framelets. Math. Comput. 79, 917–951 (2010)

17. Han, B., Jia, Q.: Quincunx fundamental refinable functions and quincunx biorthogonalwavelets. Math. Comput. 237, 165–196 (2002)

18. Han, B., Mo, Q.: Multiwavelet frames from refinable function vectors. Adv. Comput. Math.18, 211–245 (2003)

19. Han, B., Shen, Z.: Characterization of Sobolev spaces of arbitrary smoothness using nonsta-tionary tight wavelet frames. Isr. J. Math. 172, 371–398 (2009)

20. Han, B., Shen, Z.: Dual wavelet frames and Riesz bases in Sobolev spaces. Constr. Approx.29(3), 369–406 (2009)

21. Han, B., Zhuang, X.: Analysis and construction of multivariate interpolating refinable functionvectors. Acta Appl. Math. 107, 143–171 (2009)

22. Heil, C., Colella, D.: Matrix refinement equations: existence and uniqueness. J. Fourier Anal.Appl. 2, 363–377 (1996)

23. Jia, R.: Bessel sequences in Sobolev spaces. Appl. Comput. Harmon. Anal. 20, 298–311 (2006)24. Li, Y., Yang, S.: Construction of nonseparable dual �-wavelet frames in L2(Rs). Appl. Math.

Comput. 215, 2082–2094 (2009)25. Li, Y., Yang, S.: Explicit construction of symmetric orthogonal wavelet frames in L2(Rs). J.

Approx. Theory 162, 891–909 (2010)26. Li, Y., Yang, S.: Multiwavelet sampling theorem in Sobolev spaces. Sci. China Math. 53(12),

3197–3214 (2010)27. Li, Y., Yang, S.: Dual multiwavelet frames with symmetry from two-direction refinable func-

tions. Bull. Iran. Math. Soc. 37(1), 199–214 (2011)28. Petukhov, A.: Symmetric framelets. Constr. Approx. 19, 309–328 (2003)29. Ron, A., Shen, Z.: Affine systems in L2(Rn): the analysis of the analysis operator. J. Funct.

Anal. 148, 380–413 (1997)30. Ron, A., Shen, Z.: Affine systems in L2(Rn): dual systems. J. Funct. Anal. 3, 617–637 (1997)31. Yang, S., Li, Y.: Two-direction refinable functions and two-direction wavelets with high ap-

proximation order and regularity. Sci. China Ser. A. 50(12), 1687–1704 (2007)32. Yang, S., Li, Y.: Two-direction refinable functions and two-direction wavelets with dilation

factor m. Appl. Math. Comput. 188(2), 1908–1920 (2007)33. Yang, S., Li, Y.: Construction of multiwavelets with high approximation order and symmetry.

Sci. China Ser. A. 52(8), 1607–1616 (2009)

Documents

Bessel multiwavelet sequences and dual multiframelets in