# Nonhomogeneous Wavelet Dual Frames and Extension Principles in Reducing Subspaces

Authors: Jianping Zhang; Huifang Jia
Journal: Journal of Mathematics
 (2020)

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## Abstract

It can be seen from the literature that nonhomogeneous wavelet frames are much simpler to characterize and construct than homogeneous ones. In this work, we address such problems in reducing subspaces ofL2ℝd. A characterization of nonhomogeneous wavelet dual frames is obtained, and by using the characterization, an MOEP and an MEP are derived under general assumptions for such wavelet dual frames.


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## Body

## 1. Introduction

In view of the great design freedom and the potential applications in signal processing and many other fields, wavelet frames have been extensively investigated by many researchers (see [1–9] for details). In particular, the homogeneous wavelet dual frames (or called affine dual frames) in L2ℝd were original characterized by Han [10] and then characterized by Bownik [11], some of their variations can be found in [12–14]. Mixed extension principles (MEP) give us an important method to construct homogeneous wavelet dual frames from refinable functions, which were proposed by Ron and Shen [15, 16]. When the systems generated by the dilation and integer translation of two functions with refinable structure cannot form frames for their closed linear span, respectively, except two Bessel sequences, construction of homogeneous wavelet dual frames cannot be carried out like the multiresolution analysis (MRA) of Mallat. In such case, the mixed extension principles give an ideal answer to the problem and provide an effective design strategy for stable wavelet filters. Subsequently, mixed extension principles were developed by Daubechies et al. [17] in the form of mixed oblique extension principles (MOEP), which presented a more general method for constructing wavelet dual frames. From then on, the study of MEP and MOEP has interested many researchers [12, 13, 18–25]. Nonhomogeneous wavelet frames have natural connections with refinable structures and filter banks; for this type of wavelet frames, Han in [26–28] and Romero et al. in [29] extensively studied them both in theory and application. Also, notice that all of the above works are focused on the whole space L2ℝd or L2ℝ. In this work, for generality, we address nonhomogeneous wavelet dual frames and extension principles in reducing subspaces of L2ℝd. A characterization of nonhomogeneous wavelet dual frames is obtained, and by using the characterization, an MOEP and an MEP are derived under general assumptions for such wavelet dual frames.We begin with some notations and notions. We useℤ to represent the set of integers, ℕ to represent the set of positive integers, and ℕ0=ℕ∪0. Let d∈ℕ. We use Zd=0,1d to represent a d-dimensional unit torus, and, given a set E on ℝd, E to represent the Lebesgue measure of E, and χE to represent the characteristic function on E. A mapping τ:ℝd↦Td is defined by(1)τy=y−kfory∈Td+kandk∈ℤd.For a functionf∈L1ℝd∩L2ℝd, its Fourier transform f^ is defined by(2)f^·=∫ℝdfxe−2πix,·dx,and is naturally extended to L2ℝd, where ·,· is the usual inner product on ℝd. Similarly, for a function f∈L1ℝd∩L2ℝd, its inverse Fourier transform fˇ is defined by(3)fˇ·=∫ℝdfxe2πix,·dx,and is naturally extended to L2ℝd. The spectrum of a function f is defined to be(4)σf=ξ∈Td:∑k∈ℤdf^ξ+k2>0,for f∈L2ℝd, and write δ as the Dirac sequences such that δ0,0=1 and δ0,β=0 for 0≠β∈ℝd. We use A∗ to represent the conjugate transpose for a matrix A and ΓA∗ to represent the full set of A∗−1ℤd/ℤd containing the 0, i.e., a set of representatives distinct cosets of A∗−1ℤd/ℤd containing the 0. A d×d matrix A is said to be expansive if the modulus of all its eigenvalues are greater than 1. In many literature studies of wavelet frame theory, A is required to be an expansive integer matrix include A=2Id (Id is the identity matrix of order d). The shift operator Tk with k∈ℤd and the dilation operator D related to A on L2ℝd are, respectively, defined by(5)Tkf·=f·−kandDf·=detA1/2fA·forf∈L2ℝd.For a finite setF⊂L2ℝd and h∈L2ℝd, the homogeneous wavelet system XF and the nonhomogeneous wavelet system Xh,F are, respectively, defined to be(6)XF=fj,k=DjTkf:j∈ℤ,k∈ℤd,f∈F,(7)Xh,F=h0,k=Tkh:k∈ℤd∪fj,k=DjTkf:j∈ℕ0,k∈ℤd,f∈F.LetM be a closed subspace of L2ℝd. Let ϕ, ϕ˜∈M, and two finite subsets Ψ=ψ1,ψ2,…,ψL and Ψ˜=ψ˜1,ψ˜2,…,ψ˜L of M. If there are A,B>0 such that(8)Af2≤∑l=1L∑j∈ℤ∑k∈ℤdf,ψl,j,k2≤Bf2,∀f∈M,then XΨ is said to be a homogeneous wavelet frame (HWF) for M; herein A and B are called frame bounds. In particular, if A=B in (8), then XΨ is said to be a tight wavelet frames. If only the inequality on the right side of (8) is valid, then XΨ is said to be a Bessel sequence in M, herein B is called a Bessel bound. In addition, if the homogeneous wavelet systems XΨ and XΨ˜ are both Bessel sequences in M and the identity(9)f=∑l=1L∑j∈ℤ∑k∈ℤdf,ψ˜l,j,kψl,j,kforf∈M,holds in L2-sense, then XΨ,XΨ˜ is said to be a pair of homogeneous wavelet dual frames (HWDF) for M. Similarly, if there are A,B>0 such that(10)Af2≤∑k∈ℤdf,ϕ0,k2+∑l=1L∑j=0∞∑k∈ℤdf,ψl,j,k2≤Bf2,∀f∈M,then Xϕ,Ψ is said to be a nonhomogeneous wavelet frames (NWF) for M, herein A and B are called frame bounds. In particular, if A=B in (10), then Xϕ,Ψ is said to be a tight wavelet frames. If only the inequality on the right side of (10) is valid, then Xϕ,Ψ is said to be a Bessel sequence in M, herein B is called a Bessel bound. In addition, if the nonhomogeneous wavelet systems Xϕ,Ψ and Xϕ˜,Ψ˜ are both Bessel sequences in M and the identity(11)f=∑k∈ℤdf,ϕ˜0,kϕ0,k+∑l=1L∑j=0∞∑k∈ℤdf,ψ˜l,j,kψl,j,kforf∈M,holds in L2-sense, then(12)Xϕ,Ψ,Xϕ˜,Ψ˜,is said to be a pair of nonhomogeneous wavelet dual frames (NWDF) for M.Since the nonhomogeneous wavelet frames have many desired properties, which can be studied in theory and application when they can be linked to homogeneous ones. Following this idea, Han [26] and Atreas et al. [18, 19] have discussed the relations between homogeneous wavelet dual frames and nonhomogeneous ones, and their connection with the refinable structures.Definition 1.
Given a nonzero closed subspaceX of L2ℝd. If DX=X and TkX=X hold true for all k∈ℤd, then X is said to be a reducing subspace.
A characterization of reducing subspace in Fourier domain was gained by Dai et al. (Theorem1 in [30]), and it reads as follows.Proposition 1.
LetA be a d×d expansive matrix, and X≠0 a closed subspace of L2ℝd. X is a reducing subspace of L2ℝd if and only if X=FL2Ω for some Ω⊂ℝd and Ω≠0 and satisfy Ω=A∗Ω, herein(13)FL2Ω≔f∈L2ℝd:suppf^⊂Ω,and suppf^=ξ∈ℝd:f^ξ≠0.
According to Proposition1, we often use FL2Ω to represent a reducing subspace of L2ℝd rather than X. In particular, FL2ℝd=L2ℝd, and it is a reducing subspace of L2ℝd for a given expansive matrix, and Hardy space FL20,∞ is a reducing subspace of L2ℝ for a given expansive factor greater than 1. We refer the interested readers to [10, 31–34] for some related works on HWDFs in FL2Ω for the details.Definition 2.
LetA be a d×d expansive matrix. A function κ:ℤd⟶ℕ0 is defined to be(14)κn=supj≥0:A∗−jn∈ℤd,and set κ0=+∞.
Li and Zhang (Lemma 2.3 in [31]) have the following characterization for HWDF in FL2Ω.Proposition 2.
LetFL2Ω be a reducing subspace of L2ℝd, and two finite subsets Ψ=ψ1,ψ2,…,ψL and Ψ˜=ψ˜1,ψ˜2,…,ψ˜L of FL2Ω. Suppose that XΨ and XΨ˜ are both Bessel sequences in FL2Ω. Then, XΨ,XΨ˜ is a pair of HWDF in FL2Ω if and only if, for each n∈ℤd, we have(15)∑l=1L∑j=−∞κnψl^A∗−j·ψ˜^lA∗−j·+n¯=δ0,nχΩ·a.e.onℝd.
Proposition2 was also proved in [32] without any decay assumptions on the elements of Ψ and Ψ˜. Some variations of Proposition 2 can be found in [35–37].
The rest of this work is arranged as below. In Section2, we provide some necessary lemmas that will be applied in the following section. In Section 3, we give a characterization of NWDF in FL2Ω by a pair of equations, and using this characterization we derive an MOEP and an MEP under general assumptions for such wavelet dual frames.


## 2. Some Auxiliary Lemmas

By Definition2, it is easy to check the following lemmas.Lemma 1.
A∗−κn−1n:0≠n∈ℤd=∪γ∈ΓA∗∖0ℤd+γ.Lemma 2.
Forϕ,ϕ˜∈L2ℝd, we have(16)σϕ∩τσϕ˜−γ=ξ∈Td:ϕ^ξ+kϕ˜^ξ+γ+n¯≠0for somek,n∈ℤd,for γ∈ΓA∗.Proof.
For a.e.ξ∈σϕ∩τσϕ˜−γ with γ∈ΓA∗, there exist kξ,mξ∈ℤd and η∈σϕ˜ such that(17)ϕ^ξ+kξ≠0,ξ=η−γ+mξ.
Sinceη∈σϕ˜, there exists kη∈ℤd such that(18)ϕ˜^η+kη≠0,or equivalently,(19)ϕ˜^ξ+kξ+γ+nξ≠0,where nξ=kη−mξ−kξ. It follows that ϕ^ξ+kξϕ˜^ξ+γ+kξ+nξ¯≠0, and thus(20)σϕ∩τσϕ˜−γ⊂ξ∈Td:ϕ^ξ+kϕ˜^ξ+γ+n¯≠0for somek,n∈ℤd.
Now, let us prove the converse inclusion. Supposeξ∈Td satisfies that(21)ϕ^ξ+kϕ˜^ξ+γ+n¯≠0,for some γ∈ΓA∗ and k,n∈ℤd. Then, there exists n0∈ℤd such that η=ξ+γ−n0∈σϕ˜. This implies that ξ=η+n0−γ, and thus ξ∈τσϕ˜−γ. The converse inclusion therefore follows. This proof has been completed.Lemma 3.
(see Lemma 2.4 in [10]). Let ϕ∈L2ℝd. If Tkϕ:k∈ℤd is a Bessel sequence in L2ℝd and bound is B, then we have(22)∑k∈ℤdϕ^·+k2≤Ba.e.onZd.Lemma 4.
Letj∈ℤ and ϕ∈L2ℝd. Then, for f∈L2ℝd and k∈ℤd, the kth Fourier coefficient of ∑l∈ℤddetAj/2f^Aj∗·+lϕ^·+l¯ is f,ϕj,k. In particular,(23)∑l∈ℤddetAj/2f^A∗j·+lϕ^·+l¯=∑k∈ℤdf,ϕj,ke2πik,·,if ϕj,k:k∈ℤd is a Bessel sequence.Proof.
Sincef,ϕ∈L2ℝd, we have f^A∗j·ϕ^·¯∈L1ℝd, and thus(24)∫Zd∑l∈ℤddetAj/2f^A∗jξ+lϕ^ξ+l¯e−2πik,ξdξ=detAj/2∫ℝdf^A∗jξϕ^ξ¯e−2πik,ξdξ=detA−j/2∫ℝdf^ξϕ^A∗−jξ¯e−2πik,A∗−jξdξ=∫ℝdf^ξϕj,k·∧ξ¯dξ=f,ϕj,k,by the Plancherel theorem.
Ifϕj,k:k∈ℤd is a Bessel sequence, then f,ϕj,kk∈ℤd∈ℓ2ℤd, and thus, we obtain (23) directly from (24). This proof has been completed.Lemma 5.
IfXϕ,Ψ is a Bessel sequence in L2ℝd and bound is B. Then, we have(25)ϕ^·2+∑l=1L∑j=0∞ψ^lA∗−j·2≤Ba.e.onℝd.Proof.
By the Bessel sequence assumption ofXϕ,Ψ, we have(26)∑k∈ℤdf,ϕ0,k2+∑l=1L∑j=0∞∑k∈ℤdf,ψl,j,k2≤Bf2,for∀f∈L2ℝd.
By Lemma4, we have(27)∑k∈ℤdf,ϕ0,k2+∑l=1L∑j=0∞∑k∈ℤdf,ψl,j,k2=∫Zd∑k∈ℤdf^ξ+kϕ^ξ+k¯2dξ+∑l=1L∑j=0∞detAj∫Zd∑k∈ℤdf^A∗jξ+kψ^lξ+k¯2dξ,and the above equality can be written as(28)∑k∈ℤdf,ϕ0,k2+∑l=1L∑j=0∞∑k∈ℤdf,ψl,j,k2=∫Td∑k∈ℤdf^ξ+kϕ^ξ+k¯∑k∈ℤdf^ξ+k¯ϕ^ξ+kdξ+∑l=1L∑j=0∞detAj∫Td∑k∈ℤdf^A∗jξ+kψ^lξ+k¯∑k∈ℤdf^A∗jξ+k¯ψ^lξ+kdξ,=∫ℝdf^ξ¯ϕ^ξ∑k∈ℤdf^ξ+kϕ^ξ+k¯dξ+∑l=1L∑j=0∞detAj∫ℝdf^A∗jξ¯ψ^lξ∑k∈ℤdf^A∗jξ+kψ^lξ+k¯dξ.
ReplacingA∗jξ by ξ in the second part of (28) and then separating the series into two parts: k=0 and k≠0, we obtain(29)∑k∈ℤdf,ϕ0,k2+∑l=1L∑j=0∞∑k∈ℤdf,ψl,j,k2=∫ℝdf^ξ¯ϕ^ξ∑k∈ℤdf^ξ+kϕ^ξ+k¯dξ+∑l=1L∑j=0∞∫ℝdf^ξ¯ψ^lA∗−jξ∑k∈ℤdf^ξ+A∗jkψ^lA∗−jξ+k¯dξ=∫ℝdf^ξ2ϕ^ξ2+∑l=1L∑j=0∞ψ^lA∗−jξ2dξ+∫ℝdf^ξ¯ϕ^ξ∑0≠k∈ℤdf^ξ+kϕ^ξ+k¯dξ+∫ℝd∑l=1L∑j=0∞f^ξ¯ψ^lA∗−jξ∑0≠k∈ℤdf^ξ+A∗jkψ^lA∗−jξ+k¯dξ=∫ℝdf^ξ2ϕ^ξ2+∑l=1L∑j=0∞ψ^lA∗−jξ2dξ+∫ℝdf^ξ¯∑0≠k∈ℤdf^ξ+kϕ^ξϕ^ξ+k¯+∑l=1L∑j=0κkψ^lA∗−jξψ^lA∗−jξ+k¯dξ,by the definition of κ in (14).
Suppose that (25) is not valid, then there exists E⊂ℝd with E>0 such that ϕ^·2+∑l=1L∑j=0∞ψ^lA∗−j·2>B on E. As a result, we have ϕ^·2+∑l=1L∑j=0∞ψ^lA∗−j·2>B on some E′=E∩0,1d+k0 with E′>0 and k0∈ℤd. Take f such that f^=χE′ in (29); then, we obtain(30)∑k∈ℤdf,ϕ0,k2+∑l=1L∑j=0∞∑k∈ℤdf,ψl,j,k2>BE′=Bf^2=Bf2,by the Plancherel theorem, which contradict with (26). Therefore, we prove that (25) holds. This proof has been completed.Lemma 6.
(see Lemma 3.2 in [36]). Let V be dense in a Hilbert space ℋ. If fii∈I and gii∈I are both Bessel sequences in ℋ, then fii∈I,gii∈I is a pair of dual frames for ℋ if and only if(31)f,g=∑i∈If,gifi,g,for f,g∈V.Lemma 7.
(see Lemma 2.5 in [37]). Let ϕ, ϕ˜, ψ, ψ˜∈L2ℝd, we have(32)∑k∈ℤdf,ϕ˜0,kϕ0,k,g+∑j=0∞∑k∈ℤdf,ψ˜j,kψj,k,g=S1f,g+S2f,g,for f, g∈D, where(33)S1f,g=∫ℝdf^ξg^ξ¯ϕ^ξϕ˜^ξ¯+∑j=0∞ψ^A∗−jξψ˜^A∗−jξ¯dξ,S2f,g=∫ℝdg^ξ¯∑0≠k∈ℤdf^ξ+kϕ^ξϕ˜^ξ+k¯+∑j=0κkψ^A∗−jξψ˜^A∗−jξ+k¯dξ,D=f∈L2ℝd:f^∈L∞ℝdand suppf^⊂ξ∈ℝd:C−1≤ξ≤Cfor someC>1.


## 3. Main Conclusions

The following theorem generalizes Proposition2. A characterization of NWDF of form (12) in FL2Ω is obtained.Theorem 1.
LetFL2Ω be a reducing subspace of L2Rd. Let ϕ, ϕ˜∈FL2Ω, and two finite subsets Ψ=ψ1,ψ2,…,ψL and Ψ˜=ψ˜1,ψ˜2,…,ψ˜L of FL2Ω. Suppose that Xϕ,Ψ and Xϕ˜,Ψ˜ are both Bessel sequences in FL2Ω. Then, Xϕ,Ψ,Xϕ˜,Ψ˜ is a pair of NWDF for FL2Ω if and only if, for each n∈ℤd, we have(34)ϕ^·ϕ˜^·+n¯+∑l=1L∑j=0κnψ^lA∗−j·ψ˜^lA∗−j·+n¯=δ0,nχΩ·a.e.onℝd.Proof.
LetD be defined as in (33). Since D∩FL2Ω is dense in FL2Ω, we can see from Lemma 6 that Xϕ,Ψ,Xϕ˜,Ψ˜ is a pair of NWDF for FL2Ω if and only if(35)∑k∈ℤdf^χΩ∨,ϕ˜0,kϕ0,k,g^χΩ∨+∑l=1L∑j=0∞∑k∈ℤdf^χΩ∨,ψ˜l,j,kψl,j,k,g^χΩ∨=f^χΩ∨,g^χΩ∨,for f,g∈D. By Lemma 7, (35) can be written as(36)I1f,g+I2f,g=∫ℝdf^ξg^ξ¯χΩξdξ,forf,g∈D,where(37)I1f,g=∫ℝdf^ξg^ξ¯χΩξϕ^ξϕ˜^ξ¯+∑l=1L∑j=0∞ψ^lA∗−jξψ˜^lA∗−jξ¯dξ,I2f,g=∫ℝdg^ξ¯χΩξ∑0≠k∈ℤdf^χΩξ+kϕ^ξϕ˜^ξ+k¯+∑l=1L∑j=0κkψ^lA∗−jξψ˜^lA∗−jξ+k¯dξ.
Clearly, (34) implies (36). Now, we need to show that (36) also implies (34) to complete the proof.
Suppose (36) holds. Since ϕ, ϕ˜∈FL2Ω and Ψ, Ψ˜⊂FL2Ω, (34) always holds on ℝd∖Ω, so we only need to prove that (34) holds a.e. on Ω. By applying Lemma 5 and the Cauchy–Schwarz inequality, we know the series ϕ^·ϕ˜^·+n¯+∑l=1L∑j=0κnψ^lA∗−j·ψ˜^lA∗−j·+n¯ with n∈ℤd converges absolutely a.e. on ℝd, as a result of this, belongs to L∞ℝd. Therefore, almost every point in ℝd is a Lebesgue point. Let ξ0∈Ω be such a point. For 0<ϵ<1/2, take f and g such that(38)f^·=χBξ0,ϵ·Bξ0,ϵ=g^·¯in ((36), where Bξ0,ϵ=ξ∈ℝd:ξ−ξ0<ϵ. Then, we have(39)1Bξ0,ϵ∫Bξ0,ϵχΩξϕ^ξϕ˜^ξ¯+∑l=1L∑j=0∞ψ^lA∗−jξψ˜^lA∗−jξ¯dξ=1Bξ0,ϵ∫Bξ0,ϵχΩξdξ,and letting ϵ⟶0, we obtain(40)ϕ^ξ0ϕ˜^ξ0¯+∑l=1L∑j=0∞ψ^lA∗−jξ0ψ˜^lA∗−jξ0¯=1,ξ0∈Ω.
For0≠k0∈ℤd, take f and g such that(41)f^·+k0=χBξ0,ϵ·Bξ0,ϵ=g^·¯,in (36), where 0<ϵ<1/2. Then, we have(42)1Bξ0,ϵ∫Bξ0,ϵχΩξϕ^ξϕ˜^ξ+k0¯+∑l=1L∑j=0κk0ψ^lA∗−jξψ˜^lA∗−jξ+k0¯dξ=0,and letting ϵ⟶0, we obtain(43)ϕ^ξ0ϕ˜^ξ0+k0¯+∑l=1L∑j=0κk0ψ^lA∗−jξ0ψ˜^lA∗−jξ0+k0¯=0,ξ0∈Ω.
By the arbitrariness ofξ0 and 0≠k0∈ℤd, then we obtain (34). This proof has been completed.
Next, by using Theorem1, we derive an MOEP and an MEP for constructing NWDF in FL2Ω under the following setup: ϕ,ϕ˜∈FL2Ω, and satisfy.Assumption 1.
limj⟶∞ϕ^A∗−j·ϕ˜^A∗−j·¯=1 a.e. on Ω.Assumption 2.
ϕ and ϕ˜ are both A-refinable functions, that is, there are two ℤd-periodic measurable functions H0 and H˜0 such that(44)ϕ^A∗·=H0·ϕ^·,ϕ˜^A∗·=H˜0·ϕ˜^·a.e.onℝd.
Under this setup, defineΨ=ψ1,ψ2,…,ψL and Ψ˜=ψ˜1,ψ˜2,…,ψ˜L⊂FL2Ω by(45)ψ^lA∗·=Hl·ϕ^·,ψ˜^lA∗·=H˜l·ϕ˜^·,with 1≤l≤L, and Hl and H˜l being ℤd-periodic measurable functions.Theorem 2.
LetFL2Ω be a reducing subspace of L2Rd. Let ϕ,ϕ˜∈FL2Ω satisfy Assumptions 1 and 2, and two finite subsets Ψ=ψ1,ψ2,…,ψL and Ψ˜=ψ˜1,ψ˜2,…,ψ˜L of FL2Ω defined as in (45). Suppose that Xϕ,Ψ and Xϕ˜,Ψ˜ are both Bessel sequences in FL2Ω. Also, suppose there exists a ℤd-periodic function Θ∈L∞Td, and define η by η^·=Θ·ϕ^· a.e. on ℝd. Then, Xη,Ψ,Xϕ˜,Ψ˜ is a pair of NWDF for FL2Ω if and only if(46)limj⟶∞ΘA∗−j·=1,a.e.onΩ,(47)ΘA∗·H0·H˜0·+γ¯+∑l=1LHl·H˜l·+γ¯=Θ·δ0,γa.e.onσϕ∩τσϕ˜−γwithγ∈ΓA∗.Proof.
SinceXϕ,Ψ is a Bessel sequence in FL2Ω, especially, we have ϕ0,k:k∈ℤd and ψl,j,k:j∈ℕ0,k∈ℤd,1≤l≤L are both Bessel sequences in FL2Ω, respectively. By the definition of η and Lemma 3, we know η0,k:k∈ℤd is a Bessel sequence in FL2Ω. As a result, Xη,Ψ is also a Bessel sequence in FL2Ω. So, by Theorem 1, Xη,Ψ,Xϕ˜,Ψ˜ is a pair of NWDF in FL2Ω if and only if, for every n∈ℤd, we have(48)Θ·ϕ^·ϕ˜^·+n¯+∑l=1L∑j=0κnψ^lA∗−j·ψ˜^lA∗−j·+n¯=δ0,nχΩ·a.e.onℝd.
Next, we will show that (48) is equivalent to (46) and (47).
First, we suppose that (46) and (47) hold. For 0≠n∈ℤd, by Assumption 2 and (45) and the ℤd-periodicity of H˜0 and H˜l,1≤l≤L, we have(49)Θξϕ^ξϕ˜^ξ+n¯+∑l=1L∑j=0κnψ^lA∗−jξψ˜^lA∗−jξ+n¯=ϕ^A∗−1ξϕ˜^A∗−1ξ+n¯ΘξH0A∗−1ξH˜0A∗−1ξ¯+∑l=1LHlA∗−1ξH˜lA∗−1ξ¯+∑l=1L∑j=1κnψ^lA∗−jξψ˜^lA∗−jξ+n¯,due to A∗−jn∈ℤd for j=1,2,…,κn. By using (47), (49) can be written as(50)Θξϕ^ξϕ˜^ξ+n¯+∑l=1L∑j=0κnψ^lA∗−jξψ˜^lA∗−jξ+n¯=ΘA∗−1ξϕ^A∗−1ξϕ˜^A∗−1ξ+n¯+∑l=1L∑j=1κnψ^lA∗−jξψ˜^lA∗−jξ+n¯=ϕ^A∗−2ξϕ˜^A∗−2ξ+n¯ΘA∗−1ξH0A∗−2ξH˜0A∗−2ξ¯+∑l=1LHlA∗−2ξH˜lA∗−2ξ¯+∑l=1L∑j=2κnψ^lA∗−jξψ˜^lA∗−jξ+n¯,and the last equality is deduced in the same procedure as (49). Continuing this way κn+1 times, we conclude that(51)Θξϕ^ξϕ^ξ+n¯+∑l=1L∑j=0κnψ^lA∗−jξψ˜^lA∗−jξ+n¯=ϕ^A∗−κn−1ξϕ˜^A∗−κn−1ξ+n¯×ΘA∗−κnξH0A∗−κn−1ξH˜0A∗−κn−1ξ+n¯+∑l=1LHlA∗−κn−1ξH˜lA∗−κn−1ξ+n¯.
Observe thatA∗−κn−1n=n0+γ for some n0∈ℤd and γ∈ΓA∗∖0 by Lemma 1, and (51) therefore follows(52)Θξϕ^ξϕ˜^ξ+n¯+∑l=1L∑j=0κnψ^lA∗−jξψ˜^lA∗−jξ+n¯=ϕ^A∗−κn−1ξϕ˜^A∗−κn−1ξ+n0+γ¯×ΘA∗−κnξH0A∗−κn−1ξH˜0A∗−κn−1ξ+γ¯+∑l=1LHlA∗−κn−1ξH˜lA∗−κn−1ξ+γ¯=0by (47) and the ℤd-periodicity of H˜0 and H˜l,1≤l≤L.
Similarly, forn=0, we have κ0=+∞, and then(53)Θξϕ^ξϕ˜^ξ¯+∑l=1L∑j=0∞ψ^lA∗−jξψ˜^lA∗−jξ¯=ϕ^A∗−1ξϕ˜^A∗−1ξ¯ΘξH0A∗−1ξH˜0A∗−1ξ¯+∑l=1LHlA∗−1ξH˜lA∗−1ξ¯+∑l=1L∑j=1∞ψ^lA∗−jξψ˜^lA∗−jξ¯=ΘA∗−1ξϕ^A∗−1ξϕ˜^A∗−1ξ¯+∑l=1L∑j=1∞ψ^lA∗−jξψ˜^lA∗−jξ¯=⋯=ΘA∗−Nξϕ^A∗−Nξϕ˜^A∗−Nξ¯+∑l=1L∑j=N∞ψ^lA∗−jξψ˜^lA∗−jξ¯,for N∈ℕ. By applying Lemma 5 and the Cauchy–Schwarz inequality, we know the series(54)∑l=1L∑j=N∞ψ^lA∗−jξψ˜^lA∗−jξ¯,converges absolutely for a.e. ξ∈ℝd, and therefore, we have(55)limN⟶∞∑l=1L∑j=N∞ψ^lA∗−jξψ˜^lA∗−jξ=0,for a.e.ξ∈ℝd.
LettingN⟶∞ in (53), we obtain(56)Θξϕ^ξϕ˜^ξ¯+∑l=1L∑j=0∞ψ^lA∗−jξψ˜^lA∗−jξ¯=1,for a.e.ξ∈Ω,by Assumption 1 and (46).
Next, we prove the converse implication, i.e., (48) implies (46) and (47). Suppose that (48) holds. First, pick γ∈ΓA∗∖0, and we have A∗γ∈ℤd. For any p∈ℤd, set n=A∗γ+p, for the choice of n, we have κn=0 because γ+p∉ℤd. Since (48) is valid for all 0≠n∈ℤd, by applying Assumption 2 and (45), we have(57)0=Θξϕ^ξϕ˜^ξ+n¯+∑l=1Lψ^lξψ˜^lξ+n¯=ϕ^A∗−1ξϕ˜^A∗−1ξ+n¯ΘξH0A∗−1ξH˜0A∗−1ξ+n¯+∑l=1LHlA∗−1ξH˜lA∗−1ξ+n¯=ϕ^A∗−1ξϕ˜^A∗−1ξ+γ+p¯ΘξH0A∗−1ξH˜0A∗−1ξ+γ¯+∑l=1LHlA∗−1ξH˜lA∗−1ξ+γ¯,by the ℤd-periodicity of H˜0 and H˜l,1≤l≤L. Take ξ∈σϕ∩τσϕ˜−γ with γ∈ΓA∗∖0 and replace ξ with A∗ξ+λ in (57), where λ∈ℤd, and it follows that(58)ϕ^ξ+λϕ˜^ξ+γ+p¯ΘA∗ξH0ξH˜0ξ+γ¯+∑l=1LHlξH˜lξ+γ¯=0,due to the ℤd-periodicity of Θ,H0,H˜0,Hl and H˜l,1≤l≤L. Since λ and p are arbitrary integers and ξ∈σϕ∩τσϕ˜−γ with γ∈ΓA∗∖0, by Lemma 2, there exist some λ0 and p0 such that(59)ϕ^ξ+λ0ϕ˜^ξ+γ+p0¯≠0.
Then, we obtain(60)ΘA∗ξH0ξH˜0ξ+γ¯+∑l=1LHlξH˜lξ+γ¯=0a.e.ξ∈σϕ∩τσϕ˜−γwithγ∈ΓA∗∖0.
Finally, we need to prove (46) holds, and (47) holds for γ=0 to finish the proof. Taking n=0 in (48), we have κ0=+∞ and(61)Θξϕ^ξϕ˜^ξ¯+∑l=1L∑j=0∞ψ^lA∗−jξψ˜^lA∗−jξ¯=χΩξ.
Replacingξ with A∗ξ in (61), we have(62)ΘA∗ξϕ^A∗ξϕ˜^A∗ξ¯+∑l=1L∑j=0∞ψ^lA∗−j+1ξψ˜^lA∗−j+1ξ¯=χΩξ,due to Ω=A∗Ω. By applying Assumption 2 and (45), we have(63)ΘA∗ξϕ^A∗ξϕ˜^A∗ξ¯+∑l=1L∑j=0∞ψ^lA∗−j+1ξψ˜^lA∗−j+1ξ¯=ϕ^ξϕ˜^ξ¯ΘA∗ξH0ξH˜0ξ¯+∑l=1LHlξH˜lξ¯+∑l=1L∑j=0∞ψ^lA∗−jξψ˜^lA∗−jξ¯,and thus,(64)Θξϕ^ξϕ˜^ξ¯+∑l=1L∑j=0∞ψ^lA∗−jξψ˜^lA∗−jξ¯=ϕ^ξϕ˜^ξ¯ΘA∗ξH0ξH˜0ξ¯+∑l=1LHlξH˜lξ¯+∑l=1L∑j=0∞ψ^lA∗−jξψ˜^lA∗−jξ¯.
Therefore, we obtain(65)ΘA∗ξH0ξH˜0ξ¯+∑l=1LHlξH˜lξ¯=Θξ,for a.e.ξ∈σϕ∩σϕ˜.
On the contrary, collecting (53), (55), and (61), we conclude that(66)limN⟶∞ΘA∗−Nξϕ^A∗−Nξϕ˜^A∗−Nξ¯=χΩξ.
By Assumption1, then we obtain (46). This proof has been completed.Remark 1.
In the literature, the functionΘ is related to the notion of mixed fundamental function, which plays a significant role in MOEP.
LetΘ=1 a.e. on Ω in the previous theorem, an immediate consequence is generalization of the MEP, which reads as follows.Corollary 1.
LetFL2Ω be a reducing subspace of L2Rd. Let ϕ,ϕ˜∈FL2Ω satisfy Assumptions 1 and 2, and two finite subsets Ψ=ψ1,ψ2,…,ψL and Ψ˜=ψ˜1,ψ˜2,…,ψ˜L of FL2Ω defined as in (45). Suppose that Xϕ,Ψ and Xϕ˜,Ψ˜ are both Bessel sequences in FL2Ω. Then, Xϕ,Ψ,Xϕ˜,Ψ˜ is a pair of NWDF for FL2Ω if and only if(67)H0·H˜0·+γ¯+∑l=1LHl·H˜l·+γ¯=δ0,γa.e.onσϕ∩τσϕ˜−γwithγ∈ΓA∗.



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