# A Class of Weak Dual Wavelet Frames for Reducing Subspaces of Sobolev Spaces

Authors: Jianping Zhang
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

In recent years, dual wavelet frames derived from a pair of refinable functions have been widely studied by many researchers. However, the requirement of the Bessel property of wavelet systems is always required, which is too technical and artificial. In present paper, we will relax this restriction and only require the integer translation of the wavelet functions (or refinable functions) to form Bessel sequences. For this purpose, we introduce the notion of weak dual wavelet frames. And for generality, we work under the setting of reducing subspaces of Sobolev spaces, we characterize a pair of weak dual wavelet frames, and by using this characterization, we obtain a mixed oblique extension principle for such weak dual wavelet frames.


---

## Body

## 1. Introduction

LetH be a separable Hilbert space. An at most countable sequence eii∈I in H is called a frame for H if there exist two constants 0<C1≤C2<∞ such that
(1)C1f2≤∑i∈If,ei2≤C2f2forf∈H,where C1 and C2 are called lower and upper frame bounds; it is called a Bessel sequence in H if the right-hand side inequality in (1) holds, where C2 is called a Bessel bound. Given a frame eii∈I for H, a sequence e~ii∈I is called a dual of eii∈I if it is a frame such that
(2)f=∑i∈If,e~ieiforf∈H.It is well-known thateii∈I is also a dual of e~ii∈I if e~ii∈I is a dual of eii∈I. So, in this case, we also say eii∈I,e~ii∈I is a pair of dual frames. The fundamentals of frames can be found in [1–3]. We denote by ℤ, ℕ, and ℕ0 the set of integers, the set of positive integers, and the set of nonnegative integers, respectively. Let d∈ℕ and M a d×d expansive matrix (an integer matrix with all its eigenvalues being greater than 1 in modulus). The dilation operator D and the shift operator Tk with k∈ℤd are, respectively, defined by
(3)Df·=detM1/2fM·,Tkf·=f·−k,for f∈L2ℝd, and naturally extended to tempered distribution. In what follows, we denote by M∗ the transpose of M, and by
(4)ΓM∗=ε0,ε1,⋯,εdetM−1,a set of representatives of distinct cosets of ℤd/M∗ℤd with ε0=0.The Fourier transform of a functionf∈L1ℝd∩L2ℝd is defined by
(5)f^·=∫ℝdfxe−2πix,·dxand naturally extended to tempered distribution, where ·,· denotes the Euclidean inner product in ℝd. Similarly, the inverse Fourier transform of a function f∈L1ℝd∩L2ℝd is defined by
(6)f˘·=∫ℝdfxe2πix,·dxand extended to tempered distribution as usual. For a real number s, we denote by Hsℝd the Sobolev space consisting of all distributions f such that
(7)fHsℝd2=∫ℝdf∧ξ21+ξ2sdξ<∞,where ξ denotes its Euclidean norm for ξ∈Rd. It is easy to check that Hsℝd is a Hilbert space under the inner product
(8)f,gHsℝd=∫Rdf^ξg^ξ¯1+ξ2sdξforf,g∈Hsℝd.In particular,H0ℝd=L2ℝd by the Plancherel theorem. For Ω⊂ℝd with nonzero measure, we write
(9)FHsΩ=f∈Hsℝd:suppf^⊂Ω,FH0Ω=FL2Ω.Then,FHsℝd=Hsℝd. Obviously, for each g∈FH−sΩ,
(10)f,g=∫ℝdf^ξg^ξ¯dξforf∈FHsΩdefines a continuous linear function on FHsΩ. Then, FHsΩ,FH−sΩ forms a pair of dual spaces; so does Hsℝd,H−sℝd by taking Ω=ℝd. Write
(11)D=f:f^∈L∞ℝdandsuppf^isbounded.It is well known thatD is dense in Hsℝd and D∩FHsΩ is dense in FHsΩ for every s∈ℝ.Definition 1.
Given a real numbers and a d×d expansive matrix M, a nonzero closed linear subspace X of Hsℝd is called a reducing subspace if DX=X and TkX=X for each k∈ℤd, and
(12)1+M∗−1·2s/2X^=1+·2s/2X^,where X^=f^:f∈X.In particular, whens=0, (12) is trivial and Definition 1 reduces to one in [4], which is characterized in Fourier domain as follows.Proposition 2 ([4], (Theorem 1)).
For ad×d expansive matrix M, X is a reducing subspace of L2ℝd if and only if X=FL2Ω for some Ω⊂ℝd with nonzero measure satisfying Ω=M∗Ω.Proposition 3 ([4], (Theorem 2.1)).
Lets be a real number and M a d×d expansive matrix. Then, X is a reducing subspace of Hsℝd if and only if X=FHsΩ for some Ω⊂ℝd with nonzero measure satisfying Ω=M∗Ω.So, to be specific, we denote a reducing subspace ofHsℝd by FHsΩ instead of X. Given a distribution f, we write
(13)fj,k·=DjTkf·,fj,ks·=detM−js/dfj,k·,for j∈ℤ and k∈ℤd. Let FHsΩ and FH−sΩ be reducing subspaces of Hsℝd and H−sℝd, respectively, and for ψ0∈FHsΩ,ψ~0∈FH−sΩ, Ψ=ψl:1≤l≤L, and Ψ~=ψ~l:1≤l≤L finite subsets of FHsΩ and FH−sΩ, respectively, the wavelet systems Xsψ0,Ψ and X−sψ~0,Ψ~ are defined as
(14)Xsψ0,Ψ≔ψ0,0,k:k∈ℤd∪ψl,j,ks:j∈N0,k∈ℤd,1≤l≤L,X−sψ~0,Ψ~≔ψ~0,0,k:k∈ℤd∪ψ~l,j,k−s:j∈N0,k∈ℤd,1≤l≤L.We sayXsψ0,Ψ,X−sψ~0,Ψ~ is a pair of dual wavelet frames for FHsΩ,FH−sΩ if
(1)
Xsψ0,Ψ is a wavelet frame for FHsΩ and X−sψ~0,Ψ~ is a wavelet frame for FH−sΩ(2)
The identity(15)f,g=∑k∈ℤdf,ψ~0,0,kψ0,0,k,g+∑l=1L∑j=0∞∑k∈ℤdf,ψ~l,j,k−sψl,j,ks,gholds forf∈FHsΩ and g∈FH−sΩ.We sayXsψ0,Ψ,X−sψ~0,Ψ~ is a pair of weak dual wavelet frames for FHsΩ,FH−sΩ if
(1)
Tkψl:k∈ℤd,0≤l≤LandTkψ~l:k∈ℤd,0≤l≤L are Bessel sequences in Hsℝd and H−sℝd, respectively(2)
There exist dense subsetsV of FHsΩ and V~ of FH−sΩ such that(16)f,g=∑k∈ℤdf,ψ~0,0,kψ0,0,k,g+∑l=1L∑j=0∞∑k∈ℤdf,ψ~l,j,k−sψl,j,ks,gforf∈V,g∈V~,where the series converges in the following sense: both∑k∈ℤdf,ψ~0,0,kψ0,0,k,g and ∑k∈ℤdf,ψ~l,j,k−sψl,j,ks,g converge unconditionally (equivalently, both ∑k∈ℤdf,ψ~0,0,kψ0,0,k,g and ∑k∈ℤdf,ψ~l,j,k−sψl,j,ks,g converge), and
(17)∑j=0∞∑k∈ℤdf,ψ~l,j,k−sψl,j,ks,g=limJ⟶∞∑j=0J∑k∈Zdf,ψ~l,j,k−sψl,j,ks,g.It is obvious that the convergence of series in (16) is weaker than that in (15). And a pair of dual wavelet frames must be a pair of weak dual wavelet frames, whereas the converse is not true. Also, observe that in the above definition of weak dual wavelet frames, ψl,0≤l≤L need not belong to FHsΩ, and ψ~l,0≤l≤L need not belong to FH−sΩ.Due to the great design freedom and the potential applications in signal denoising, image restoration, numerical analysis, etc., the study of wavelet frames forL2ℝd and Sobolev spaces has been attracting many researchers and seen great achievements (see [5–16] for details). In particular, Bownik in [6] obtained the following important characterization for homogeneous dual wavelet frames:Proposition 4.
LetXΨ and XΨ~ be Bessel sequences in L2ℝd. Then, XΨ,XΨ~ is a pair of dual wavelet frames for L2ℝd if and only if
(18)∑l=1L∑j=−∞κkψ^lM∗−j·ψ^lM∗−j·+k¯=δ0,ka.e.onℝdfork∈ℤd,where κk is defined by Definition 6 in Section 2.Li and Zhang in [17] generalized Proposition 4 to Sobolev space pairs Hsℝd,H−sRd for nonhomogeneous dual wavelet frames:Proposition 5.
Givens∈R, let Xsψ0,Ψ and X−sψ~0,Ψ~ be Bessel sequences in Hsℝd and H−sℝd, respectively. Then, Xsψ0,Ψ,X−sψ~0,Ψ~ is a pair of dual frames in Hsℝd,H−sℝd if and only if
(19)ψ^0·ψ~^0·+k¯+∑l=1L∑j=0κkψ^lM∗−j·ψ^lM∗−j·+k¯=δ0,ka.e.onℝdfork∈ℤd.An important method to construct (dual) wavelet frames from refinable functions is extension principles. Ron and Shen in [15, 16] prosed the unitary extension principle (UEP) and the mixed extension principle (MEP). Subsequently, Daubechies et al. in [10] developed them in the form of the oblique extension principle (OEP) and the mixed oblique extension principle (MOEP). From then on, the study of the extension principles has interested many researchers [4, 5, 7, 8, 11, 18–20].Observe that all above works, the wavelet systems (or the refinable functions) are required to be Bessel sequences. In order to achieve the Bessel property, some conditions have to be imposed on the wavelet systems (or the refinable functions) that are too technical and artificial. It is natural to ask what are expected from general refinable functions without too many restrictions. For this purpose, Jia and Li in [21] introduced the nation of weak wavelet biframes (weak dual wavelet frames). Starting from a pair of general refinable functions without smoothness restrictions, they obtained a construction of weak dual wavelet frames for reducing subspace FL2Ω of L2ℝd.Inspired by all these works, in present paper, we investigate a class of weak dual wavelet frames for reducing subspaces of Sobolev spaces. In Section2, we first give some necessary lemmas, and then, we give a Fourier-domain characterization of weak dual wavelet frames in FHsΩ,FH−sΩ associated with D∩FHsΩ,D∩FH−sΩ. In Section 3, by using the above characterization, we derive a mixed oblique extension principle for such weak dual wavelet frames.Before proceeding, we introduce some necessary notations. We denote byTd=0,1d the d-dimensional torus and, for a Lebesgue measurable set E in ℝd, by E its Lebesgue measure and χE its characteristic function, respectively, by δ the Dirac sequence such that δ0,0=1 and δ0,k=0 for 0≠k∈ℤd, and denote by τ the mapping from ℝd to Td defined by
(20)τx=x−kforx∈Td+kwithk∈ℤd.For functionsf and g on ℤd, we define
(21)f,gt·=∑k∈ℤdf·+kg·+k¯1+·+k2t,t∈ℝ,if it is well defined, and the spectrum σtf by
(22)σtf=ξ∈Td:f^,f^tξ>0.It is obvious thatξ∈σtf if and only if ξ∈Td and f^ξ+k≠0 for some k∈Zd. So σtf is independent of t. For simplicity, we use σf to replace σtf.


## 2. The Characterization of Weak Dual Wavelet Frames

This section is devoted to characterizing weak dual wavelet frames inFHsΩ,FH−sΩ. Fist, we give some necessary lemmas for later use.Definition 6.
LetM be a d×d expansive matrix. Define a function κ:ℤd⟶N0 by
(23)κn=supj∈N0:M∗−jn∈ℤd,and set κ0=+∞.By a standard argument, we have the following three lemmas:Lemma 7.
M∗−κn−1n:0≠n∈ℤd=⋃γ∈ΓM∗\0ℤd+γ.Lemma 8.
Forϕ∈Hsℝd and ϕ~∈H−sℝd, we have
(24)σϕ∩τσϕ~−γ=ξ∈Td:ϕ^ξ+kϕ~^ξ+γ+n¯≠0forsomek,n∈ℤdforγ∈ΓM∗.Lemma 9.
Letakk∈ℤd and bkk∈ℤd be two complex sequences, and ∑j=0∞∑0≠k∈ℤdaM∗jkbk<∞. Then,
(25)∑j=0∞∑0≠k∈ℤdaM∗jkbk=∑0≠k∈ℤd∑j=0κkakbM∗−jk.Lemma 10 ([4], (Lemma 3.1)).
Givens∈R, a d×d expansive matrix M, and ϕ∈HsRd, we have the following:
(i)
Tkϕ:k∈ℤd is a Bessel sequence in Hsℝd if and only if ϕ^,ϕ^s∈L∞ℝd. In this case, ϕ^,ϕ^L∞ℝd is a Bessel bound(ii)
IfTkϕ:k∈Zd is a Bessel sequence in Hsℝd, then ϕn,k:k∈ℤd is a Bessel sequence in Hsℝd with the Bessel bound Cnϕ^,ϕ^sL∞Td for n∈Z, and(26)∑k∈ℤdf,ϕn,k2≤Cnϕ^,ϕ^sL∞TdfH−sℝd2,for f∈H−sℝd, where Cn=supξ∈Rd1+ξ2/1+M∗−nξ2s.Lemma 11.
Lets∈R, f∈Hsℝd, ψ∈H−sℝd, and j∈Z. Then, for k∈Zd, the k-th Fourier coefficient of detMj/2f^M∗j·,ψ^·0ξ is f,ψj,k. In particular,
(27)detMj/2f^M∗j·,ψ^·0ξ=∑k∈ℤdf,ψj,ke2πik,ξ,if Tkψ:k∈ℤd is a Bessel sequence in H−sℝd.Proof.
Sincef∈Hsℝd and ψ∈H−sℝd, we have f^M∗j·ψ^·¯∈L1ℝd, and thus,
(28)∫ℝddetMj/2f^M∗j·,ψ^·0ξe−2πik,ξdξ=detMj/2∫ℝdf^M∗jξψ^ξ¯e−2πik,ξdξ=detM−j/2∫ℝdf^ξψ^M∗−jξ¯e−2πik,M∗−jξdξ=∫ℝdf^ξψj,k·^ξ¯dξ=f,ψj,k,by the Plancherel theorem. So the k-th Fourier coefficient of detMj/2f^M∗j·,ψ^·0ξ is f,ψj,k.
IfTkψ:k∈ℤd is a Bessel sequence in H−sℝd, then ψj,k:k∈ℤd is a Bessel sequence in H−sℝd by Lemma 10 (ii). It follows that f,ψj,kk∈ℤd∈ℓ2ℤd, and thus, (27) holds.☐☐Lemma 12 ([4], (Lemma 3.5)).
LetS be a bounded set in ℝd. Then, there exist finite sets F1⊂N0 and F2⊂Zd\0 such that
(29)S∩S+M∗jk=∅forj,k∈N0×Zd\0\F1×F2.Lemma 13.
Givens∈ℝ, let Tkψl:k∈ℤd,0≤l≤L and Tkψ~l:k∈ℤd,0≤l≤L be Bessel sequences in Hsℝd and H−sℝd, respectively. Then,
(30)∑k∈ℤdf,ψ~0·−kψ0·−k,g+∑l=1L∑j=0∞∑k∈ℤdf,ψ~l,j,k−sψl,j,ks,g=∫ℝdf^ξg^ξ¯ψ^0ξψ~^0ξ¯+∑l=1L∑j=0∞ψ^lM∗−jξψ~^lM∗−jξ¯dξ+∫ℝdg^ξ¯∑0≠k∈ℤdf^ξ+kψ^0ξψ~^0ξ+k¯+∑l=1L∑j=0κkψ^lM∗−jξψ~^lM∗−jξ+k¯dξ,for f,g∈D, where D is defined by (11).Proof.
By Lemma11, we have
(31)∑k∈ℤdf,ψ~0·−kψ0·−k,g+∑l=1L∑j=0∞∑k∈ℤdf,ψ~l,j,k−sψl,j,ks,g=∫Td∑k∈ℤdf^ξ+kψ~^0ξ+k¯∑k∈ℤdψ^0ξ+kg^ξ+k¯dξ+∑l=1L∑j=0∞detMj∫Td∑k∈ℤdf^M∗jξ+kψ~^lξ+k¯∑k∈ℤdψ^lξ+kg^M∗jξ+k¯dξ.
Write(32)G·=∑k∈ℤdψ^0·+kg^·+k¯,Gl,j·=∑k∈ℤdψ^l·+kg^M∗j·+k.¯
Then, by the Cauchy-Schwarz inequality, we have(33)G·≤∑k∈ℤdψ^0·+kg^·+k¯≤∑k∈ℤdψ∧0·+k21+·+k2s1/2∑k∈ℤdg∧·+k21+·+k2−s1/2=ψ^0,ψ^0s1/2·g^,g^−s1/2·.
By a similar procedure, we also have(34)Gl,j·≤ψ^l,ψ^ls1/2·g^M∗j·,g^M∗j·−s1/2·.
Ifg∈D, then Tkg:k∈ℤd and Tkg~:k∈ℤd with g~^·=g^M∗j· are Bessel sequences in H−sℝd. Also, observe that Tkψ0:k∈ℤd and Tkψl:k∈ℤd,1≤l≤L are Bessel sequences in Hsℝd, and thus, G,Gl,j∈L∞Td by Lemma 10 (i). It follows that
(35)∫Td∑k∈ℤdf^ξ+kψ~^0ξ+k¯∑k∈ℤdψ^0ξ+kg^ξ+k¯dξ=∫ℝdψ^0ξg^ξ¯∑k∈ℤdf^ξ+kψ~^0ξ+k¯dξ,∫Td∑k∈ℤdf^M∗jξ+kψ~^lξ+k¯∑k∈ℤdψ^lξ+kg^M∗jξ+k¯dξ=∫ℝdψ^lξg^M∗jξ¯∑k∈ℤdf^M∗jξ+kψ~^lξ+k¯dξ,by the Fubini-Tonelli theorem. Therefore, we get
(36)∑k∈ℤdf,ψ~0·−kψ0·−k,g+∑l=1L∑j=0∞∑k∈ℤdf,ψ~l,j,k−sψl,j,ks,g=∫ℝdψ^0ξg^ξ¯∑k∈ℤdf^ξ+kψ~^0ξ+k¯dξ+∑l=1L∑j=0∞detMj∫ℝdψ^lξg^M∗jξ¯∑k∈ℤdf^M∗jξ+kψ~^lξ+k¯dξ.
By using the Cauchy-Schwarz inequality, we have(37)ψ^0·g^·¯∑k∈ℤdf^·+kψ~^0·+k¯≤∑k∈ℤdψ^0·+kg^·+k¯∑k∈ℤdf^·+kψ~^0·+k¯≤∑k∈ℤdψ∧0·+k21+·+k2s1/2∑k∈ℤdg∧·+k21+·+k2−s1/2×∑k∈ℤdf∧·+k21+·+k2s1/2∑k∈ℤdψ~∧0·+k21+·+k2−s1/2=ψ^0,ψ^0s1/2·g^,g^−s1/2·f^,f^s1/2·ψ~^0,ψ~^0−s1/2·,which belongs to L∞Rd by Lemma 10 (i) since Tkf:k∈Zd is also a Bessel sequence in HsRd if f∈D, and thus,
(38)∫Rdψ^0ξg^ξ¯∑k∈Zdf^ξ+kψ~^0ξ+k¯dξ=∫Rdψ^0ξg^ξ¯f^ξψ~^0ξ¯dξ+∫Rdψ^0ξg^ξ¯∑0≠k∈Zdf^ξ+kψ~^0ξ+k¯dξ.
Sincef,g∈D, then there exists a bounded set S in Rd such that suppf^∪suppg^⊂S. By Lemma 12, there exist finite sets F1⊂N0 and F2⊂Zd\0 such that
(39)S∩S+M∗jk=∅forj,k∈N0×Zd\0\F1×F2.
Therefore, we have(40)∑l=1L∑j=0∞detMj∫Rdψ^lξg^M∗jξ¯∑0≠k∈Zdf^M∗jξ+kψ~^lξ+k¯dξ=∑l=1L∑j∈F1detMj∫Rdψ^lξg^M∗jξ¯∑k∈F2f^M∗jξ+kψ~^lξ+k¯dξ.
Write(41)K=⋃k∈F2∪0∪j∈F1M∗−jS−k.
Then, forj∈F1 and k∈F2, we have
(42)detMj∫Rd∣ψ^lξg^M∗jξ¯f^M∗jξ+kψ~^lξ+k¯∣dξ=∫Rd∣ψ^lM∗−jξg^ξ¯f^ξ+M∗jkψ~^lM∗−jξ+k¯∣dξ≤∥g^∥L∞Rd∥f^∥L∞Rd∫M∗−jSψ∧lξ2dξ1/2∫M∗−jSψ~∧lξ+k2dξ1/2≤∥g^∥L∞Rd∥f^∥L∞Rd∫Kψ∧lξ2dξ1/2∫Kψ~∧lξ2dξ1/2.
Also, observe thatmaxξ∈K1+ξ2−s/21+ξ2s/2≥1 and maxξ∈K1+ξ2s/21+ξ2−s/2≥1. It follows that
(43)detMj∫Rd∣ψ^lξg^M∗jξ¯f^M∗jξ+kψ~^lξ+k¯∣dξ≤∥g^∥L∞Rd∥f^∥L∞Rdmaxξ∈K1+ξ2−s/2maxξ∈K1+ξ2s/2∫Kψ∧lξ21+ξ2sdξ1/2×∫Kψ~∧lξ21+ξ2−sdξ1/2≤∥g^∥L∞Rd∥f^∥L∞Rdmaxξ∈K1+ξ2−s/2maxξ∈K1+ξ2s/2∥ψ^l∥HsRd∥ψ~^l∥H−sRd<∞.
Therefore, we can change the order of summation and integration in (40).
(44)∑l=1L∑j=0∞detMj∫Rdψ^lξg^M∗jξ¯∑0≠k∈Zdf^M∗jξ+kψ~^lξ+k¯dξ=∑l=1L∑j=0∞∫Rdψ^lM∗−jξg^ξ¯∑0≠k∈Zdf^ξ+M∗jkψ~^lM∗−jξ+k¯dξ=∫Rd∑l=1L∑j=0∞ψ^lM∗−jξg^ξ¯∑0≠k∈Zdf^ξ+M∗jkψ~^lM∗−jξ+k¯dξ=∫Rdg^ξ¯∑0≠k∈Zdf^ξ+k∑l=1L∑j=0κkψ^lM∗−jξψ~^lM∗−jξ+k¯dξ,where Lemma 9 is used in the last equality. And thus, the conclusion follows by collecting (36), (38), and (44).☐☐The following theorem characterizes a pair of weak dual wavelet frames forFHsΩ,FH−sΩ.Theorem 14.
LetFHsΩ and FH−sΩ be reducing subspaces of HsRd and H−sRd, respectively, and ψ0∈FHsΩ, ψ~0∈FH−sΩ and Ψ=ψl:1≤l≤L, Ψ~=ψ~l:1≤l≤L finite subsets of FHsΩ and FH−sΩ. Suppose Tkψl:k∈Zd,0≤l≤L and Tkψ~l:k∈Zd,0≤l≤L are both Bessel sequences in HsRd and H−sRd, respectively. Then, Xsψ0,Ψ,X−sψ~0,Ψ~ is a pair of weak dual wavelet frames for FHsΩ,FH−sΩ if and only if
(45)limJ⟶∞∫Rdψ^0ξψ~^0ξ¯+∑l=1L∑j=0Jψ^lM∗−jξψ~^lM∗−jξ¯hξdξ=∫RdχΩξhξdξforh∈L∞K,∀compactK⊂Rd\0,(46)ψ^0·ψ~^0·+k¯+∑l=1L∑j=0κkψ^lM∗−j·ψ~^lM∗−j·+k¯=0for0≠k∈Zdanda.e.onΩ.Proof.
SinceD∩FHsΩ is dense in FHsΩ for every s∈R, then Xsψ0,Ψ,X−sψ~0,Ψ~ is a pair of weak dual wavelet frames for FHsΩ,FH−sΩ if and only if
(47)∑k∈Zdf,ψ~0,0,kψ0,0,k,g+∑l=1L∑j=0∞∑k∈Zdf,ψ~l,j,k−sψl,j,ks,g=f,g,for f∈D∩FHsΩ,g∈D∩FH−sΩ, or equivalently
(48)∑k∈Zdf∧χΩ∨,ψ~0,0,kψ0,0,k,g∧χΩ∨+∑l=1L∑j=0∞∑k∈Zdf∧χΩ∨,ψ~l,j,k−sψl,j,ks,g∧χΩ∨=f∧χΩ∨,g∧χΩ∨=f^χΩ,g^χΩ,for f,g∈D due to D∩FHsΩ=h∧χΩ∨:h∈D. By the Bessel assumptions, we know that the series ∑k∈Zdf∧χΩ∨,ψ~0,0,kψ0,0,k,g∧χΩ∨ and ∑k∈Zdf∧χΩ∨,ψ~l,j,k−sψl,j,ks,g∧χΩ∨ are absolutely convergent for f,g∈D since D is dense both in HsRd and H−sRd. By Lemma 13, (48) can be written as
(49)limJ⟶∞∫Rdf^ξg^ξ¯χΩξψ^0ξψ~^0ξ¯+∑l=1L∑j=0Jψ^lM∗−jξψ~^lM∗−jξ¯dξ+∫Rdg^ξ¯χΩξ∑0≠k∈Zdf^χΩξ+kψ^0ξψ~^0ξ+k¯+∑l=1L∑j=0κkψ^lM∗−jξψ~^lM∗−jξ+k¯dξ=∫Rdf^ξg^ξ¯χΩξdξforf,g∈D.
Obviously, (45) and (46) imply (49). Next, we prove the converse implication to complete the proof. Suppose (49) holds. For 0≠k∈Zd, the function
(50)ψ^0·ψ~^0·+k¯+∑l=1L∑j=0κkψ^lM∗−j·ψ~^lM∗−j·+k¯∈L1Rd,and thus, almost every point in Rd is a Lebesgue point of all function
(51)ψ^0·ψ~^0·+k¯+∑l=1L∑j=0κkψ^lM∗−j·ψ~^lM∗−j·+k,¯with 0≠k∈Zd. Let ξ0∈Rd be such a point. For 0<ε<1/2 and 0≠k0∈Zd, take f and g such that f^·+k0=g^·¯=χBξ0,ε·/∣Bξ0,ε∣ in (49), where Bξ0,ε=ξ∈Rd:ξ−ξ0<ε. Then, we have
(52)1Bξ0,ε∫Bξ0,εXΩξXΩξ+k0ψ^0ξψ~^0ξ+k0¯+∑l=1L∑j=0κk0ψ^lM∗−1ξψ~^lM∗−jξ+k0¯dξ=0,or equivalently,
(53)1Bξ0,ε∫Bξ0,εψ^0ξψ~^0ξ+k0¯+∑l=1L∑j=0κk0ψ^lM∗−1ξψ~^lM∗−jξ+k0¯dξ=0,since ψ^0ξψ~^0ξ+k0¯+∑l=1L∑j=0κk0ψ^lM∗−jξψ~^lM∗−jξ+k0¯=0 if ξ and ξ+k0 belong to Ω when ξ∈Bξ0,ε. Letting ε⟶0 in (53), we obtain
(54)ψ^0ξ0ψ~^0ξ0+k0¯+∑l=1L∑j=0κk0ψ^lM∗−jξ0ψ~^lM∗−jξ0+k0¯=0.
By the arbitrariness ofξ0∈Rd and 0≠k0∈Zd, then we have
(55)ψ^0·ψ~^0·+k¯+∑l=1L∑j=0κkψ^lM∗−j·ψ~^lM∗−j·+k¯=0a.e.onRdfor0≠k∈Zd.
And thus, (46) holds since the above function vanishes out of Ω. Now, we prove (45) holds.
Arbitrarily fixh∈L∞K, ∀ compact K⊂ℝd\0, then there exists a finite set k1,k2,⋯,km⊂ℤd such that K=∪i=1m0,1d+ki∩K=∪i=1mSi. Define hi=hχSi for 1≤i≤m. Then, we have h=∑i=1mhi, and thus, (45) holds for h if it holds for each hi with 1≤i≤m. Next, we prove (45) holds for each hi with 1≤i≤m. Take f and g in (49) such that
(56)f^·=hi·hi·,ifhi·≠01,ifhi·=0andg^·¯=hi·for1≤i≤m.
Then, we obtain(57)limJ⟶∞∫Rdhiξψ^0ξψ~^0ξ¯+∑l=1L∑j=0Jψ^lM∗−jξψ~^lM∗−jξ¯dξ=∫RdhiξχΩξdξ.
(45) therefore holds for hi and holds for h. By the arbitrariness of h, we obtain (45). The proof is completed.☐☐


## 3. Constructing Weak Dual Wavelet Frames from a Pair of General Refinable Functions

This section is devoted to constructing weak dual wavelet frames forFHsΩ,FH−sΩ from a pair of general refinable functions under the following assumptions:Assumption 1.
ψ0∈HsRd and ψ~0∈H−sRd are M-refinable functions with symbols in L∞Td, i.e., there exists b0,b~0∈L∞Td such that
(58)ψ^0·=b0M∗−1·ψ^0M∗−1·andψ~^0·=b~0M∗−1·ψ~^0M∗−1·a.e.onRd.Assumption 2.
ψ^0,ψ^0s,ψ~^0,ψ~^0−s∈L∞Td.Assumption 3.
ψ^0ψ~^0¯χB0,δ∈L∞Rd for some δ>0, where B0,δ denotes the δ-neighborhood of the origin 0.GivenL∈N, let bl,b~l∈L∞Td with 1≤l≤L define Ψ=ψl:1≤l≤L and Ψ~=ψ~l:1≤l≤L by
(59)ψ^l·=blM∗−1·ψ^0M∗−1·,ψ~^l·=b~lM∗−1·ψ~^0M∗−1·.Remark 15.
By Lemma10 (i), Assumption 2 is equivalent to the fact that Tkψ0:k∈Zd and Tkψ~0:k∈Zd are Bessel sequences in HsRd and H−sRd, respectively. It is easy to check that Tkψl:k∈Zd,1≤l≤L and Tkψ~l:k∈Zd,1≤l≤L are also Bessel sequences in HsRd and H−sRd, respectively.Theorem 16.
LetFHsΩ and FH−sΩ be reducing subspaces of HsRd and H−sRd, respectively, ψ0∈FHsΩ and ψ~0∈FH−sΩ two functions satisfy Assumptions 1–3, and let Ψ=ψl:1≤l≤L and Ψ~=ψ~l:1≤l≤L be two finite subsets of FHsΩ and FH−sΩ, respectively, defined by (59). Assume that there exists a function Θ∈L∞Td such that
(60)limj⟶∞∫ΩΘM∗−jξhξdξ=∫Ωhξdξforh∈L∞K,∀compactK⊂Rd\0,(61)ΘM∗·b0·b~0·+q¯+∑l=1Lbl·b~l·+q¯=Θ·δ0,qa.e.onσψ0∩τσψ~0−qwithq∈ΓM∗.Then,Xη,Ψ,Xψ~0,Ψ~ is a pair of weak dual wavelet frames for FHsΩ,FH−sΩ, where η is defined by η^·=Θ·ψ^0·.Proof.
First, we claim that the following equation holds:(62)ΘM∗−j·ψ^0M∗−j·ψ~^0M∗−j·+n¯+∑l=1Lψ^lM∗−j·ψ~^lM∗−j·+n¯=ΘM∗−j−1·ψ^0M∗−j−1·ψ~^0M∗−j−1·+n¯,
a.e. onℝd for n∈ℤd and 0≤j<κn. Indeed, by Assumption 1 and (59) and the Zd-periodicity of b~l with 1≤l≤L, we have
(63)ΘM∗−j·ψ^0M∗−j·ψ~^0M∗−j·+n¯+∑l=1Lψ^lM∗−j·ψ~^lM∗−j·+n¯=ΘM∗−j·b0M∗−j−1·b~0M∗−j−1·¯+∑l=1LblM∗−j−1·b~lM∗−j−1·¯ψ^0M∗−j−1·ψ~^0M∗−j−1·+n¯.
So (62) holds if
(64)ΘM∗−j·b0M∗−j−1·b~0M∗−j−1·¯+∑l=1LblM∗−j−1·b~lM∗−j−1·¯=ΘM∗−j−1ξ,
a.e. onℝd with ψ^0M∗−j−1·ψ~^0M∗−j−1·+n¯≠0. Fix ξ with ψ^0M∗−j−1ξψ~^0M∗−j−1ξ+n¯≠0. Suppose M∗−j−1ξ=ξ′+nξ for some ξ′∈Td and nξ∈ℤd. Then, ξ′∈σψ0∩σψ~0 by Lemma 8, and thus,
(65)ΘM∗ξ′b0ξ′b~0ξ′¯+∑l=1Lblξ′b~lξ′¯=Θξ′,by (61). This is equivalent to (64) by the Zd-periodicity of Θ, bl, and b~l with 0≤l≤L. Therefore, (62) holds.
By Theorem14, to complete the proof, it is enough to prove that
(66)limJ⟶∞∫RdΘξψ^0ξψ~^0ξ¯+∑l=1L∑j=0Jψ^lM∗−jξψ~^lM∗−jξ¯hξdξ=∫RdχΩξhξdξ,for h∈L∞K, ∀ compact K⊂Rd\0, and
(67)Θ·ψ^0·ψ~^0·+n¯+∑l=1L∑j=0κnψ^lM∗−j·ψ~^lM∗−j·+n¯=0a.eonΩfor0≠n∈ℤd.
We first prove (67). For 0≠n∈Zd, by using (62), we have
(68)Θ·ψ^0·ψ~^0·+n¯+∑l=1L∑j=0κnψ^lM∗−j·ψ~^lM∗−j·+n¯=ΘM∗−1·ψ^0M∗−1·ψ~^0M∗−1·+n¯+∑l=1L∑j=1κnψ^lM∗−j·ψ~^lM∗−j·+n¯=ΘM∗−2·ψ^0M∗−2·ψ~^0M∗−2·+n¯+∑l=1L∑j=2κnψ^lM∗−j·ψ~^lM∗−j·+n¯=⋯=ΘM∗−κn·ψ^0M∗−κn·ψ~^0M∗−κn·+n¯+∑l=1Lψ^lM∗−κn·ψ~^lM∗−κn·+n¯.
By Lemma7, we have M∗−κn−1n=n′+q for some n′∈Zd and 0≠q∈ΓM∗. It follows that
(69)ΘM∗−κn·ψ^0M∗−κn·ψ~^0M∗−κn·+n¯+∑l=1Lψ^lM∗−κn·ψ~^lM∗−κn·+n¯=ΘM∗−κn·b0M∗−κn−1·b~0M∗−κn−1·+q¯+∑l=1LblM∗−κn−1·b~lM∗−κn−1·+q¯×ψ^0M∗−κn−1·ψ~^0M∗−κn−1·+n¯=0,by (61), and (67) therefore holds.
Next, we prove (66) holds. By using (62), we have
(70)Θ·ψ^0·ψ~^0·¯+∑l=1L∑j=0Jψ^lM∗−j·ψ~^lM∗−j·¯=Θ·ψ^0·ψ~^0·¯+∑l=1Lψ^l·ψ~^l·¯+∑l=1L∑j=1Jψ^lM∗−j·ψ~^lM∗−j·¯=ΘM∗−1·ψ^0M∗−1·ψ~^0M∗−1·¯+∑l=1L∑j=1Jψ^lM∗−j·ψ~^lM∗−j·¯=ΘM∗−2·ψ^0M∗−2·ψ~^0M∗−2·¯+∑l=1L∑j=2Jψ^lM∗−j·ψ~^lM∗−j·¯=⋯=ΘM∗−J·ψ^0M∗J·ψ~^0M∗−J·¯+∑l=1Lψ^lM∗−J·ψ~^lM∗−J·¯=ΘM∗−J·b0M∗−J−1·b~0M∗−J−1·¯+∑l=1LblM∗−J−1·b~lM∗−J−1·¯ψ^0M∗−J−1·ψ~^0M∗−J−1·¯=ΘM∗−J−1·ψ^0M∗−J−1·ψ~^0M∗−J−1·¯,where (61) was used in the last equality. And thus,
(71)limJ⟶∞∫RdΘξψ^0ξψ~^0ξ¯+∑l=1L∑j=0Jψ^lM∗−jξψ~^lM∗−jξ¯hξdξ=limJ⟶∞∫RdΘM∗−J−1ξψ^0M∗−J−1ξψ~^0M∗−J−1ξ¯hξdξ=limJ⟶∞∫ΩΘM∗−J−1ξψ^0M∗−J−1ξψ~^0M∗−J−1ξ¯hξdξ,for h∈L∞K with compact K⊂Rd\0. Since K is compact and K⊂Rd\0, then there exists J0>0 such that M∗−J−1ξ∈B0,δ for J>J0+1 and ξ∈K. And thus, there exists a constant M>0 such that
(72)ΘM∗−J−1·h·ψ^0M∗−J−1·ψ~^0M∗−J−1·¯−1≤Mh·,by Assumption 3. This implies that
(73)limJ⟶∞∫ΩΘM∗−J−1ξhξψ^0M∗−J−1ξψ~^0M∗−J−1ξ¯−1dξ=0,by Assumptions 1 and 3. Therefore, (66) holds by collecting (71) and (60).☐☐Remark 17.
(i)
In the literature, the functionΘ is related to the notion of mixed fundamental function, which plays a significant role in MOEP.(ii)
In the above theorem, (60) is trivial and Xψ0,Ψ,Xψ~0,Ψ~ is a pair of weak dual wavelet frames for FHsΩ,FH−sΩ if Θ·=1 a.e. on Td


## 4. Conclusion

In this paper, we introduce the notion of weak dual wavelet frames. And for generality, we work under the setting of reducing subspaces of Sobolev spaces, we characterize a pair of weak dual wavelet frames, and by using this characterization, we obtain a mixed oblique extension principle for such weak dual wavelet frames.



---

*Source: 1372184-2022-01-19.xml*