This paper introduces multiresolution analyses with composite dilations (AB-MRAs) and addresses frame multiresolution analyses with composite dilations in the setting
of reducing subspaces of L2(ℝn) (AB-RMRAs). We prove that an AB-MRA can induce an AB-RMRA on a given reducing subspace L2(S)∨. For a general expansive matrix, we obtain
the characterizations for a scaling function to generate an AB-RMRA, and the main theorems
generalize the classical results. Finally, some examples are provided to illustrate the general
theory.

1. Introduction

As well known, multiresolution analyses (MRAs) play a significant role in the construction of wavelets for L2(ℝ) [1, 2]. Up to now, different characterizations of the scaling function for an MRA have been presented. It is shown in [1] that a function φ∈L2(ℝ) is a generator for an MRA if and only if

∑k∈ℤ|φ̂(ξ+k)|2=1, a.e. ξ∈[-1/2,1/2];

limj→+∞|φ̂(2-jξ)|2=1, a.e. ξ∈ℝ;

there exists m0∈L2([0,1]) such that φ̂(2ξ)=m0(ξ)φ̂(ξ), a.e. ξ∈ℝ.

If condition (2) is replaced by ℝ=⋃j∈ℤ2jsupp(φ̂) or another condition that the function F(x,y)=(1/(y-x))∫xy|φ̂(ω)|2dω is dyadicaly away from zero at the origin, then the two different characterizations of the scaling functions for MRAs are obtained in [3, 4], respectively.

Similarly, under certain conditions, wavelet with composite dilations can be constructed by AB-MRAs which is the generalized definition of MRAs and permits the existence of fast implementation algorithm [5]. Given an n×n invertible matrix a, f∈L2(ℝn), and k∈ℤn, we define the dilation operator D and the shift operator Tk on L2(ℝn) by Daf(⋅)∶=|deta|1/2f(a⋅),Tkf(⋅)∶=f(⋅-k).
The affine system with composite dilations is defined by 𝒜AB(Ψ)={DaDbTkΨ:k∈ℤn,b∈B,a∈A} where Ψ={ψ1,ψ2,…,ψL}⊂L2(ℝn). By choosing Ψ, A, and B appropriately, we can make 𝒜AB(Ψ) an orthonormal basis or, more generally, a Parseval frame (PF) for L2(ℝn) [5–7]. In this case, Ψ is called an AB-multiwavelet or a PF AB-multiwavelet, respectively. Since not all of the AB-multiwavelet come from AB-MRAs, we only focus on the AB-multiwavelet which come from AB-MRAs. For convenience, we denote the operator DbTk by ℬ.

Before proceeding, we need some conventions. We denote by Tn=[-1/2,1/2]n the n-dimensional torus. For a Lebesgue measurable set E in ℝn, we denote by |E| its measure, denote by χE the characteristic function of E, and define E~∶=E+ℤn. An n×n matrix A is called an expansive matrix if it is an integer matrix with all its eigenvalues greater than 1 in the module. G denotes the set of all expansive matrices. We denote by GLn(ℤ) the set {a:a is an n×n integral matrix and |deta|≠0}, by SLñ(ℤ) the set {a:a is an n×n integral matrix and |deta|=1}, and by B the set of the subgroups of SLñ(ℤ), respectively. For a Lebesgue measurable function f, we define its support by
supp(f)∶={x∈Rn:f(x)≠0}.
The Fourier transform of f∈L1(ℝn)∩L2(ℝn) is defined by
f̂(ξ)∶=∫Rnf(x)e-i2π〈ξ,x〉dx
on ℝn, where 〈ξ,x〉 denotes the inner product in ℝn. Let S be a Lebesgue nonzero measurable set in ℝn. We denote by L2(S)∨ the closed subspace of L2(ℝn) of the form
L2(S)∨∶={f∈L2(Rn):supp(f̂)⊆S}.

Definition 1.1 (see [<xref ref-type="bibr" rid="B8">8</xref>, <xref ref-type="bibr" rid="B9">9</xref>]).

The sequence {xk,l}k,l in a separable Hilbert space H is called a semiorthogonal PF for H if {xk,l}k,l is a PF for H and satisfies 〈xk1,l1,xk2,l2〉=0 for any k1,k2∈Λ1, l1,l2∈Λ2, and k1≠k2, where Λ1, Λ2 are two countable index sets. In particular, if {xk,l}k,l is a semiorthogonal PF for span{xk,l}k,l¯, it is called a semiorthogonal sequence.

Definition 1.2 (see [<xref ref-type="bibr" rid="B4">4</xref>, <xref ref-type="bibr" rid="B10">10</xref>]).

A closed subspace X of L2(ℝn) is called a reducing subspace if DaX=X and TkX=X for any k∈ℤn, a∈G.

The following proposition provides a characterization of reducing subspace.

Proposition 1.3 (see [<xref ref-type="bibr" rid="B4">4</xref>, <xref ref-type="bibr" rid="B10">10</xref>]).

A closed subspace X of L2(ℝn) is a reducing subspace if and only if
X={f∈L2(Rn):supp(f̂)⊆S}
for some measurable set S⊆ℝn with ãS=S. So, to be specific, one denotes a reducing subspace by L2(S)∨ instead of X. In particular, L2(ℝn) is a reducing subspace of L2(ℝn).

Definition 1.4 (see [<xref ref-type="bibr" rid="B5">5</xref>–<xref ref-type="bibr" rid="B7">7</xref>]).

Let B⋉ℤn be a subgroup of the integral affine group SLñ(ℤ)⋉ℤn (the semidirect product of SLñ and ℤn). The closed subspace V of L2(ℝn) is called a B⋉ℤn invariant subspace if ℬV=V for any (b,k)∈B⋉ℤn.

Definition 1.5 (see [<xref ref-type="bibr" rid="B2">2</xref>–<xref ref-type="bibr" rid="B4">4</xref>]).

Let B be a countable subset of SLñ(Z) and A={ai:i∈ℤ} where a∈GLn(ℤ). We say that a sequence {Vj}j∈ℤ of closed subspaces of L2(ℝn) is an AB-MRA if the following holds:

V0 is a B⋉ℤn invariant space;

for each j∈ℤ, Vj⊂Vj+1, and Vj=DajV0;

⋃j∈ℤVj¯=L2(ℝn);

⋂j∈ℤVj={0};

there exists φ∈V0 such that ΦB={DbTkφ:b∈B,k∈ℤn} is a semiorthogonal PF for V0.

The space V0 is called an AB scaling space, and the function φ is an AB scaling function for V0 or a generator of AB-MRA.

Similarly, we say that a sequence {Vj}j∈ℤ is an AB-RMRA if it is an AB-MRA on L2(S)∨, that is, conditions (1), (2), (4), (5), and (3)′ ⋃j∈ℤVj¯=L2(S)∨ are satisfied.

The fact that an AB-MRA can induce an AB-RMRA will be demonstrated by the obvious following results.

Proposition 1.6.

Let I be a countable index set and P the orthogonal projection operator from a Hilbert space H to its proper subspace K. If Ψ={ψi:i∈I} is a Parseval frame on H, then P(Ψ)={P(ψi):i∈I} is a Parseval frame on K.

Proposition 1.7.

Let P be the orthogonal projection operator from a Hilbert space H to its reducing subspace K. Then P can commutate with the shift and dilation operators Tk and Da, respectively.

Theorem 1.8.

Suppose that {φ;Vj} is an AB-MRA, then {φ̃;Ṽj} is an AB-RMRA for L2(S)∨, where φ̃∶=Pφ, Ṽ0∶=span¯{DbTkφ̃:b∈B,k∈ℤn}, Ṽj∶=span¯{DajDbTkφ̃:b∈B,k∈ℤn}, and P is the orthogonal projection operator from L2(ℝn) to L2(S)∨.

The rest of this paper is organized as follows. Theorem 1.8 and some properties of an AB-RMRA will be proved in Section 2. In Section 3, the characterization of the generator for an AB-RMRA will be established, which is the main purpose of this paper. Finally, some examples are provided to illustrate the general theory.

2. Preliminaries

In this section, we will firstly prove Theorem 1.8 as follows.

We can easily prove that {DbTkφ̃:b∈B,k∈ℤn} is a Parseval frame sequence by Propositions 1.6 and 1.7. Naturally, {DbTkφ̃:b∈B,k∈ℤn} is a semi-Parseval frame for Ṽ0. Let φ∈V0⊂L2(ℝn). Thus Pφ=φ1. For any f∈Ṽ0, we have
f=∑b∑k〈f,DbTkφ̃〉DbTkφ̃=∑b∑k〈f,DbTkφ1〉DbTkφ1,Db′Tk′f=∑b∑k〈f,DbTkφ1〉φ1(b′bx-bk′-k)=∑b∑k〈f,DbTkφ1〉φ1(bx-k),
namely, Db′Tk′f∈Ṽ0. So Ṽ0 is a B⋉ℤn invariant space. On the other hand, f=∑b∑k〈f,DbTkPφ〉DbTkPφ=∑b∑k〈Pf,DbTkφ〉PDbTkφ=∑b∑k〈f,DbTkφ〉P∑b′∑k′〈DbTkφ,DaDbTkφ〉DaDb′Tk′φ=∑b∑kcb,kDaDbTkφ̃∈Ṽ1.
So Ṽ0⊂Ṽ1. Notice that Ṽi=DajṼ0. Then Ṽi⊂Ṽi+1. Thus, conditions (1), (2), and (5) in Definition 1.5 have been proved. However, condition (3)′ is the natural consequence of the later Lemma 3.1 in Section 3. Therefore, we complete the proof of Theorem 1.8.

Some properties of AB-RMRA, which were not discussed in [5–7], will be presented. The first one can be obtained obviously by the definition of AB-RMRA as follows.

Proposition 2.1.

Suppose that {Vj}j∈ℤ is an AB-RMRA. Then

for each j∈ℤ, {DajDbTkφ:b∈B,k∈ℤn} is a semiorthogonal PF on Vj;

V0 is a B⋉ℤn invariant subspace, while Vj is a B⋉a-jℤn invariant subspace.

Condition (5) of AB-RMRA can be characterized by the following proposition.

Proposition 2.2.

Let φ∈L2(S)∨. Then ΦB={DbTkφ:b∈B,k∈ℤn} is a semiorthogonal PF sequence if and only if

∑k∈ℤn|φ̂(ξ+k)|2=χF(ξ), a.e., where F={ξ∈Tn∩Ω:φ̂(ξ+k)≠0,k∈ℤn};

∑k∈ℤnφ̂(ξ+k)φ̂(b̃-1(ξ+k))¯=0, a.e. ξ∈Ω, for each b∈B and b≠In.

Proof.

Necessity. For any f(x)∈span¯{Tkφ:k∈ℤn}, we have
f(x)=∑b,k〈f,Bφ〉Bφ(x)=∑k〈f,Tkφ〉Tkφ(x)+∑b≠In∑k〈f,Bφ〉Bφ(x)=∑k〈f,Tkφ〉Tkφ(x).
By Theorem 1.6 in [1] and Theorem 7.2.3 in [8], conclusion (1) holds clearly. Using Parseval theorem, we can deduce
〈Tkφ,DbTk′φ〉=∫Ωφ(x-k)φ(bx-k′)¯dx=∫Ωφ̂(ξ)φ̂(b̃-1ξ)¯e-2πi(k-b̃-1k′)⋅ξdξ=∫Ω∑lφ̂(ξ+l)φ̂(b̃-1(ξ+l))¯e-2πi(k-k1)⋅ξdξ,
where k1=b̃-1k′. Note that for any k,k′∈ℤn, b≠b′∈B, 〈DbTkφ,Db′Tk′φ〉=0 if and only if for any b∈B and b≠In,
〈Tkφ,DbTk′φ〉=0.

Then, we have
∑lφ̂(ξ+l)φ̂(b̃-1(ξ+l))¯=0,a.e.ξ∈Ω.

Sufficiency.

By Theorem 7.2.3 in [8] and conclusion (1), {Tkφ:k∈ℤn} is a PF sequence. So is {DbTkφ:k∈ℤn} for any b∈B. It follows from (2.4), (2.5), and conclusion (2) that for any k,k′∈ℤn, b,b′∈B, and b≠b′, we get 〈DbTkφ,Db′Tk′φ〉=0. Thus, for any f(x)∈span¯{DbTkφ:b∈B,k∈ℤn}, there exists b∈B and fb(x)∈span¯{DbTkφ:k∈ℤn} such that
‖f‖2=∑b∈B‖fb‖2=∑b∈B∑k∈Zn|〈fb,DbTkφ〉|2=∑b∈B∑k∈Zn|〈f,DbTkφ〉|2.
By Theorem 1.6 in [1], the proof of Proposition 2.2 is completed.

Proposition 2.3.

Let {Vj}j∈ℤ be a sequence of closed subspace of L2(S)∨, where
Vj∶=span¯{DajDbTkφ:b∈B,k∈Zn}.
If conditions (1), (2), and (5) of AB-RMRA are satisfied, then one has the following.

There exists {cb,k}∈l2(B×ℤn) such that
φ(x)=∑b∑kcb,k|deta|1/2φ(bax-k).

There exists {hb(ξ)}b∈B⊆L∞(Tn) such that
φ̂(ξ)=∑bhb[(ba)̃-1ξ]φ̂[(ba)̃-1ξ],
for any b∈B, where hb(ξ)=|deta|-1/2∑kcb,ke-2πik·ξ.

Proof.

By conditions (2) and (5) of AB-RMRA and the fact that φ∈V0⊂V1, we obtain
φ(x)=∑b,k〈φ,DaBφ〉DaBφ(x)=∑b∑k∈Zncb,k|deta|1/2φ(bax-k),
where cb,k=〈φ,Daℬφ〉 and {cb,k}∈l2(B×ℤn). Therefore (2.9) holds. Taking Fourier transform on both sides of (2.9), we obtain (2.10), where for any b∈B, hb(ξ)=|deta|-1/2∑kcb,ke-2πik·ξ. In what follows, we will only prove {hb(ξ)}b∈B⊆L∞(Tn). Indeed, for a∈GLn(ℤ),ãℤn is a subgroup of ℤn and the quotient group ℤn/ãℤn has order M=|deta|. Thus, we can choose a complete set of representatives of ℤn/ãℤn, that is, the set {α0,α1,…,αM-1} so that each k∈ℤn can be uniquely expressed in the form k=ãk′+αi with k′∈ℤn, 0≤i≤M-1. For simplicity, we denote (ba)̃-1 and (b′a)̃-1 by b* and b1*, respectively. Then we have
∑k|φ̂(ξ+k)|2=∑k∈Zn|∑bhb[b*(ξ+k)]φ̂[b*(ξ+k)]|2=∑k∈Zn∑b,b1hb[b*(ξ+k)]hb1[b1*(ξ+k)]¯φ̂[b*(ξ+k)]φ̂[b1*(ξ+k)]¯=∑i=0M-1∑k′,b,b1hb[b*(ξ+αi)]hb1[b1*(ξ+αi)]¯φ̂[b*(ξ+αi)+b̃-1k′]×φ̂[b1*(ξ+αi)+b1̃-1k′]¯=∑i=0M-1∑b,b1hb[b*(ξ+αi)]hb1[b1*(ξ+αi)]¯×∑k′∈Znφ̂[b*(ξ+αi)+b̃-1k′]φ̂[b1*(ξ+αi)+b1̃-1k′]¯=∑i=0M-1∑b|hb[b*(ξ+αi)]|2∑k′∈Zn|φ̂[b*(ξ+αi)+b̃-1k′]|2,
where (2.12) is obtained by the periodicity of function sequence {hb(ξ)}b and (2.13) is proved by conclusion (2) in Proposition 2.2. In addition, using Proposition 2.2 again and (2.13) above, for any ξ∈F, we get ∑k∈ℤn|φ̂(ξ+k)|2=∑i=0M-1∑b|hb[b*ξ+b*αi)]|2=1. Then, for ξ∈ℝn, |hb(ξ)|2≤1, so hb(ξ)∈L∞(Tn). Therefore, the proof of Proposition 2.3 is completed.

3. Characterization of the Generator for an AB-RMRA

In this section, we will characterize the scaling function of AB-RMRA which will determine a multiresolution structure and AB-wavelets and the obtained results can be easily extended to the whole space L2(ℝn).

Lemma 3.1.

Let {Vj}j∈ℤ be a sequence of closed subspaces of L2(S)∨ and defined by (2.8). Assume that conditions (1), (2), and (5) in an AB-RMRA are satisfied. Then the following results are equivalent:

⋃j∈ℤVj¯=L2(S)∨;

limj→+∞∑b|φ̂[(baj)̃-1ξ]|2=1, a.e. ξ∈ℝn.

Proof.

Theorems 1.7 and 5.2 in [1] imply that for any f∈L2(ℝn), limj→+∞∥Pjf∥2=∥f∥2 is equivalent to ⋃j∈ℤVj¯=L2(ℝn). Thus, for any f∈L2(S)∨, limj→+∞∥Pjf∥2=∥f∥2 is equivalent to ⋃j∈ℤVj¯=L2(S)∨. Hence, we have to prove that limj→+∞∥Pjf∥2=∥f∥2 is equivalent to limj→+∞∑b|φ̂[(baj)̃-1ξ]|2=1, a.e. ξ∈Ω. First, we prove (1)⇒(2) For any f∈L2(S)∨, f=Pjf+Qjf, where Qj:L2(ℝ)→(Vj)⊥ is the orthogonal projection operator. Set f̂(ξ)=χTn∩S(ξ). Then, when j is large enough, we have
‖Pjf‖2=∑b∑k|〈f,DajDbTkφ〉|2=∑b∑k|∫Rn|detaj|-1/2f̂(ξ)φ̂[(baj)̃-1ξ]¯e2πik⋅(baj)̃-1ξdξ|2=|detaj|∑b∑k|∫(baj)̃-1Tnφ̂(ξ)¯e2πik⋅ξ|2dξ=|detaj|∑b∫Tn|χ(baj)̃-1Tn(ξ)φ̂(ξ)¯|2dξ=∑b∫Tn|φ̂[(baj)̃-1(ξ)]|2dξ=∫Tn∑b|φ̂[(baj)̃-1(ξ)]|2dξ.
Before proving the equivalence, we need to prove two assertions as follows:

∑b|φ̂[(baj)̃-1ξ]|2∈L1(Tn);

limj→+∞∑b|φ̂[(baj)̃-1ξ]|2 makes sense.

Since
∫Tn∑b|φ̂[(baj)̃-1ξ]|2dξ=∑b∫Tn|φ̂[(baj)̃-1ξ]|2dξ=|detaj|∑b∫(baj)̃-1Tn|φ̂(ξ)|2dξ=|detaj|∫⋃b(baj)̃-1Tn|φ̂(ξ)|2dξ≤|detaj|∫Rn|φ̂(ξ)|2dξ<∞,
it follows that (i) holds.

For (ii), we will only prove that {∑b|φ̂[(baj)̃-1ξ]|2}j∈ℤ is a monotonic bounded sequence when ξ(∈S) is fixed. Indeed, by the orthogonality, for each b≠b′∈B, we have supp(φ̂(b̃-1ξ))∩supp(φ̂(b′̃-1ξ))=∅. In addition, we deduce from (2.10) that, for any b∈B,
φ̂[(baj)̃-1ξ]=∑b′∈Bhb′[(b′a)̃-1(baj)̃-1ξ]φ̂[(b′a)̃-1(baj)̃-1ξ]=∑b′∈Bhb′(b′̃-1b̃-1aj+1̃-1ξ)φ̂(b′̃-1b̃-1aj+1̃-1ξ).

Set b*=b̃-1aj+1̃-1. Then by the orthogonality and Proposition 2.3, we obtain
∑b∈B|φ̂[(baj)̃-1ξ]|2=∑b∈B|∑b′∈Bhb′(b′̃-1b*ξ)φ̂(b′̃-1b*ξ)|2=∑b∈B∑b1∈Bhb1(b1̃-1b*ξ)φ̂(b1̃-1b*ξ)∑b2∈Bhb2(b2̃-1b*ξ)¯φ̂(b2̃-1b*ξ)¯=∑b∈B|hb((baj+1)̃-1ξ)φ̂((baj+1)̃-1ξ)|2≤∑b∈B|φ̂((baj+1)-1̃ξ)|2.
Hence, {∑b∈B|φ̂((baj)̃-1ξ)|2}j∈ℤ is a monotonic sequence when ξ is fixed. On the other hand, by the property of B, we deduce that DbTkφ(x)=φ(bx-k)=φ(b(x-b-1k))=φ(b(x-k′))=Tk′Dbφ(x). Note that {DbTkφ:k∈ℤn} is a PF for span¯{DbTkφ:k∈ℤn} for any b∈B. Then {Tk′Dbφ:k′∈ℤn} is a PF on span¯{DbTkφ:k∈ℤn}, which implies that there exists a set Mb such that ∑k|φ̂(b̃-1ξ+k)|2=χMb(ξ). By the orthogonality, ∑b∑k|φ̂(b̃-1ξ+k)|2=χ⋃bMb(ξ), consequently ⋃bMb=F. Hence, ∑b|φ̂(b̃-1ξ)|2≤1 holds for ξ∈S. So limj→+∞∑b∈B|φ̂((baj)̃-1ξ)|2 exists. Now we have proved the two assertions. By the Lebesgue dominant convergence theorem, we get limj→+∞∥Pjf∥2=∫Tnlimj→+∞∑b|φ̂[(baj)̃-1ξ]|2dξ=∥f∥2=|Tn|=1. Thus, limj→+∞∑b∈B|φ̂[(baj)̃-1ξ]|2=1, a.e. ξ∈ℝn.

Next, we prove (2)⇒(1). Let D be the class of all functions f∈L2(ℝn) such that f̂∈L∞(ℝn) and f̂ is compactly supported in ℝn∖{0}. If we can show that limj→+∞∥Pjf∥2=∥f∥2 for all f∈D, then, by Lemma 1.10 in [1], the proof is finished. Indeed, denoting (baj)̃-1 by b*, we have
‖Pjf‖2=∑b∑k|〈f,DajDbTkφ〉|2=∑b∑k|∫Rn|detaj|-1/2f̂(ξ)φ̂(b*ξ)¯e2πik⋅b*ξdξ|2=∑b∑k|∑m∫aj̃Tn|detaj|-1/2f̂(ξ+aj̃m)φ̂[b*(ξ+aj̃m)]¯e2πik⋅b*ξdξ|2=∑b∑k|∫aj̃Tn|detaj|-1/2∑mf̂(ξ+aj̃m)φ̂[b*(ξ+aj̃m)]¯e2πik⋅b*ξdξ|2=∑b∫aj̃Tn|∑mf̂(ξ+aj̃m)φ̂[b*(ξ+ajTm)]¯|2dξ=∑b∫aj̃Tn∑mf̂(ξ+aj̃m)φ̂[b*(ξ+aj̃m)]¯∑nf̂(ξ+aj̃n)¯φ̂[b*(ξ+aj̃n)]dξ=∑b∑m∫aj̃Tn+m∑pf̂(η)f̂(η+aj̃p)¯φ̂[b*η+b̃p]φ̂(b*η)¯dη=∑b∫Rn∑pf̂(η)f̂(η+aj̃p)¯φ̂(b*η+b̃p)φ̂(b*η)¯dη=∫Rn|f̂(η)|2∑b|φ̂(b̃-1aj̃η)|2dη+Rf,
where Rf=∑b∑p≠0∫ℝnf̂(η)f̂(η+aj̃p)¯φ̂(b*η+b̃p)φ̂(b*η)¯dη. Since f has compact support, when j is large enough, supp(f̂(η))∩supp(f̂(η+aj̃p))=∅, consequently, Rf=0. Thus, taking j→+∞ in (3.5), we obtain
limj→+∞‖Pjf‖2=limj→+∞∑b∑k|〈f,DajDbTkφ〉|2=limj→+∞∑b∫Rn|f̂(η)|2∑b|φ̂(b̃-1aj̃-1η)|2dη=∫Rn|f̂(η)|2limj→+∞∑b|φ̂(b̃-1aj̃-1η)|2dη=∫Rn|f̂(η)|2dη=‖f‖2.

Lemma 3.2.

Let a∈G, φ∈L2(ℝn) satisfy (2.10), and let {Vj}j∈ℤ be defined by (2.8). Then
⋃j∈ZVj¯=L2(S)∨,
where S=⋃j∈ℤ⋃b∈Bãjb̃supp(φ̂).

Proof.

By the definition of {Vj}j∈ℤ, we have DaVj=Vj+1 for any j∈ℤ. It follows that Da(⋃j∈ℤVj)=⋃j∈ℤVj. Note that Da is a unitary operator. Hence Da(⋃j∈ℤVj¯)=Da(⋃j∈ℤVj)¯=⋃j∈ℤVj¯. It is obvious to see that Vj={f:f̂=∑bFb[(baj)̃-1ξ]φ̂[(baj)̃-1ξ]}, where {Fb}b∈L2(Tn). Then, for any f∈V0, we have f̂(ξ)=∑bFb(b̃-1ξ)φ̂(b̃-1ξ), Fb∈L2(Tn). Notice that for any b∈B, φ̂(b̃-1ξ)=∑bhb(b̃-1ã-1ξ)φ̂(b̃-1ã-1ξ). Thus
f̂(ξ)=∑b1Fb1(b1-Tξ)∑b2hb2(b2̃-1ã-1ξ)φ̂(b2̃-1ã-1ξ)=∑b1∑b2Fb1(b1̃-1ξ)hb2(b2̃-1ã-1ξ)φ̂(b2̃-1ã-1ξ).
Put Hb2(b̃-1ã-1ξ)=∑b1Fb1(b1̃-1ξ)Mb2(b2̃-1ã-1ξ). Then, we obtain f̂(ξ)=∑b2Hb2(b2̃-1ã-1ξ)φ̂(b2̃-1ã-1ξ). Recalling that Fb(ξ)∈L2(Tn) and hb(ξ)∈L∞(Tn), we have Mb2(ξ)∈L2(Tn), so f(x)∈V1. Thus, for any f(x), we get V0⊂V1, so Vj⊂Vj+1. Therefore, for any f(x)∈⋃j∈ℤVj, we can choose jf>0 such that f(x)∈Vjf, that is, f(x)=∑b∑kcb,kφ(bajfx-k). Hence, for any m∈ℤn, we have
Tmf(x)=∑b∑kcb,kφ(bajf(x-m)-k)=∑b∑kcb,kφ(bajfx-bajfm-k)=∑b∑kcb,kφ(bajfx-k),
which implies Tm(⋃j∈ℤVj)⊆⋃j∈ℤVj. Thus, Tm(⋃j∈ℤVj)=⋃j∈ℤVj holds. Note that Tk is also a unitary operator. Hence, Tm(⋃j∈ℤVj¯)=Tm(⋃j∈ℤVj)¯=⋃j∈ℤVj¯. By Proposition 1.3, ⋃j∈ℤVj¯ is a reducing subspace. We notate ⋃j∈ℤVj¯=L2(S)∨. Next we have to prove S=⋃j∈ℤ⋃baj̃b̃supp(φ̂). By φ(baj·)∈Vj, we have supp(φ̂((baj)̃-1ξ))⊂S. Obviously, we will only prove that S∖⋃j∈ℤ⋃baj̃b̃supp(φ̂) is a zero measurable set. If S∖⋃j∈ℤ⋃baj̃b̃supp(φ̂) is a set with nonzero measure, then a contradiction is led. In fact, choosing a set M⊂S∖⋃j∈ℤ⋃baj̃b̃supp(φ̂) with 0<|M|<+∞ and using Plancherel theorem, we have
‖Pjf‖2=〈Pjf,Pjf〉=〈f,Pjf〉=〈f̂,f̂j′〉,
for any f∈L2(ℝn), where fj′∶=Pjf. In particular, setting f̂(ξ)=χM(ξ) and taking j→+∞ in (3.10), we get ∥Pjf∥2=〈f̂,f̂j′〉=0, but ∥f∥2=|M|>0, and this is a contradiction. Therefore, we complete the proof of Lemma 3.2.

By Lemmas 3.1 and 3.2 and Theorems 1.7 and 5.2 in [1], we characterize the density condition of AB-RMRA as follows.

Theorem 3.3.

Let a∈G and {Vj}j∈ℤ be defined by (2.8). If conditions (1), (2), and (5) of AB-RMRA are satisfied, then the following results are equivalent:

⋃j∈ℤVj¯=L2(S)∨;

limj→+∞∥Pjf∥2=∥f∥2, where Pj:L2(ℝn)→Vj denotes an orthogonal projection operator;

limj→+∞∑b∈B|φ̂((baj)̃-1ξ)|2=1, a.e. ξ∈S;

S=⋃j∈ℤ⋃bãjb̃suppφ̂.

It is well known that ⋂j∈ℤVj={0} can be deduced by the other conditions of MRA. Similarly, the condition ⋂j∈ℤVj={0} of AB-RMRA can be also deduced by the others. Thus, using Proposition 2.2 and Theorem 3.3, we can get the main theorem as follows.

Theorem 3.4.

Let a∈G, and let B be a subgroup of SLñ(ℤ) with aBa-1⊆B. Suppose that φ∈L2(S)∨ and {Vj}j∈ℤ is defined by (2.8). Then φ is a generator of AB-RMRA if and only if

there exists {hb(ξ)}b∈B⊂L∞(Tn) such that
φ̂(ξ)=∑bMb(b̃-1ã-1ξ)φ̂(b̃-1ã-1ξ);

∑k|φ̂(ξ+k)|2=χF(ξ), a.e., where F={ξ∈Tn∩S:φ̂(ξ+k)≠0,k∈ℤn};

for any b∈B and b≠In, ∑kφ̂(ξ+k)φ̂(b̃-1(ξ+k))¯=0, a.e. ξ∈S;

S=⋃j∈ℤ⋃bãjb̃supp(φ̂).

Remarks 1.

(1) Condition (4) in Theorem 3.4 can be replaced by any one of conditions in Theorem 3.3.

(2) If S=ℝn in Theorem 3.4, then we obtain the characterization of generator of AB-MRA.

(3) If n=1, a=(2), and B=(1) in Theorem 3.4, then we obtain the characterization of {Vj}j∈ℤ as a generator of MRA on L2(ℝ).

Corollary 3.5.

Let a∈G, and let B be a subgroup of SLñ(ℤ) with aBa-1⊆B. Suppose that E is a bounded nonzero measurable set satisfying ã-1E⊂E with S=⋃j∈ℤ⋃bãjb̃E. Define φ̂(ξ)=χE(ξ) and F={ξ∈Tn∩Ω:φ̂(ξ+k)≠0,k∈ℤn}. Then φ generates an AB-RMRA if and only if

F=⋃k∈ℤn(E+k) with |E∩(E+k)|=0, |bTE∩b′TE|=0(b≠b′);

there exists b∈B such that (ba)̃-1E⊂E and [(ba)̃-1E]~∩E=(ba)̃-1E;

Tkχ(ba)̃-1E(ξ)=χ(ba)̃-1E(ξ) on (ba)̃-1E∩[(ba)̃-1E+k]≠∅ for k∈ℤn∖{0}.

4. Some Examples

Three examples are provided to illustrate the general theory in this section.

Example 4.1.

Let φ̂(ξ)=χI(ξ), where I=I+∪I-, I-={ξ∈ℝ2∣-ξ∈I+}, and I+ is a triangle region with vertices (0,0),(α,0),(α,α). Let a=(2002), B={(1i01):i∈ℤ}, S0={(ξ1,ξ2)∈ℝ2:0≤ξ1≤α,ξ2∈ℝ}, Sj=(aj̃)S0. Define Vj=L2(Sj)∨. Then by Corollary 3.5, we get the following:

when α≤1, φ(x) is a generator for an AB-RMRA;

when α>1, φ(x) is not a generator.

Example 4.2.

Given k∈ℤ, let E=[0,β0]2∪(⋃j=-k-1[2jα,2jβ]2), where 0<β0≤2-kα, α<β≤min{2α,2k+1β0,2}. If φ̂=χE(·), then φ generates an AB-RMRA, where S={(x,y):x≥0,y≥0}, a=(0220).

Example 4.3.

Let E=[0,1/4]2∪[3/8,3/8+ε]2, where 0<ε≤1/16. Assume that φ̂=χE(·). Then φ generates an AB-RMRA, where S={(x,y):x≥0,y≥0}, a=(2002).

Acknowledgments

The paper was supported by the National Natural Science Foundation of China (no. 61071189) and the Innovation Scientists and Technicians Troop Construction Projects of Henan Province of China (no. 084100510012).

HernándezE.WeissG.WalnutD. F.LianQ.-F.LiY.-Z.Reducing subspace frame multiresolution analysis and frame waveletsZhangH. Y.ShiX. L.Characterization of generators for multiresolution analysisGuoK.LabateD.LimW.-Q.WeissG.WilsonE.Wavelets with composite dilations and their MRA propertiesGuoK.LabateD.LimW.-Q.WeissG.WilsonE.Wavelets with composite dilationsGuoK.LabateD.LimW.-Q.WeissG.WilsonE.The theory of wavelets with composite dilationsChristensenO.LiD. F.XueM. Z.DaiX.DiaoY.GuQ.HanD.The existence of subspace wavelet sets