We present a notion of frame multiresolution analysis on local fields of positive characteristic based on the theory of shift-invariant spaces. In contrast to the standard setting, the associated subspace V0 of L2(K) has a frame, a collection of translates of the scaling function φ of the form φ(·-u(k)):k∈N0, where N0 is the set of nonnegative integers. We investigate certain properties of multiresolution subspaces which provides the quantitative criteria for the construction of frame multiresolution analysis (FMRA) on local fields of positive characteristic. Finally, we provide a characterization of wavelet frames associated with FMRA on local field K of positive characteristic using the shift-invariant space theory.
1. Introduction
Multiresolution analysis is considered as the heart of wavelet theory. The concept of multiresolution analysis provides a natural framework for understanding and constructing discrete wavelet systems. A multiresolution analysis is an increasing family of closed subspaces {Vj:j∈Z} of L2(R) such that ⋂j∈ZVj={0} and ⋃j∈ZVj is dense in L2(R) which satisfies f∈Vj if and only if f(2·)∈Vj+1. Furthermore, there exists an element φ∈V0 such that the collection of integer translates of function φ, {φ(·-k):k∈Z}, represents a complete orthonormal system for V0. The function φ is called the scaling function or the father wavelet. The concept of multiresolution analysis has been extended in various ways in recent years. These concepts are generalized to L2(Rd), to lattices different from Zd, allowing the subspaces of multiresolution analysis to be generated by Riesz basis instead of orthonormal basis, admitting a finite number of scaling functions, replacing the dilation factor 2 by an integer M≥2 or by an expansive matrix A∈GLd(R) as long as A⊂AZd (see [1, 2]).
On the other hand, this elegant tool for the construction of wavelet bases has been extensively studied by several authors on the various spaces, namely, Cantor dyadic groups [3], locally compact Abelian groups [4], p-adic fields [5], zero-dimensional groups [6], and Vilenkin groups [7]. Recently, R. L. Benedetto and J. J. Benedetto [8] developed a wavelet theory for local fields and related groups. They did not develop the multiresolution analysis (MRA) approach; their method is based on the theory of wavelet sets. The local fields are essentially of two types: zero and positive characteristic (excluding the connected local fields R and C). Examples of local fields of characteristic zero include the p-adic field Qp whereas local fields of positive characteristic are the Cantor dyadic group and the Vilenkin p-groups. The structures and metrics of the local fields of zero and positive characteristic are similar, but their wavelet and MRA theory are quite different. The concept of multiresolution analysis on a local field K of positive characteristic was introduced by Jiang et al. [9]. They pointed out a method for constructing orthogonal wavelets on local field K with a constant generating sequence. Subsequently, tight wavelet frames on local fields of positive characteristic were constructed by Shah and Debnath [10] using extension principles. As far as the characterization of wavelets on local fields is concerned, Behera and Jahan [11] have given the characterization of all wavelets associated with multiresolution analysis on local field K based on results on affine and quasiaffine frames. Recently, Shah and Abdullah [12] have introduced the notion of nonuniform multiresolution analysis on local field K of positive characteristic and obtained the necessary and sufficient condition for a function φ to generate a nonuniform multiresolution analysis on local fields. More results in this direction can also be found in [13, 14] and the references therein.
Since the use of multiresolution analysis has proven to be a very efficient tool in wavelet theory mainly because of its simplicity, it is of interest to try to generalize this notion as much as possible while preserving its connection with wavelet analysis. In this connection, Benedetto and Li [15] considered the dyadic semiorthogonal frame multiresolution analysis of L2(R) with a single scaling function and successfully applied the theory in the analysis of narrow band signals. The characterization of the dyadic semiorthogonal frame multiresolution analysis with a single scaling function admitting a single frame wavelet whose dyadic dilations of the integer translates form a frame for L2(R) was obtained independently by Benedetto and Treiber by a direct method [16] and by Kim and Lim by using the theory of shift-invariant spaces [17]. Later on, Yu [18] extended the results of Benedetto and Li’s theory of FMRA to higher dimensions with arbitrary integral expansive matrix dilations and has established the necessary and sufficient conditions to characterize semiorthogonal multiresolution analysis frames for L2(Rn).
In this paper, we introduce the notion of frame multiresolution analysis (FMRA) on local field K of positive characteristic by extending the above described methods. We first investigate the properties of multiresolution subspaces, which will provide the quantitative criteria for the construction of FMRA on local fields of positive characteristic. We also show that the scaling property of an FMRA also holds for the wavelet subspaces and that the space L2(K) can be decomposed into the orthogonal sum of these wavelet subspaces. Finally, we study the characterization of wavelet frames associated with FMRA on local field K of positive characteristic using the shift-invariant space theory.
The paper is organized as follows. In Section 2, we discuss some preliminary facts about local fields of positive characteristic including the definition of a frame. The notion of frame multiresolution analysis of L2(K) is introduced in Section 3 and its quantitative criteria are given by means of Theorem 12. In Section 4, we establish a complete characterization of wavelet frames generated by a finite number of mother wavelets on local field K of positive characteristic.
2. Preliminaries on Local Fields
Let K be a field and a topological space. Then, K is called a local field if both K+ and K* are locally compact Abelian groups, where K+ and K* denote the additive and multiplicative groups of K, respectively. If K is any field and is endowed with the discrete topology, then K is a local field. Further, if K is connected, then K is either R or C. If K is not connected, then it is totally disconnected. Hence, by a local field, we mean a field K which is locally compact, nondiscrete, and totally disconnected. The p-adic fields are examples of local fields. More details are referred to in [19, 20]. In the rest of this paper, we use N,N0, and Z to denote the sets of natural and nonnegative integers and integers, respectively.
Let K be a fixed local field. Then, there is an integer q=pr, where p is a fixed prime element of K and r is a positive integer, and a norm |·| on K such that for all x∈K we have |x|≥0 and for each x∈K∖{0} we get |x|=qk for some integer k. This norm is non-Archimedean; that is, |x+y|≤max{|x|,|y|} for all x,y∈K and |x+y|=max{|x|,|y|} whenever |x|≠|y|. Let dx be the Haar measure on the locally compact, topological group (K,+). This measure is normalized so that ∫Ddx=1, where D={x∈K:|x|≤1} is the ring of integers in K. Define B={x∈K:|x|<1}. The set B is called the prime ideal in K. The prime ideal in K is the unique maximal ideal in D, and hence as a result B is both principal and prime. Therefore, for such an ideal B in D, we have B=〈p〉=pD.
Let D*=D∖B={x∈K:|x|=1}. Then, it is easy to verify that D* is a group of units in K* and if x≠0, then we may write x=pkx′, x′∈D*. Moreover, each Bk=pkD={x∈K:|x|<q-k} is a compact subgroup of K+ and is known as the fractional ideals of K+ (see [19]). Let U={ai}i=0q-1 be any fixed full set of coset representatives of B in D; then, every element x∈K can be expressed uniquely as x=∑l=k∞clpl with cl∈U. Let χ be a fixed character on K+ that is trivial on D but is nontrivial on B-1. Therefore, χ is constant on cosets of D, implying that if y∈Bk, then χy(x)=χ(yx) for x∈K. Suppose that χu is any character on K+; then, clearly the restriction χu|D is also a character on D. Therefore, if {u(n):n∈N0} is a complete list of distinct coset representatives of D in K+, then, as it was proved in [20], the set {χu(n):n∈N0} of distinct characters on D is a complete orthonormal system on D.
The Fourier transform f^ of a function f∈L1(K)∩L2(K) is defined by
(1)f^(ξ)=∫Kf(x)χξ(x)¯dx.
It is noted that
(2)f^(ξ)=∫Kf(x)χξ(x)¯dx=∫Kf(x)χ(-ξx)dx.
Furthermore, the properties of Fourier transform on local field are much similar to those on the real line. In particular, Fourier transform is unitary on L2(K).
We now impose a natural order on the sequence {u(n):n∈N0}. Since D/B≅GF(q), where GF(q) is a c-dimensional vector space over the field GF(q) (see [20]), we choose a set {1=ζ0,ζ1,ζ2,…,ζc-1}⊂D* such that span {ζj}j=0c-1≅GF(q). For n∈N0 such that 0≤n<q, we have
(3)n=a0+a1p+⋯+ac-1pc-1,0≤ak<p,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiik=0,1,…,c-1.
Define
(4)u(n)=(a0+a1ζ1+⋯+ac-1ζc-1)p-1.
For n∈N0 and 0≤bk<q, k=0,1,2,…,s, we write
(5)n=b0+b1q+b2q2+⋯+bsqs,
such that
(6)un=ub0+ub1p-1+⋯+u(bs)p-s.
Also, for r,k∈N0 and 0≤s<qk, we have
(7)u(rqk+s)=u(r)p-k+u(s).
Further, it is easy to verify that u(n)=0 if and only if n=0 and {u(l)+u(k):k∈N0}={u(k):k∈N0} for a fixed l∈N0. Hereafter, we use the notation χn:=χu(n), n≥0.
Let the local field K be of characteristic p>0 and let ζ0,ζ1,ζ2,…,ζc-1 be as above. We define a character χ on K as follows:
(8)χ(ζμp-j)=exp2πip,μ=0,j=1,1,μ=1,…,c-1orj≠1.
Definition 1.
Let H be a separable Hilbert space. A sequence {fk:k∈N0} in H is called a frame for H if there exist constants A and B with 0<A≤B<∞ such that
(9)Af22≤∑k∈N0f,fk2≤Bf22,∀f∈H.
The largest constant A and the smallest constant B satisfying (9) are called the upper and the lower frame bound, respectively. A frame is said to be tight if it is possible to choose A=B and a frame is said to be exact if it ceases to be a frame when any one of its elements is removed. An exact frame is also known as a Riesz basis.
The following theorem gives us an elementary characterization of frames.
Theorem 2 (see [15]).
A sequence {fk:k∈N0} in a Hilbert space H is a frame for H if and only if there exists a sequence a={ak}∈l2(N0) with al2(N0)≤Cf, C>0, such that
(10)f=∑k∈N0akfk,
and ∑k∈N0|〈f,fk〉|2<∞, for every f∈H.
For j∈Z and y∈K, we define the dilation operator δj and the translation operator τy as follows:
(11)δjfx=qj/2fp-jx,τyfx=fx-y,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiif∈L2K.
Our study uses the theory of shift-invariant spaces developed in [21, 22] and the references therein. A closed subspace S of L2(K) is said to be shift-invariant if τkf∈S whenever f∈S and k∈N0. A closed shift-invariant subspace S of L2(K) is said to be generated by Φ⊂L2(K) if S=span¯{τkφ(x):=φ(x-u(k)):k∈N0,φ∈Φ}. The cardinality of the smallest generating set Φ for S is called the length of S which is denoted by |S|. If |S|= finite, then S is called a finite shift-invariant space (FSI) and if |S|=1, then S is called a principal shift-invariant space (PSI). Moreover, the spectrum of a shift-invariant space is defined to be
(12)σ(S)=ξ∈D:S^(ξ)≠{0},
where S^ξ={f^(ξ+u(k))∈l2(N0):f∈S,k∈N0}.
3. Frame Multiresolution Analysis on Local Fields
We first introduce the notion of a frame multiresolution analysis (FMRA) of L2(K).
Definition 3.
Let K be a local field of positive characteristic p>0 and let p be a prime element of K. A frame multiresolution analysis of L2(K) is a sequence of closed subspaces {Vj:j∈Z} of L2(K) satisfying the following properties:
Vj⊂Vj+1 for all j∈Z;
⋃j∈ZVj is dense in L2(K) and ⋂j∈ZVj={0};
f(·)∈Vj if and only if f(p-1·)∈Vj+1 for all j∈Z;
the function f lying in V0 implies that the collection f(·-u(k))∈V0, for all k∈N0;
the sequence {τkφ:=φ(·-u(k)):k∈N0} is a frame for the subspace V0.
The function φ is known as the scaling function while the subspaces Vj’s are known as approximation spaces or multiresolution subspaces. A frame multiresolution analysis is said to be nonexact and, respectively, exact if the frame for the subspace V0 is nonexact and, respectively, exact. In multiresolution analysis studied in [9], the frame condition is replaced by that of an orthonormal basis or an exact frame.
Next, we establish several properties of multiresolution subspaces that will help in the construction of frame multiresolution analysis on local field K of positive characteristic. The following proposition shows that, for every j∈Z, the sequence {φj,k:k∈N0}, where
(13)φj,k(x)=qj/2φp-jx-u(k),
is a frame for Vj.
Proposition 4.
Let {τkφ:k∈N0} be a frame for V0=span¯{τkφ:k∈N0} and
(14)Vj={f∈L2(K):fpj·∈V0},j∈Z.
Then, the sequence {φj,k:k∈N0} defined in (13) is a frame for Vj with the same bounds as those for V0.
Proof.
For any f∈Vj, we have
(15)∑k∈N0δ-jf,τkφ2=∑k∈N0∫Kq-j/2f(pjx)φ(x-u(k))¯dx2(16)∑k∈N0δ-jf,τkφ2=∑k∈N0∫Kf(x)qj/2φ(p-jx-u(k))¯dx2(17)∑k∈N0δ-jf,τkφ2=∑k∈N0f,φj,k2.
Since {τkφ:k∈N0} is a frame for V0, therefore we have
(18)Af22=Aδ-jf22≤∑k∈N0f,φj,k2≤Bδ-jf22=Bf22.
This completes the proof of the proposition.
We now characterize all functions of FSI space by virtue of its Fourier transforms.
Proposition 5.
Let {τkφ:k∈N0,φ∈Φ} be a frame for its closed linear spanV, where Φ={φ1,φ2,…,φL}⊂L2(K). Then, f∈L2(K) lies in V if and only if there exist integral periodic functions hl∈L2(D), l=1,…,L, such that
(19)f^(ξ)=∑l=1Lhl(ξ)φ^l(ξ).
Proof.
Since the system {τkφ:k∈N0,φ∈Φ} is a frame for V, then, by Theorem 2, there exists a sequence {akl}∈l2(N0), for l=1,…,L, such that
(20)f(x)=∑l=1L∑k∈N0aklφlx-uk.
Taking Fourier transform on both sides of (20), we obtain
(21)f^(ξ)=∑l=1Lhl(ξ)φ^l(ξ),
where hl(ξ)=∑k∈N0aklχk(ξ) are the integral periodic functions in L2(D). The converse is established by taking hl as above and applying the inverse Fourier transform on both sides of (19).
We now study some properties of the multiresolution subspaces Vj of the form (14) by means of the Fourier transform.
Proposition 6.
Let {τkφ:k∈N0} be a frame for V0=
span
¯{τkφ:k∈N0} and, for j∈Z, define Vj by (14). Then, for any function ψ∈V1, there exists periodic function G∈L2(D) such that
(22)ψ^(p-1ξ)=q1/2G(ξ)φ^(ξ).
Proof.
By the definition of Vj, it follows that ψ(p·)∈V0. By Proposition 5, there exists a periodic function G∈L2(D) such that (ψ(p·))∧=ψ^(p-1ξ)=q1/2G(ξ)φ^(ξ) lies in L2(K).
The following theorem establishes a sufficient condition to ensure that the nesting property holds for the subspaces Vj’s.
Theorem 7.
Let {τkφ:k∈N0} be a frame for V0=
span
¯{τkφ:k∈N0} and, for j∈Z, define Vj by (14). Assume that there exists a periodic function H∈L∞(D) such that
(23)φ^(ξ)=q1/2H(pξ)φ^pξ.
Then, Vj⊆Vj+1, for every j∈Z.
Proof.
Given any f∈Vj, there exists a sequence {ak}k∈N0∈l2(N0) such that
(24)f(x)=qj/2∑k∈N0akφ(p-jx-u(k)).
Let m0(ξ)=∑k∈N0akχk(ξ)∈L2(D) and let m1(pξ)=m0(ξ)H(pξ). Then, clearly m1 lies in L2(D) as H lies in L∞(D). Therefore, by Parseval’s identity, there exists a sequence {bk}k∈N0∈l2(N0) such that m1(ξ)=∑k∈N0bkχk(ξ) lies in L2(K).
Taking Fourier transform of (24) and using assumption (23), we obtain
(25)f^(ξ)=qj/2m0pjξφ^pjξ=q(j+1)/2m0pjξHpj+1ξφ^pj+1ξ=q(j+1)/2m1pj+1ξφ^pj+1ξ.
By implementing inverse Fourier transform to (25), we have
(26)f(x)=q(j+1)/2∑k∈N0bkφp-j-1x-u(k).
Using Proposition 4, we observe that f∈Vj+1. Moreover, it is easy to verify that the function H in (23) is not unique.
The following theorem is the converse to Theorem 7.
Theorem 8.
Let {τkφ:k∈N0} be a frame for V0=
span
¯{τkφ:k∈N0} and, for j∈Z, define Vj by (14). Assume that V0⊆V1 and Φ(ξ)=φ^(ξ-u(k))l2(N0)2. Then, there exists periodic function H∈L∞(D) such that (23) holds.
Proof.
Since {τkφ:k∈N0} is a frame for V0, therefore there exist positive constants A and B such that
(27)A≤Φξ≤Ba.e.onσ(V0).
Since V0⊆V1, we have φ∈V1. By Proposition 6, there exists a periodic function H0∈L2(D) such that
(28)φ^p-1ξ=q1/2H0(ξ)φ^(ξ).
Therefore, we have
(29)φ^ξ2=qH0(pξ)2φ^(pξ)2a.e.
Let S=B∖σ(V0) and H∈L2(D) be a periodic function such that H=H0, a.e. on σ(V0), and H is bounded on S by a positive constant C. Then, it follows from the above fact that H is not unique so that (29) also holds for H; that is,
(30)φ^ξ2=qHpξ2φ^pξ2a.e.
Taking n=kp+r, where k∈N0 and r=0,1,…,q-1, we have
(31)φ^(ξ+u(n))2=qH(pξ+pu(r))2|φ^pξ+pur+uk2a.e.
Summing up (31) for all k∈N0 and r=0,1,…,q-1, we have
(32)∑n∈N0φ^ξ+un2=q∑r=0q-1Hpξ+pur2∑k∈N0|φ^pξ+pur+uk2a.e.,
which is equivalent to
(33)Φξ=q∑r=0q-1Hpξ+pur2Φpξ+pura.e.
or
(34)Φp-1ξ=q∑r=0q-1H(ξ+pu(r))2Φ(ξ+pu(r))a.e.
Note that Φ(p-1ξ)≤B a.e. and, hence, (34) becomes
(35)∑r=0q-1Hξ+pur2Φ(ξ+pu(r))≤qBa.e.
This implies that, for almost every ξ∈B-1 and r=0,1,…,q-1, we have
(36)H(ξ+pu(r))2Φ(ξ+pu(r))≤qB.
Also, if Φ(ξ+pu(r))=0, then H(ξ+pu(r))≤C and if Φ(ξ+pu(r))>0, then we may assume that A≤Φ(ξ+pu(r))≤B. Thus, for almost every ξ∈B-1 and r=0,1,…,q-1, we have
(37)H(ξ+pu(r))2≤maxC2,qBA-1.
Hence, H is essentially bounded on D. This proves the theorem completely.
The following two propositions are proved in [23].
Proposition 9.
Suppose V0=
span
¯{τkφ:k∈N0} and, for each j∈Z, define Vj by (14) such that V0⊆V1. Assume that |φ^|>0,a.e on a neighborhood of zero. Then, the union ⋃j∈ZVj is dense in L2(K).
Proposition 10.
Let φ∈L2(K) and define V0=
span
¯{τkφ:k∈N0}. For each j∈Z, define Vj by (14). Then, one has ⋂j∈ZVj={0}.
Lemma 11.
Let Vj be the family of subspaces defined by (14) with Vj⊆Vj+1, for each j∈Z. Suppose φ∈L2(K) is a nonzero function with V0=
span
¯{τkφ:k∈N0}. Then, for every j∈Z,Vj is a proper subspace of Vj+1.
Proof.
Suppose that Vl=Vl+1 for some l∈Z. Let f∈Vj+1; then, for any given j∈Z, we have f(pj+1-l-1x)∈Vj+1. Since f(pj-lx)∈Vl, therefore f lies in Vj and Vj=Vj+1. Hence, ⋂j∈ZVj=V0. By Proposition 10, it follows that Vj={0}, which is a contradiction.
Combining all our results so far, we have the following theorem.
Theorem 12.
Let φ∈L2(K) and define V0=
span
¯{τkφ:k∈N0}. For each j∈Z, define Vj by (14) and Φ(ξ)=φ^(ξ-u(k))l2(N0)2. Suppose that the following hold:
A≤Φ(ξ)≤B a.e. on σ(V0),
there exists a periodic function H∈L∞(D) such that
(38)φ^(ξ)=q1/2H(pξ)φ^(pξ),a.e.
|φ^|>0, a.e. on a neighborhood of zero.
Then, {Vj:j∈Z} defines a frame multiresolution analysis of L2(K).
Proof.
Since V0 is a shift-invariant subspace of L2(K), therefore the system {τkφ:k∈N0} forms a frame for V0 with frame bounds A and B. By Theorem 7 and Lemma 11, it follows that Vj⊂Vj+1, for every j∈Z. Hence, by the definition of Vj,f lies in Vj if and only if f(pj.) lies in V0, while f(p-1·) lies in Vj+1 if and only if f(pj+1·) lies in V0. Thus, f lies in Vj if and only if f(p-1·) lies in Vj+1. Moreover, by assumption (iii) and Proposition 10, it follows that ⋃j∈ZVj is dense in L2(K) and ⋂j∈ZVj={0}. Thus, the sequence {Vj:j∈Z} satisfies all the conditions to be a frame multiresolution analysis of L2(K).
In order to construct wavelet frames associated with frame multiresolution analysis on local fields K of positive characteristic, we introduce the orthogonal complement subspaces {Wj:j∈Z} of Vj in Vj+1. It is easy to verify that the sequence of subspaces {Wj:j∈Z} also satisfies the scaling property; that is,
(39)Wj={f∈L2(K):f(pj·)∈W0},j∈Z.
Theorem 13.
Let {Vj:j∈Z} be an increasing sequence of closed subspaces of L2(K) such that ⋃j∈ZVj is dense in L2(K) and ⋂j∈ZVj={0}. Let Wj be the orthogonal complement of Vj in Vj+1, for each j∈Z. Then, the subspaces Wj are pairwise orthogonal and
(40)L2(K)=⨁j∈ZWj.
Proof.
Assume that i<j; then, 〈fi,fj〉=0, for any fi∈Wj as Wi⊂Vi+1⊂Vj. Let Pj be the orthogonal projection operators from L2(K) onto Vj; then, limj→∞Pjf=f, limj→-∞Pjf=0, and Wj={f-Pjf:f∈Vj+1}. Therefore, for any f∈L2(K), we have
(41)f=∑j∈Z(Pj+1f-Pjf).
Thus, the result of the direct sum follows since Pj+1-Pj is the orthogonal projector from L2(K) onto Wj.
4. Characterization of Wavelet Frames on Local Fields
In this section, we give the characterization of wavelet frames associated with frame multiresolution analysis on local fields of positive characteristic. First, we will characterize the existence of a function ψ in W0, where W0 is the orthogonal complement of V0 in V1, by virtue of the analysis filters G and H, defined as in Section 3.
Theorem 14.
Let H be a periodic function associated with the frame multiresolution analysis {Vj:j∈Z} satisfying the condition (23). Define W0 as the orthogonal complement of V0 in V1. Let ψ∈V1 such that
(42)ψ^ξ=q1/2Gpξφ^pξ,
where G is a periodic function in L2(D). Then, ψ lies in W0 if and only if
(43)∑r=0q-1Hpξ+purΦpξ+purGpξ+pur¯=0a.e.ξ.
Proof.
We note that ψ lies in W0 if and only if
(44)ψ,τkψ=〈ψ,ψ(·-u(k))〉=0,∀k∈N0.
Define
(45)F(ξ)=∑k∈N0φ^(ξ+u(k))ψ^(ξ+u(k))¯.
Then, it is easy to verify that F lies in L1(D) by using Monotonic Convergence Theorem and the Plancherel Theorem as
(46)∫D|F(ξ)|dξ≤∫D∑k∈N0φ^(ξ+u(k))ψ^(ξ+u(k))dξ=∑k∈N0∫Dφ^(ξ+u(k))ψ^(ξ+u(k))dξ=∫Kφ^(ξ)ψ^(ξ)dξ≤φ^2ψ^2=φ2ψ2.
For a fixed n∈N0, we define FM as
(47)FM(ξ)=∑k=0Mφ^(ξ+u(k))ψ^(ξ+u(k))¯χn(ξ).
Then, in view of (23) and (42), we have
(48)FMξ=q∑r=0q-1∑qk+r≤MH(pξ+pu(r))φ^(pξ+pu(r)+u(k))2∑r=0q-1∑qk+r≤M·G(pξ+pu(r))¯χn(ξ).
Using Monotonic Convergence Theorem and the Cauchy-Schwartz inequality, we obtain
(49)FM-FχnL2D≤∫D∑k≥M+1φ^(ξ+u(k))ψ^(ξ+u(k))dξ=∑k≥M+1∫Dφ^(ξ+u(k))ψ^(ξ+u(k))dξ=∑k≥M+1∫x+Dφ^(ξ)ψ^(ξ)dξ≤∫|ξ|>Mφ^(ξ)ψ^(ξ)dξ≤∫ξ>M|φ^(ξ)|2dξ1/2∫ξ>Mψ^ξ2dξ1/2⟶0asM⟶∞.
Hence,
(50)limM→∞FM-FχnL2(D)=0.
Therefore, there exists a subsequence {FMj} such that
(51)limj→∞FMj-FχnL2(D)=0,a.e.
Hence,
(52)Fξ=q∑r=0q-1H(pξ+pu(r))·Φ(pξ+pu(r))G(pξ+pu(r))¯a.e.
Using (50) and the Dominated Convergence Theorem, we have, for all n∈N0,
(53)〈ψ,τ-nφ〉=∫Kψ^(ξ)φ^(ξ)¯χn(ξ)dξ=∑k∈N0∫x+Dψ^(ξ)φ^(ξ)¯χn(ξ)dξ=limM→∞∑k=0M∫Dψ^(ξ+u(k))limM→∞∑k=0M∫D·φ^(ξ+u(k))¯χn(ξ)χk(ξ)dξ=limM→∞∫DFM(ξ)dξ=∫DF(ξ)χn(ξ)dξ.
Consequently, F=0, a.e., is the necessary and sufficient condition for (44) to hold for all n∈N0.
Lemma 15.
Let {Wj:j∈Z} be a sequence of pairwise orthogonal closed subspaces of L2(K) such that L2(K)=⨁j∈ZWj. Then, for every f∈L2(K), there exist fj∈Wj, j∈Z, such that f(x)=∑j∈Zfj(x). Furthermore,
(54)f22=∑j∈Zfj22.
Proof.
For any arbitrary function f∈L2(K), we have
(55)limn→∞f-∑j=-nnfj2=0,
where fj∈Wj, for each j∈Z. Moreover, for a fixed n∈N, we have
(56)∑j=-nnfj22=∑j=-nnfj22.
Since the norm ·2 is continuous, therefore the desired result is obtained by taking n→∞ on both sides of the above equality.
Theorem 16.
Let φ be the scaling function for a frame multiresolution analysis {Vj:j∈Z} and suppose that Wj is the orthogonal complement of Vj in Vj+1. Let Ψ={ψ1,ψ2,…,ψL}⊂W0. Then, the collection
(57)FΨ=ψj,kl(x)∶=qj/2ψl(p-jx-u(k)),j∈Z,k∈N0,l=1,…,Lψj,kl
constitutes a wavelet frame for L2(K) with frame bounds A and B if and only if
(58){τkψl:k∈N0,l=1,…,L}
forms a frame for W0 with frame bounds A and B.
Proof.
Suppose that the system FΨ given by (57) is a wavelet frame for L2(K) with bounds A and B. Then, it follows from (39) that the family of functions ψj,kl lies in Wj, for l=1,…,L, j∈Z, and k∈N0.
By applying Theorem 13 to an arbitrary function f∈W0, we have
(59)∑j∈Z∑k∈N0f,ψj,kl2=∑k∈N0f,τkψl2.
Using the frame property of the system FΨ, we have
(60)Af22≤∑l=1L∑k∈N0f,τkψl2≤Bf22,
and it follows that the collection {τkψl:k∈N0,l=1,…,L} is a frame for W0.
Conversely, suppose that the collection {τkψl:k∈N0,l=1,…,L} is a frame for W0 with bounds A and B. For any fixed j∈Z and f∈Wj, we have from (39) that f(pj·)∈W0. Moreover, by making use of the fact that
(61)f,ψj,kl=qj/2∫Kfxψlp-jx-uk¯dx,q-j/2f(pj·)22=q-j∫Kfpjx2dx=f22,
we have
(62)Aq-j/2f(pj·)22≤∑l=1L∑k∈N0f,ψj,kl2≤Bq-j/2f(pj·)22.
Thus, for a given j∈Z, the collection {ψj,kl:k∈N0,l=1,…,L} constitutes a frame for Wj with frame bounds A and B.
Let f be an arbitrary function in L2(K); then, by Theorem 13 and Lemma 15, there exist fj∈Wj such that
(63)f=∑j∈Zfj,fi,ψj,kl=0,i≠j.
Therefore, we have
(64)∑l=1L∑j∈Z∑k∈N0f,ψj,kl2=∑l=1L∑j∈Z∑k∈N0∑i∈Zfi,ψj,kl2=∑l=1L∑j∈Z∑k∈N0fj,ψj,kl2.
Using (62), we obtain
(65)A∑j∈Zfj22≤∑l=1L∑j∈Z∑k∈N0〈fj,ψj,kl〉2≤B∑j∈Zfj22.
Combining (64), (65), and Lemma 15, we have
(66)Afj22≤∑l=1L∑j∈Z∑k∈N0〈fj,ψj,kl〉2≤Bfj22.
This completes the proof of the theorem.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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