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The characterization of a class of multivariate MRA and semi-orthogonal parseval frame wavelets
Applied Mathematics and Computation 217 (2011) 9151–9164
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The characterization of a class of multivariate MRA and semi-orthogonal parseval frame wavelets q Yun-Zhang Li ⇑, Feng-Ying Zhou College of Applied Sciences, Beijing University of Technology, Beijing 100124, PR China
a r t i c l e
i n f o
Keywords: Parseval frame wavelet Semi-orthogonal Parseval frame wavelet MRA Parseval frame wavelet Semi-orthogonal MRA Parseval frame wavelet
a b s t r a c t Let A be a d d expansive matrix with jdetAj = 2. This paper addresses Parseval frame wavelets (PFWs) in the setting of reducing subspaces of L2 ðRd Þ. We prove that all semiorthogonal PFWs (semi-orthogonal MRA PFWs) are precisely the ones with their dimension functions being non-negative integer-valued (0 or 1). We also characterize all MRA PFWs. Some examples are provided. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Let H be a separable Hilbert space. An at most countable sequence {fn:n 2 I} in H is called a frame for H if there exist 0 < C1 6 C2 < 1 such that
C 1 kf k2 6
X
jhf ; fn ij2 6 C 2 kf k2
ð1Þ
n2I
for f 2 H, where C1, C2 are called lower frame bound and upper frame bound, respectively; is called a tight frame for H with frame bound C1 if C1 = C2 in (1); is called a Parseval frame for H if C1 = C2 = 1 in (1), which is, in particular, an orthonormal basis for H if kfnk = 1 for n 2 I in addition. {fn:n 2 I} is called a Bessel sequence in H if the right inequality in (1) holds for f 2 H, where C2 is called the Bessel bound; is called a frame sequence if it is a frame for its closed linear span in H. The fundamentals of frames can be found in [1] and [2]. We denote by Z; Zþ and N the set of integers, the set of nonnegative integers and the set of positive integers, respectively. For d 2 N, we denote by Id the d d identity matrix, by M⁄ the transpose of M for a d d real matrix M, by xk the k-th component of x for x 2 Rd , by Td ¼ ½ 12 ; 12 Þd the d-dimensional torus in Rd , and by L2 ðTd Þ R the Hilbert space of Zd -periodic square-integrable functions with the inner product hf ; gi ¼ Td f ðxÞgðxÞdx for f ; g 2 L2 ðTd Þ. The Fourier transform is defined by
^f ðÞ ¼
Z
f ðxÞe2pihx;i dx for f 2 L1 ðRd Þ \ L2 ðRd Þ
Rd
and extended to L2 ðRd Þ by Plancheral’s theorem. Let S be a measurable set in Rd . A collection {Si: i 2 I} of countably many S measurable sets is called a partition of S if S = i2ISi, and Si \ Sj = ; up to a set of measure zero for i, j 2 I with i – j. A d d matrix A is called an expansive matrix if it is an integer matrix with all its eigenvalues being greater than 1 in module. Let A be a d d expansive matrix. Define the dilation operator D and the shift operator Tk with k 2 Zd on L2 ðRd Þ respectively by q Supported by the National Natural Science Foundation of China, Beijing Natural Science Foundation (Grant No. 1092001), the Scientific Research Common Program of Beijing Municipal Commission of Education, the Project-sponsored by SRF for ROCS, SEM of China. ⇑ Corresponding author. E-mail address: [email protected] (Y.-Z. Li).
0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.150
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Df ðÞ :¼ j det Aj2 f ðAÞ; T k f ðÞ :¼ f ð kÞ for f 2 L2 ðRd Þ. Obviously, they are both unitary operators on L2 ðRd Þ. For an arbitrary measurable set X in Rd , we denote by FL2(X) the closed subspace of L2 ðRd Þ of the form
n o FL2 ðXÞ :¼ f 2 L2 ðRd Þ : ^f ðÞ ¼ 0 a:e: on Rd n X : A closed subspace X of L2 ðRd Þ is called a reducing subspace if DX = X and TkX = X for each k 2 Zd . The following proposition provides us with a characterization of reducing subspaces: Proposition 1 ([3], Theorem 1). Let A be a d d expansive matrix. A closed subspace X of L2 ðRd Þ is a reducing subspace if and only if X = FL2(X) for some measurable set X in Rd with the property X = A⁄X. Let A be a d d expansive matrix, and let X be a nonzero reducing subspace of L2 ðRd Þ. By Proposition 1, X corresponds a set X Rd with nonzero measure for which
A X ¼ X and X ¼ FL2 ðXÞ: So, to be specific, we denote a reducing subspace by FL2(X) instead of X. In particular, L2 ðRd Þ is a reducing subspace of L2 ðRd Þ for an arbitrary expansive matrix A, and FL2([0, 1)) (Hardy space) is also a reducing subspace of L2 ðRÞ for an arbitrary 2 6 A 2 N. Throughout this paper, we always assume that A is a d d expansive matrix with jdetAj = 2. For such A, we have the following proposition: Proposition 2. Let A be a d d expansive matrix with jdetAj = 2. Then (i) f( + (A⁄)1e) = f( + (A⁄)1d) for an arbitrary Zd -periodic function f, e and d with {0, e} and {0, d} being both the sets of representatives of distinct cosets in Zd =A Zd ; (ii) there exists 1 6 k0 6 d such that 2ððA Þ1 eÞk0 is odd for each e with {0, e} being a set of representatives of distinct cosets in Zd =A Zd . Proof (i) Since ðA Þ1 ðe dÞ 2 Zd , we have (i) by Zd -periodicity of f. (ii) Fix e with {0, e} being a set of representatives of distinct cosets in Zd =A Zd . Since jdetAj = 2, we have 2ðA Þ1 e 2 Zd . Also observing that ðA Þ1 e R Zd leads to 2ðA Þ1 e 2 Zd n 2Zd . This implies that 2ððA Þ1 eÞk0 is odd for some 1 6 k0 6 d. Let {0, d} be another set of representatives of distinct cosets in Zd =A Zd . We always have ðA Þ1 e ðA Þ1 d 2 Zd , which implies that 2ððA Þ1 e ðA Þ1 dÞ 2 2Zd , and thus 2ððA Þ1 dÞk0 is odd. So (ii) holds. The proof is completed. h Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. A function w 2 FL2(X) is called a Parseval frame wavelet (PFW) for FL2(X) if fDj T k w : j 2 Z; k 2 Zd g is a Parseval frame for FL2(X); is called a semi0 0 0 orthogonal PFW for FL2(X) if it is a PFW for FL2(X) and hDjTkw, Dj Tk0 wi = 0 for (j, k), ðj ; k Þ 2 Z Zd with j – j0 ; is called a wavej 2 2 d let for FL (X) if fD T k w : j 2 Z; k 2 Z g is an orthonormal basis for FL (X), which is obviously a semi-orthogonal PFW. Definition 1. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. A sequence fV j gj2Z of closed subspaces of FL2(X) is called a multiresolution analysis (an MRA) for FL2(X) associated with A if the following conditions are satisfied: (i) (ii) (iii) (iv)
V Vj+1 for j 2 Z; T Sj 2 j2Z V j ¼ FL ðXÞ and j2Z V j ¼ f0g; f 2 V0 if and only if Djf 2 Vj for j 2 Z; there exists / 2 FL2(X) such that fT k / : k 2 Zd g is an orthonormal basis for V0.
Some other MRA structures were introduced in [4–7] for the construction of wavelet frames in L2 ðRd Þ. This definition is a natural generalization of the usual MRA for L2 ðRd Þ. Some more general notions can be found in [8–10]. We can refer to [11–16] for some related results on wavelets in reducing subspaces. In Definition 1, we call / a scaling function of the T MRA. From Theorem 1.1 in [17], it follows that the condition j2Z V j ¼ f0g in (ii) is trivial, a special case of which can be obtained by Corollary 4.14 in [18]. By the definition, V j ¼ spanfDj T k / : k 2 Zd g (so we say / generates the MRA), and there ^ Þ ¼ mðÞ/ðÞ ^ a.e. on Rd . It is easy to check that exists a unique m 2 L2 ðTd Þ such that /ðA
jmðÞj2 þ jmð þ ðA Þ1 eÞj2 ¼ 1
ð2Þ
Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
9153
a.e. on Rd , where {0, e} is a set of representatives of distinct cosets in Zd =A Zd . Let k0 be as in Proposition 2. Define m1 2 L2 ðTd Þ by
m1 ðnÞ ¼ e2pink0 lðA nÞmðn þ ðA Þ1 eÞ 2
ð3Þ
d
and w 2 L ðR Þ via its Fourier transform by
^ ^ Þ1 nÞ; wðnÞ ¼ m1 ððA Þ1 nÞ/ððA
ð4Þ
d
where l is an arbitrary Z -periodic, unimodular and measurable function. Note that
mðnÞ
mðn þ ðA Þ1 eÞ
!
m1 ðnÞ m1 ðn þ ðA Þ1 eÞ is a unitary matrix. By the same procedure as in [[19],Proposition 2.3], we can prove that fT k w : k 2 Zd g is an orthonormal basis for W0 = V1 V0 (W0 is the orthogonal complement of V0 in V1), and thus w is a wavelet for FL2(X), which is independent of the choice of e by Proposition 2. Such w is called an MRA wavelet since it is associated with an MRA. In this paper, we focus on expansive matrices A satisfying jdet Aj = 2. Many applications, such as image compression, employ wavelet bases in R2 . Tensor-product wavelets impose an unnecessary product structure on the plane, which is artificial for natural images. For a 2 2 expansive matrix A with jdet Aj = 2, wavelets are nonseparable, and thus offer the hope of a more isotropic analysis. For a general d d expansive matrix A, a wavelet frame for L2 ðRd Þ induced by a refinable function requires at least jdetAj 1 wavelets. However, wavelet frame systems with few generators can reduce the complexity of the associated wavelet transform. The restriction jdetAj = 2 may admit few wavelets. In addition, by the above argument, MRA wavelets have an explicit expression when jdetAj = 2. See [20–24] for details. MRA wavelets have many desirable features, but they impose some restrictions. Even if for the one-dimensional case 6pin A = 2 in the setting of L2 ðRÞ, there are useful ‘‘filters’’, such as mðnÞ ¼ 1þe2 , that do not produce orthonormal bases, nevertheless, they do produce systems that have the reconstruction property, as well as many useful features. Therefore, it is natural to develop a theory involving more general filters that produce systems having these properties. A natural setting for such a theory is provided by frames (see [[19], Chapter 8]). For the one-dimensional case A = 2, this problem was studied in [4,5,25] and [26] under the setting of L2 ðRÞ. However, the results there do involve certain restrictions and technical 6pin assumptions such as semi-orthogonality, and they exclude the filter mðnÞ ¼ 1þe2 . Fortunately, the authors in [27] and [28] introduced the notions of generalized filter, pseudo-scaling function and MRA PFW (which includes the filter mentioned above), and characterized PFWs being respectively semi-orthogonal PFWs, MRA PFWs and semi-orthogonal MRA PFWs for L2 ðRÞ. Some related results were also obtained in [29]. For d d expansive matrices A satisfying Ad = 2Id, some results on the construction of wavelet frames, wavelet bi-frames and sibling frames for L2 ðRd Þ were obtained in [30–35]. It is obvious that jdetAj = 2 when Ad = 2Id. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Li and Zhou jointly in [9] and [10] obtained an explicit expression of wavelet frames and semi-orthogonal wavelet frames for FL2(X) induced by refinable functions. This paper extends the results in [28] to the case of a general d d expansive matrix A with jdetAj = 2 under the setting of FL2(X). Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. A function m defined on d R is called a generalized filter if it is a Zd -periodic measurable function satisfying (2) a.e. on Rd . We denote by e F the set of all generalized filters. A function / 2 L2 ðRd Þ is called a pseudo-scaling function if there exists m 2 e F (not necessarily unique) such that
^ Þ ¼ mðÞ/ðÞ ^ /ðA
ð5Þ
a.e. on Rd . A PFW w for FL2(X) is called an MRA PFW for FL2(X) if there exist m 2 e F, a Zd -periodic unimodular measurable function l and a pseudo-scaling function / such that both (3) and (4) hold a.e. on Rd . These notions are a natural generalization of those in [27], where the one-dimensional case A = 2 was treated under the setting of L2 ðRÞ. Let us first see some examples. Example 1. Let A be a d d expansive matrix with Ad = 2Id, let L be a positive odd number, and let {0, e} be a set of representatives of distinct cosets in Zd =A Zd with ð2ðA Þ1 eÞk0 being odd. Define w 2 L2 ðRd Þ via its Fourier transform by
^ wðnÞ ¼ ie
0 1 l1 d1 piLððA Þ nÞk0 Y e sinðpLððA Þl1 nÞk0 Þ 1 A 0 sinðpLððA Þ nÞk0 Þ @ pLððA Þl1 nÞk0 l¼0
ð2þLÞpiððA Þ1 nÞk
for n 2 Rd . Then w is an MRA PFW for L2 ðRd Þ. Proof. Define
mðnÞ ¼
1 þ e2piLnk0 ; 2
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and / 2 L2 ðRd Þ via its Fourier Transform by d1 Y
^ /ðnÞ ¼
epiLððA
l
Þ nÞk
0
sinðpLððA Þl nÞk0 Þ
l¼0
pLððA Þl nÞk0
for n 2 Rd . Then it is easy to check that m 2 e F,
^ ^ Þ1 nÞ /ðnÞ ¼ mððA Þ1 nÞ/ððA and
^ wðnÞ ¼ e2piððA
1
Þ
nÞk
0
^ Þ1 nÞ mððA Þ1 n þ ðA Þ1 eÞ/ððA
ð6Þ
for n 2 Rd . This implies that / is a pseudo-scaling function, and that w is a PFW for L2 ðRd Þ by Theorem 4.1 in [9]. Therefore w is an MRA PFW for L2 ðRd Þ by the definition of MRA PFW. The proof is completed. h 6pin
Remark 1. When d = 1, A = 2 and L = 3 in Example 1, mðnÞ ¼ 1þe2 , which is mentioned in the above. There are many expan0 1 1 1 0 1 1 d sive matrices satisfying A = 2Id. For example, A1 ¼ and A2 ¼ @ 0 1 1 A satisfy A21 ¼ 2I2 and A32 ¼ 2I3 , respec1 1 1 1 0 tively. Ehler in [30] provided such a class of examples satisfying Ad = 2Id: for d = 2, 3, let
A¼
0
1 1 ; 1 1
0
2
1
1
1
1
1
B C A ¼ @ 1 1 0 A
and for d > 3, let
1 0 2 1 1 C B . .. B .. . 1 0 0C C B B . .. .. .. .. C C B .. . . . . C: A¼B C B B 0 0 1 0C C B C B @ 1 1 0 A 1 1 0
Example 2. Let A ¼
1 , and let L be a positive odd number. Define w 2 L2 ðR2 Þ by 1
1 1
81 x 2 D1 ; > < L2 ; wðxÞ ¼ 12 ; x 2 D2 ; > : L 0; otherwise for x 2 R2 , where
1 L1 ; 0 6 x1 x 2 < L ; x 2 R2 : 6 x2 < 2 2 Lþ1 1 2 6 x2 < ; 0 6 x 1 x 2 < L : D2 ¼ x 2 R : 2 2
D1 ¼
Then w is an MRA wavelet for L2 ðR2 Þ when L = 1, and is a non-semi-orthogonal MRA PFW for L2 ðR2 Þ when L P 3. Proof. Take
e ¼ 10
and k0 = 1 in Example 1. A simple computation shows that
ð2þLÞpiððA ^ wðnÞ ¼ ie
1
Þ
nÞ1
sinðpLððA Þ1 nÞ1 Þ epiLn1
sinðpLn1 Þ piLððA Þ1 nÞ1 sinðpLððA Þ1 nÞ1 Þ e pLn1 pLððA Þ1 nÞ1
for n 2 R2 . So, by Example 1, w is an MRA PFW for L2 ðR2 Þ. When L = 1, by the same procedure as in Example 1, we can prove that w is associated with / as in (6) with
mðnÞ ¼
1 þ e2pin1 2
and /ðxÞ ¼ v½0;1Þ ðx1 x2 Þv½0;1Þ ðx2 Þ:
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It is easy to check that k/k = 1. This implies that / generates an MRA for L2 ðR2 Þ by Theorem 1 and Theorem 2 in [36], and thus w is an MRA wavelet for L2 ðR2 Þ. When L P 3, it is easy to check that
hw; wðAÞi ¼
2 L1 2L2
– 0:
Therefore w is a non-semi-orthogonal MRA PFW for L2 ðR2 Þ. The proof is completed. h 0 0 1 0 1 0 1 1 0 1 B 0 ⁄ 1 ⁄ 1 @ A @ A B Note that (2(A ) e)1 = 1 for A ¼ 0 1 1 and e ¼ 0 , and that (2(A ) e)1 = 1 for A ¼ @ 1 1 1 0 0 0 1 1 1 B0C C e¼B @ 0 A . By the same procedure as in Example 2, we have the following two examples: 0 0 1 1 1 0 Example 3. Let A ¼ @ 0 1 1 A, and let L be a positive odd number. Define w 2 L2 ðR3 Þ by 1 1 0
2 0 1 1
1 1 1 1
1 1 0C C and 0A 1
81 x 2 D1 ; > < L3 ; 1 wðxÞ ¼ 3 ; x 2 D2 ; > : L 0; otherwise
for x 2 R3 , where
L1 1 < x2 6 ; 0 6 x1 x3 < L; 0 6 x2 þ x3 < L ; 2 2 1 L þ 1 ; 0 6 x1 x3 < L; 0 6 x2 þ x3 < L : D2 ¼ x 2 R3 : < x2 6 2 2
D1 ¼
x 2 R3 :
Then w is an MRA wavelet 0 0 B 0 Example 4. Let A ¼ B @ 1 1
for L2 ðR3 Þ when L = 1, and is a non-semi-orthogonal MRA PFW for L2 ðR3 Þ when L P 3. 1 2 1 1 0 1 0C C, and let L be a positive odd number. Define w 2 L2 ðR4 Þ by 1 1 0 A 1 1 1
81 x 2 D1 ; > < L4 ; 1 wðxÞ ¼ 4 ; x 2 D2 ; > : L 0; otherwise
for x 2 R4 , where
D1 ¼ fx 2 R4 : 0 6 x4 < L; 0 6 x3 þ x4 < L; 0 6 x1 þ x2 þ x3 þ x4 < L; 1 6 2ðx2 þ x3 þ x4 Þ < L 1g; D2 ¼ fx 2 R4 : 0 6 x4 < L; 0 6 x3 þ x4 < L; 0 6 x1 þ x2 þ x3 þ x4 < L; 1 L 6 2ðx2 þ x3 þ x4 Þ < 1g: Then w is an MRA wavelet for L2 ðR4 Þ when L = 1, and is a non-semi-orthogonal MRA PFW for L2 ðR4 Þ when L P 3. For examples in the setting of reducing subspaces, see Example 5 and Example 6 in Section 3. This paper focuses on the characterization of PFWs being respectively semi-orthogonal PFWs, MRA PFWs and semi-orthogonal MRA PFWs in the setting of reducing subspaces. Let w 2 L2 ðRd Þ, and let A be a d d expansive matrix with jdetAj = 2. Define its periodization function rw by
rw ðÞ ¼
X
^ þ kÞj2 jwð
ð7Þ
k2Zd
a.e on Rd , its dimension function Dw by
Dw ðÞ ¼
2 1 X X ^ j wððA Þ ð þ kÞÞ
ð8Þ
j¼1 k2Zd
a.e. on Rd , and define Wj for each j 2 N by
^ Þj ð þ kÞÞ : k 2 Zd g Wj ðÞ ¼ fwððA
ð9Þ
a.e. on Rd , and Fw() by
F w ðÞ ¼ spanfWj ðÞ : j 2 Ng
ð10Þ
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a.e. on Rd , where the closure is taken in l ðZd Þ. Note that
Z Td
Dw ðnÞdn ¼
1 Z X j¼1
^ Þj nÞj2 dn ¼ jwððA
1 X
Rd
^ 2 ¼ kwk ^ 2¼ 2j kwk
j¼1
Z Td
rw ðnÞdn:
2
ð11Þ 2
We have rw(), Dw() < 1 a.e. on Rd by their Zd -periodicity, Wj ðÞ 2 l ðZd Þ a.e. on Rd , and Fw() is a closed subspace of l ðZd Þ a.e. on Rd . Define
tk ðÞ ¼
1 X
^ Þj ÞwððA ^ Þj ð þ kÞÞ wððA
ð12Þ
j¼0
a.e. on Rd for k 2 Zd . Note that
1 X ^ j ^ j ^ 2 þ Dw ðÞÞ12 ðrw ðÞ þ Dw ðÞÞ12 < 1 wððA Þ ÞwððA Þ ð þ kÞÞ 6 ðjwðÞj j¼0
a.e. on Rd by Cauchy–Schwartz inequality. It follows that tk is well defined a.e. on Rd for k 2 Zd . In this paper, we obtain the following results: Theorem 1. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X). Then w is a semi-orthogonal PFW for FL2(X) if and only if Dw ðÞ 2 Zþ a.e. on Rd . Theorem 2. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X). Then w is an MRA PFW for FL2(X) if and only if dimFw() 2 {0, 1} a.e. on Rd . Theorem 3. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X). Then w is a semi-orthogonal MRA PFW for FL2(X) if and only if Dw() 2 {0, 1} a.e. on Rd . This paper is organized as follows. Section 2 is devoted to proof of Theorem 1; Section 3 is devoted to proof of Theorem 2 and Theorem 3. 2. Proof of Theorem 1 The following lemma is borrowed from [16], the L2 ðRd Þ version (i.e. X ¼ Rd ) of which was obtained independently by Bownik in [37] and Calogero in [38]. Lemma 1 ([16], Theorem 3). Let A be a d d expansive matrix, let FL2(X) be a reducing subspace of L2 ðRd Þ, and let w 2 FL2(X). Then w is a PFW for FL2(X) if and only if
X
^ Þj Þj2 ¼ v ðÞ and tk ðÞ ¼ 0 jwððA X
j2Z
a.e. on Rd for k 2 Zd n A Zd . Lemma 2 ([12], Lemma 3.1). Let A be a d d expansive matrix. Then, for an arbitrary f 2 L1 ðRd Þ, limj?1f(Aj) = 0 a.e. on Rd . The following lemma is a well-known result, some variations of which can be found in [2,4,39] and [40]. Lemma 3. For w 2 L2 ðRd Þ, the following hold: (i) fT k w : k 2 Zd g is a Bessel sequence with Bessel bound C if and only if rw() 6 C a.e. on Rd ; (ii) fT k w : k 2 Zd g is a frame sequence with frame bound C1 and C2 if and only if C1vU() 6 rw() 6 C2vU() a.e. on Rd , where U ¼ fn 2 Rd : rw ðnÞ > 0g. Lemma 4. Let w 2 L2 ðRd Þ, let A be a d d expansive matrix with jdetAj = 2, and let {0, e} be a set of representatives of distinct cosets in Zd =A Zd . Then
Dw ðÞ þ Dw ð þ ðA Þ1 eÞ ¼ Dw ðA Þ þ rw ðA Þ a.e. on Rd . Proof. Since Zd ¼ A Zd þ f0; eg, we have
Dw ðA Þ ¼
1 X X j¼1 k2Zd
^ Þjþ1 ð þ kÞÞj2 þ jwððA
1 X X j¼1 k2Zd
^ Þjþ1 ð þ k þ ðA Þ1 eÞÞj2 jwððA
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a.e. on Rd , and consequently, by the definition of Dw and rw,
Dw ðA Þ ¼ Dw ðÞ
X
^ ð þ kÞÞj2 þ Dw ð þ ðA Þ1 eÞ jwðA
k2Zd
X
^ ð þ k þ ðA Þ1 eÞÞj2 ¼ Dw ðÞ þ Dw ð þ ðA Þ1 eÞ rw ðA Þ jwðA
k2Zd
a.e. on Rd . The proof is completed. h Lemma 5. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X). Then the following are equivalent: (i) w is a semi-orthogonal PFW for FL2(X); (ii) rw() = vU() a.e. on Rd , where U ¼ fn 2 Rd : rw ðnÞ > 0g; P (iii) kwk2 ¼ k2Zd jhw; T k wij2 ; P d ^ j ^ for j 2 N. (iv) d wððA Þ ð þ kÞÞwð þ kÞ ¼ 0 a.e. on R k2Z
Proof. Write W j ¼ spanfDj T k w : k 2 Zd g for j 2 Z. Since w is a PFW for FL2(X), we have
f ¼
XX
hf ; Dj T k wiDj T k w for f 2 W 0 ;
ð13Þ
j2Z k2Zd
kwk2 ¼
2 X X j hw; D T k wi :
ð14Þ
j2Z k2Zd
Note that w being a semi-orthogonal PFW for FL2(X) is equivalent to W0\Wj for 0 – j 2 Z. Combined with (13), it follows that (i) is equivalent to fT k w : k 2 Zd g being a Parseval frame for W0, which implies the equivalence between (i) and (ii) by Lemma 3. By Lemma 3, we also have (ii) implies (iii). Next we prove that ðiiiÞ ) ðiÞ () ðiv Þ. Since 0 0 0 0 hDj T k w; Dj T k0 wi ¼ hw; Dj j T k0 Aj0 j k wi for k, k 2 Zd and j, j 2 Z with j0 > j, (i) is equivalent to hw, DjTkwi = 0 for j > 0 and j d k 2 Z . Suppose (iii) holds. Then hw, D Tkwi = 0 for 0 – j 2 Z by (14), and thus (i) holds. Now we turn to the equivalence between (i) and (iv). Note that
Z j 2pihk;ni ^ wððA ^ Þj nÞe2pihk;ðA Þj ni dn ¼ 22 ^ Þj nÞwðnÞe ^ wðnÞ wððA dn Rd Rd ! Z X j ^ Þj ðn þ lÞÞwðn ^ þ lÞ e2pihk;ni dn wððA ¼ 22 j
hw; Dj T k wi ¼ 22
Z
Td
l2Zd
for j > 0 and k 2 Zd . By the equivalence between (i) and hw, DjTkwi = 0 for j > 0 and k 2 Zd , we obtain the equivalence between (i) and (iv). The proof is completed. h Lemma 6. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X). Define
Hn ðÞ ¼
1 X X
^ Þj ÞwððA ^ Þj ð þ kÞÞ ^ Þn ð þ kÞÞwððA wððA
j¼0 k2Zd
a.e. on Rd for each n 2 N. Then
^ Hn ðÞ ¼ Hn1 ðA Þ þ wðÞ
X
^ þ kÞwððA ^ Þn ð þ kÞÞ wð
k2Zd
a.e. on Rd for 1 < n 2 N. Proof. We first prove that Hn() is well-defined a.e. on Rd . Note that, for n 2 N and a.e. n 2 Rd , !1 !12 2 2 X 1 1 X X X X ^ Þj nÞj ^ Þn ðn þ kÞÞjjwððA ^ Þj ðn þ kÞÞj 6 ^ Þj nÞj ^ Þn ðn þ kÞÞj2 ^ Þj ðn þ kÞÞ jwððA jwððA jwððA jwððA wððA j¼0
j¼0
k2Zd
6
1 X j¼0
k2Zd
^ Þj nÞj jwððA
X
k2Zd
!12 ^ Þj ðn þ kÞÞj2 jwððA
n
1 2
ðrw ððA Þ nÞÞ 6
j¼0
k2Zd 1
1 X
n
1
1
ðrw ðnÞ þ Dw ðnÞÞ2 ðrw ððA Þ nÞÞ2 6 ð1 þ Dw ðnÞÞ2 < 1
!12 ^ Þj nÞj2 jwððA
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by Lemma 1 and Lemma 3. So 1 X X
^ Þj ÞwððA ^ Þj ð þ kÞÞ ^ Þn ð þ kÞÞwððA wððA
j¼0 k2Zd
converges absolutely a.e. on Rd . It follows that Hn() is well-defined, and that
Hn ðÞ ¼
X
^ Þn ð þ kÞÞtk ðÞ wððA
ð15Þ
k2Zd
a.e. on Rd for n 2 N, where tk is defined as in (12). Also observing that
^ Þ1 þkÞ þ t A k ðÞ ^ Þ1 ÞwððA tk ððA Þ1 Þ ¼ wððA leads to
X
Hnþ1 ððA Þ1 Þ ¼
^ Þnþ1 ððA Þ1 þkÞÞt k ððA Þ1 Þ wððA
k2Zd
^ Þ1 Þ ¼ wððA
X nþ1 1 ^ ðA Þ ððA Þ þkÞ wððA ^ Þ1 þkÞ w k2Zd
þ
X n
^ ðA Þ ð þ A kÞ t A k ðÞ w k2Zd
X nþ1 1 ^ ðA Þ ððA Þ þkÞ wððA ^ Þ1 þkÞ þ Hn ðÞ ^ w Þ Þ ¼ wððA 1
k2Zd d
a.e. on R for n 2 N by Lemma 1. The proof is completed.
h
Proof of Theorem 1. Sufficiency. Suppose Dw ðÞ 2 Zþ a.e. on Rd . Then rw ðÞ 2 Zþ a.e. on Rd by Lemma 4. Also observing that w is a PFW for FL2(X) leads to
X
jhf ; T k wij2 6
XX
jhf ; Dj T k wij2 6 kf k2
j2Z k2Zd
k2Zd
for f 2 spanfT k w : k 2 Zd g, which implies that rw() 6 1 a.e. on Rd by Lemma 3. It follows that rw() = vU() a.e. on Rd , where U ¼ fn 2 Rd : rw ðnÞ > 0g. So w is a semi-orthogonal PFW for FL2(X) by Lemma 5. Necessity. Suppose w is a semi-orthogonal PFW for FL2(X). Then
X
^ Þj ð þ kÞÞwð ^ þ kÞ ¼ 0 wððA
k2Zd
a.e. on Rd for j 2 N by Lemma 5. Write
Gn ðÞ ¼
1 X X
^ Þj ÞwððA ^ Þj ð þ kÞÞ ^ Þn ð þ kÞÞwððA wððA
j¼1 k2Zd
a.e. on Rd for n 2 N. Then Gn() = Hn(), and Gn() = Gn1(A⁄) a.e. on Rd for 1 < n 2 N by Lemma 6. It follows that
Gn ðÞ ¼ G1 ððA Þn1 Þ a:e: on Rd for n 2 N:
ð16Þ
Note that
G1 ðÞ ¼
X
^ þA kÞtA k ðA Þ wðA
k2Zd
a.e. on Rd . By Lemma 1, we have
G1 ðÞ ¼
X
^ þkÞt k ðA Þ ¼ wðA
k2Zd
1 X
^ Þjþ1 Þ wððA
j¼0
which together with Lemma 5 implies that
^ Þrw ðA Þ G1 ðÞ ¼ wðA a.e. on Rd . Therefore,
^ Þn Þrw ððA Þn Þ Gn ðÞ ¼ G1 ððA Þn1 Þ ¼ wððA
X k2Zd
^ þkÞwððA ^ Þj ðA þkÞÞ; wðA
Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
9159
^ Þn Þ ¼ 0 when a.e. on Rd for n 2 N. Also observing that rw((A⁄)n) is either 0 or 1 a.e. on Rd by Lemma 5, and that wððA ^ Þn Þ a.e. on Rd for n 2 N. So rw((A⁄)n) = 0, we obtain that Gn ðÞ ¼ wððA
Wn ðÞ ¼
1 X hWn ðÞ; Wj ðÞil2 ðZd Þ Wj ðÞ j¼1
P 2 d a.e. on R for n 2 N, where Wj is defined as in (9) for j 2 N. Note that 1 n¼1 kWn ðÞkl2 ðZd Þ ¼ Dw ðÞ < 1 a.e. on R , where Dw is defined as in (8). Applying [[19], Chapter 7, Lemma 3.7], we obtain that d
Dw ðÞ ¼ dimF w ðÞ a.e. on Rd , where Fw is defined as in (10). Also observing Dw() < 1 a.e. on Rd leads to Dw ðÞ 2 Zþ a.e. on Rd . The proof is completed. h
3. Proof of Theorem 2 and Theorem 3 In this section, we focus on the proof of Theorem 2 and Theorem 3. For this purpose, we first establish two lemmas. Let w 2 L2 ðRd Þ, let A be a d d expansive matrix with jdetAj = 2, and let Dw and Wj with j 2 N be defined as in (8) and (9), respectively. Define
Z :¼ fn 2 Rd : Dw ðnÞ ¼ 0g;
ð17Þ
d
ð18Þ
P 1 :¼ fn 2 R : kW1 ðnÞkl2 ðZd Þ – 0g; d
P j :¼ fn 2 R : kWj ðnÞkl2 ðZd Þ – 0 and kWl ðnÞkl2 ðZd Þ ¼ 0;
for 1 6 l 6 j 1g for j > 1:
ð19Þ
Obviously, the sets Z and P j ’s are Zd -periodic and measurable, and they form a partition of Rd . Lemma 7. Let A be a d d expansive matrix with jdetAj = 2, let FL2(X) be a reducing subspace of L2 ðRd Þ, and let w 2 FL2(X). Assume that dim Fw() 2 {0, 1} a.e. on Rd . Define /0 by
8 if n 2 Z; < 0; pffiffiffiffiffiffiffiffiffi ^ 0 ðnÞ :¼ / Dw ðnÞ j ^ : kW ðnÞk wððA Þ nÞ; if n 2 P j for some j 2 N; 2 j
ð20Þ
l ðZd Þ
where Z and P j are defined as in (17)–(19), respectively. Then (i) /0 2 FL2(X); ^ 0 ðÞj2 ¼ P1 jwððA ^ Þj Þj2 a.e. on Rd ; (ii) j/ j¼1 d ^ 0 ðA Þ ¼ m0 ðÞ/ ^ 0 ðÞ (iii) there exists a Z -periodic measurable function m0 : Rd ! C such that jm0()j 6 1 a.e. on Rd , and that / a.e. on Rd .
Proof (i) It is easy to check that
X
^ 0 ð þ kÞj2 ¼ Dw ðÞ j/
ð21Þ
k2Zd
a.e. on Rd , which implies that /0 2 L2 ðRd Þ by (11). Also observing (20) gives (i). (ii) Note that (ii) is trivial a.e. on Z. Next we prove that it holds a.e. on each P l with l 2 N. Since dimFw() = 1 a.e. on P l , to each j 2 N there corresponds a Zd -periodic measurable function klj : P l ! C such that
Wj ðÞ ¼ klj ðÞWl ðÞ a.e. on P l . It follows that 1 X
^ Þj Þj2 ¼ jwððA
j¼1
Dw ðÞ ¼
1 X
!
jklj ðÞj2
j¼1 1 X j¼1
kWj ðÞk2l2 ðZd Þ ¼
1 X j¼1
which leads to (ii) a.e. on P l .
^ Þl Þj2 ; jwððA ! jklj ðÞj2 kWl ðÞk2l2 ðZd Þ ;
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P ^ Þl ð þ ðA Þ1 kÞÞj2 P kWl ðÞk22 d for 2 6 l 2 N, to each 2 6 l 2 N there corresponds a (iii) Since kWl1 ðA Þk2l2 ðZd Þ ¼ k2Zd jwððA l ðZ Þ d Z -periodic measurable function cl : P l ! f1; 2; . . . ; l 1g such that A P l P cl ðÞ . Also observing that dimFw(A⁄) = 1 a.e. on P l , we conclude that, for each 2 6 l 2 N, there exists a Zd -periodic measurable function kl : P l ! C such that kl() – 0 a.e. on P l and
Wl1 ðA Þ ¼ kl ðÞWcl ðÞ ðA Þ
ð22Þ
a.e. on P l . Similarly, for each l 2 N, there exists a Zd -periodic measurable function ~ kl : P 1 \ ðA Þ1 P l ! C such that
Wlþ1 ðÞ ¼ ~kl ðÞW1 ðÞ
ð23Þ
1
d
a.e. on P 1 \ ðA Þ P l . Define m0 : R ! C by
m0 ðnÞ ¼
8 pffiffiffiffiffiffiffiffiffiffiffi ffi Dw ðA nÞkWl ðnÞk 2 d > l ðZ Þ > p ffiffiffiffiffiffiffiffi ffi > ; for n 2 P l with l P 2; > Dw ðnÞkWc ðnÞ ðA nÞk 2 d kl ðnÞ > > l ðZ Þ l > > p ffiffiffiffiffiffiffiffiffiffiffi ffi > D ðA nÞkW ðnÞk ~k ðnÞ > w 1 > l2 ðZd Þ l > ffi > ; for n 2 P 1 \ ðA Þ1 P l with l 2 N; > pffiffiffiffiffiffiffiffi Dw ðnÞkWl ðA nÞk 2 d > l ðZ Þ > < e
2pink 0
kW1 ðnþðA Þ1 eÞk 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > > Dw ðnþðA Þ1 eÞ > > > > > p1ffiffi ; > > 2 > > > > > > 0; :
l ðZd Þ
;
for n 2 Z \ ðP 1 ðA Þ1 eÞ;
ð24Þ
for n 2 Z \ ðZ ðA Þ1 eÞ; 1 S T S for n 2 ðP 1 \ ðA Þ1 ZÞ ðZ ð P l ðA Þ1 eÞÞ; l¼2 d
where {0, e} is a set of representatives of distinct cosets in Z =A Z , and k0 is as in Proposition 2. Then m0 is Zd -periodic ^ 0 ðA Þ ¼ m0 ðÞ/ ^ 0 ðÞ a.e. on Z c by (20), (22) and (23), / ^ 0 ðA Þ ¼ 0 ¼ m0 ðÞ/ ^ 0 ðÞ a.e. on Z by (ii) and (20). and measurable, / c d ^ Also observing that there exists a mapping s : Z ! Z such that /0 ð þ sðÞÞ – 0 a.e. on Z c by (21), we have
jm0 ðÞj ¼ jm0 ð þ sðÞÞj ¼
d
^ 0 ðA ð þ sðÞÞÞj j/ ^ 0 ð þ sðÞÞj j/
^ 0 ðA ð þ sðÞÞÞj2 6 j/ ^ 0 ð þ sðÞÞj2 a.e. on Rd by (ii). Therefore we have jm0()j 6 1 a.e. on Z c . It is a.e. on Z c . However j/ obvious that jm0()j 6 1 a.e. on Z by (24). The proof is completed. h Lemma 8. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X), and that dimFw() 2 {0, 1} a.e. on Rd . Define the Zd -periodic measurable function m1 : Rd ! C by
m1 ðnÞ ¼
8 1 pffiffi ; for n 2 Z \ ðZ ðA Þ1 eÞ; > > > 2 > > > < m0 ðn þ ðA Þ1 eÞ; for n 2 Z n ðZ ðA Þ1 eÞ; 2pink
0 kW1 ðnÞk 2 e > > pffiffiffiffiffiffiffiffiffi l ðZd Þ ; > > Dw ðnÞ > > : 0;
ð25Þ
for n 2 P 1 ; for n 2 P l with l P 2;
where {0, e} is a set of representatives of distinct cosets in Zd =A Zd , k0 is as in Proposition 2, Z and P j are respectively defined as in (17)–(19), and m0 is defined as in (24). Then (i) (ii) (iii) (iv)
^ nÞ ¼ e2pink0 m1 ðnÞ/ ^ 0 ðnÞ for a.e. n 2 Rd ; jm1(n)j 6 1, and wðA 2 2 d jm0()j + jm1()j = 1 a.e. on R ; m0 ðÞm0 ð þ ðA Þ1 eÞ ¼ m1 ð þ ðA Þ1 eÞm1 ðÞ a.e. on Rd ; jm0()j = jm1( + (A⁄)1e))j, and jm1()j2 + jm1( + (A⁄)1e)j2 = 1 a.e. on Rd .
Proof ^ Þ ¼ 0 a.e. on P c . We only need to prove that m1 ðÞ/ ^ 0 ðÞ ¼ 0 a.e. on (i) By (25) and (20), (i) holds a.e. on P 1 . Note that wðA 1 c c ^ 0 ðÞ ¼ 0 a.e. on Z. So (i) holds. P 1 . It is true since m1() = 0 a.e. on ðP 1 [ ZÞ , and / (ii) By (24) and (25), m0 ðÞ ¼ m1 ðÞ ¼ p1ffiffi2 a.e. on Z \ ðZ ðA Þ1 eÞ, which implies (ii) holds a.e. on Z \ ðZ ðA Þ1 eÞ. Note that
jm0 ðÞj2 þ jm1 ðÞj2 ¼ jm0 ð þ ðA Þ1 eÞj2 þ jm1 ð þ ðA Þ1 eÞj2 a.e. on Z n ðZ ðA Þ1 eÞ by (24) and (25). So, to prove (ii) a.e. on Rd , we only need to prove that
Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
jm0 ðÞj2 þ jm1 ðÞj2 ¼ 1 a:e: on Z c :
9161
ð26Þ
By (ii) and (iii) in Lemma 7 and (i) in Lemma 8, we have
^ 0 ðÞj2 ¼ jwðA ^ Þj2 þ j/ ^ 0 ðA Þj2 ¼ ðjm0 ðÞj2 þ jm1 ðÞj2 Þj/ ^ 0 ðÞj2 j/ a.e. on Rd , which implies that
X
^ 0 ð þ kÞj2 ¼ ðjm0 ðÞj2 þ jm1 ðÞj2 Þ j/
k2Zd
X
^ 0 ð þ kÞj2 j/
k2Zd
a.e. on Rd . Also applying Lemma 7 (ii) leads to
Dw ðÞ ¼ jm0 ðÞj2 þ jm1 ðÞj2 Dw ðÞ a.e. on Rd . It gives (26) due to the fact that Dw() – 0 a.e. on Z c . (iii) By (24) and (25), we have
m0 ðÞ ¼ m1 ð þ ðA Þ1 eÞ;
m0 ð þ ðA Þ1 eÞ ¼ m1 ðÞ
ð27Þ 1
a.e. on Z, which gives (iii) a.e. on Z. Observing n þ ðA Þ leads to
1
⁄ 1
e 2 Z for n 2 Z ðA Þ e, and applying (27) to n + (A ) e
m0 ðn þ ðA Þ1 eÞ ¼ m1 ðn þ 2ðA Þ1 eÞ ¼ m1 ðnÞ; m0 ðnÞ ¼ m0 ðn þ ðA Þ1 e þ ðA Þ1 eÞ ¼ m1 ðn þ ðA Þ1 eÞ: Therefore (iii) holds a.e. on Z [ ðZ ðA Þ1 eÞ. Next we prove that it holds a.e. on ðZ [ ðZ ðA Þ1 eÞÞc . By (i) and Zd periodicity of m1,
^ Þj ÞwððA ^ Þj ð þ kÞÞ ¼ jm1 ððA Þj1 Þj2 / ^ 0 ððA Þj1 Þ/ ^ 0 ððA Þj1 ð þ kÞÞ wððA a.e. on Rd for j 2 N and k 2 Zd . Combined with (ii) and Lemma 7 (iii), it follows that
^ Þj ÞwððA ^ Þj ð þ kÞÞ ¼ ð1 jm0 ððA Þj1 Þj2 Þ/ ^ 0 ððA Þj1 ð þ kÞÞ ^ 0 ððA Þj1 Þ/ wððA ^ 0 ððA Þj1 ð þ kÞÞ m0 ððA Þj1 Þ/ ^ 0 ððA Þj1 Þm0 ððA Þj1 ð þ kÞÞ/ ^ 0 ððA Þj1 ð þ kÞÞ ^ 0 ððA Þj1 Þ/ ¼/ ^ 0 ððA Þj1 Þ/ ^ 0 ððA Þj1 ð þ kÞÞ / ^ 0 ððA Þj ð þ kÞÞ ^ 0 ððA Þj Þ/ ¼/ a.e. on Rd for j 2 N and k 2 Zd , which implies that 1 X
^ Þj ÞwððA ^ Þj ð þ kÞÞ ¼ / ^ 0 ð þ kÞ ^ 0 ðÞ/ wððA
j¼1
a.e. on Rd for k 2 Zd by Lemma 2. So, by Lemma 1,
0¼
1 X
^ Þj ÞwððA ^ Þj ð þ qÞÞ ¼ wðÞ ^ þ qÞ þ / ^ 0 ðÞ/ ^ 0 ð þ qÞ ^ wð wððA
ð28Þ
j¼0
a.e. on Rd for q 2 Zd n A Zd . Since ðA Þ1 q ðA Þ1 e 2 Zd , we have
e2piððA
1
Þ
qÞk
0
¼ e2piððA
1
Þ
eÞk0
¼ 1;
and thus
^ wð ^ þ qÞ ¼ m1 ððA Þ1 Þm1 ððA Þ1 ð þ qÞÞ/ ^ 0 ððA Þ1 ð þ qÞÞ ^ 0 ððA Þ1 Þ/ wðÞ a.e. on Rd for q 2 Zd n A Zd by (i). Also from (28) and Lemma 7, it follows that
^ 0 ððA Þ1 Þ/ ^ 0 ððA Þ1 ð þ qÞÞ ¼ 0 m0 ððA Þ1 Þm0 ððA Þ1 ð þ qÞÞ m1 ððA Þ1 Þm1 ððA Þ1 ð þ qÞÞ / a.e. on Rd for q 2 Zd n A Zd , equivalently,
^ 0 ðÞ/ ^ 0 ð þ ðA Þ1 qÞ ¼ 0 m0 ðÞm0 ð þ ðA Þ1 qÞ m1 ðÞm1 ð þ ðA Þ1 qÞ /
ð29Þ
a.e. on Rd for q 2 Zd n A Zd . Since Dw()Dw( + (A⁄)1e) – 0 a.e. on ðZ [ ðZ ðA Þ1 eÞÞc , there exist two mappings s1, s2: ðZ [ ðZ ðA Þ1 eÞÞc ! Zd such that
^ 0 ð þ s1 ðÞÞ/ ^ 0 ð þ s2 ðÞ þ ðA Þ1 eÞ – 0 /
ð30Þ
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Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
a.e. on ðZ [ ðZ ðA Þ1 eÞÞc by (21). Combined with (29), it follows that
^ 0 ð þ s1 ðÞÞ/ ^ 0 ð þ s1 ðÞ þ ðA Þ1 ððA Þðs2 ðÞ s1 ðÞÞ þ eÞÞ ¼ 0; m0 ðÞm0 ð þ ðA Þ1 eÞ m1 ðÞm1 ð þ ðA Þ1 eÞ / namely,
^ 0 ð þ s1 ðÞÞ/ ^ 0 ð þ s2 ðÞ þ ðA Þ1 eÞ ¼ 0 m0 ðÞm0 ð þ ðA Þ1 eÞ m1 ðÞm1 ð þ ðA Þ1 eÞ / a.e. on ðZ [ ðZ ðA Þ1 eÞÞc . Combined with (30), it follows that
m0 ðÞm0 ð þ ðA Þ1 eÞ ¼ m1 ðÞm1 ð þ ðA Þ1 eÞ a.e. on ðZ [ ðZ ðA Þ1 eÞÞc . So (iii) holds. (iv) By (iii) and (ii), we have
jm0 ðÞj2 ¼ jm0 ðÞj2 jm0 ð þ ðA Þ1 eÞj2 þ jm0 ðÞj2 jm1 ð þ ðA Þ1 eÞj2 ¼ jm1 ðÞj2 jm1 ð þ ðA Þ1 eÞj2 þ jm0 ðÞj2 jm1 ð þ ðA Þ1 eÞj2 ¼ jm1 ð þ ðA Þ1 eÞj2 a.e. on Rd , and thus
jm0 ðÞj ¼ jm1 ð þ ðA Þ1 eÞj
ð31Þ
d
a.e. on R . Combined with (ii), it follows that
jm1 ðÞj2 þ jm1 ð þ ðA Þ1 eÞj2 ¼ jm1 ðÞj2 þ jm0 ðÞj2 ¼ 1 a.e. on Rd . The proof is completed.
h
Proof of Theorem 2. Sufficiency. Define C ¼ fn 2 Rd : m0 ððA Þ1 n þ ðA Þ1 eÞ ¼ 0g, and
lðÞ ¼
8 > > <
m1 ððA Þ1 Þ m0 ððA Þ1 þðA Þ1 eÞ
on Cc ; ð32Þ
1 1 > Þ eÞ > : m1 ððA Þ þðA on C; 1
m0 ððA Þ
Þ
where m0 and m1 are defined as in (24) and (25), respectively. Then l is well-defined, unimodular and measurable by Lemma 8 (ii) and (iv). We also claim that l is Zd -periodic. By Zd -periodicity of m0 and m1, l is A Zd -periodic. Also observing that Zd ¼ A Zd þ f0; eg, to prove l being Zd -periodic, we only need to prove that
lðÞ ¼ lð þ eÞ a:e: on Rd :
ð33Þ
By Lemma 8 (ii) and (iv), we have
jm0 ðÞj2 þ jm0 ð þ ðA Þ1 eÞj2 ¼ 1 a:e: on Rd :
ð34Þ
It follows that C \ (C e) = ;, which implies that (33) holds a.e. on C [ (C e) by (32). Now we turn to (C [ (C e))c. By Lemma 8 (iii) and (32), we have
0 – m0 ððA Þ1 Þm0 ððA Þ1 þðA Þ1 eÞ ¼ m1 ððA Þ1 þðA Þ1 eÞm1 ððA Þ1 Þ ¼ lð þ eÞm0 ððA Þ1 ÞlðÞm0 ððA Þ1 þðA Þ1 eÞ ¼ lð þ eÞlðÞm0 ððA Þ1 Þm0 ððA Þ1 þðA Þ1 eÞ a.e. on (C [ (C e))c. It follows that lð þ eÞlðÞ ¼ 1 a.e. on (C [ (C e))c, and thus l() = l( + e) a.e. on (C [ (C e))c due to the fact that l is unimodular. Therefore (33) holds, and thus l is a unimodular, Zd -periodic and measurable function. Note ^ 0 ðA Þ ¼ m0 ðÞ/ ^ 0 ðÞ a.e. on Rd by Lemma 7 that jm0()j2 + jm0( + (A⁄)1e)j2 = 1 a.e. on Rd by Lemma 8 (ii) and (iv), and that / (iii). To prove the sufficiency, we only need to prove that
^ nÞ ¼ e2pink0 lðA nÞm0 ðn þ ðA Þ1 eÞ/ ^ 0 ðnÞ wðA for a.e. n 2 Rd . Also observing that
^ nÞ ¼ e2pink0 m1 ðnÞ/ ^ 0 ðnÞ wðA for a.e. n 2 Rd , it suffices to prove that
m1 ðÞ ¼ lðA Þm0 ð þ ðA Þ1 eÞ
ð35Þ
Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
9163
a.e. on Rd . By Lemma 8 (iv),
jm1 ðÞj ¼ jm0 ð þ ðA Þ1 eÞj a.e. on Rd , which implies that (35) holds a.e. on (A⁄)1C. However, (35) also holds a.e. on ((A⁄)1C)c by (32). Therefore (35) holds a.e. on Rd , and the sufficiency follows. Necessity. Suppose w is an MRA PFW for FL2(X). Then there exist / 2 FL2(X), m 2 L2 ðTd Þ, and a unimodular Zd -periodic function l such that m 2 e F and 2piððA ^ wðnÞ ¼e
1
Þ
nÞ
^ Þ1 nÞ lðnÞmððA Þ1 n þ ðA Þ1 eÞ/ððA
k0
for a.e. n 2 Rd . It follows that
^ Þj ðn þ kÞÞ ¼ e2piððA wððA ¼e
j1
Þ
nÞ
k0
2piððA Þj1 nÞ
k0
^ Þj1 ðn þ kÞÞ lððA Þj nÞmððA Þj1 n þ ðA Þ1 eÞ/ððA j
j1
1
lððA Þ nÞmððA Þ n þ ðA Þ eÞ
j2 Y
!
n
^ þ kÞ mððA Þ nÞ /ðn
n¼0
for a.e. n 2 Rd , j 2 N and k 2 Zd , where Zd -periodicity of l and m is used in the first equality, and we make the convention that Q1 n d n¼0 mððA Þ nÞ ¼ 1 for a.e. n 2 R . Therefore, we have
Wj ðnÞ ¼ e
2piððA Þj1 nÞk
0
j
j1
1
lððA Þ nÞmððA Þ n þ ðA Þ eÞ
j2 Y
! n
mððA Þ nÞ U0 ðnÞ
ð36Þ
n¼0
^ þ kÞ : k 2 Zd g. By (36), dimFw() 2 {0, 1} a.e. on Rd . The proof is completed. h for a.e. n 2 Rd and j 2 N, where U0 ðÞ ¼ f/ð Proof of Theorem 3. By Theorem 1 and its proof, w is a semi-orthogonal PFW for FL2(X) if and only if Dw ðÞ ¼ dimF w ðÞ 2 Zþ a.e. on Rd . Combined with Theorem 2, it follows that w is a semi-orthogonal MRA PFW for FL2(X) if and only if Dw() = dimFw() 2 {0, 1} a.e. on Rd . Note that Dw() 2 {0, 1} a.e. on Rd implies that w is a semi-orthogonal PFW for FL2(X) by Theorem 1, and thus implies that Dw() = dimFw() 2 {0, 1} a.e. on Rd by the proof of the necessity of Theorem 1. Therefore, w is a semi-orthogonal MRA PFW for FL2(X) if and only if Dw() 2 {0, 1} a.e. on Rd . The proof is completed. h Applying Lemma 1 and Theorem 3, we have the following example: Example 5. Let A = 2, and let FL2(X) be a reducing subspace of L2 ðRÞ. Define w via its Fourier transform by
^ ¼ epi v ðÞ wðÞ E a.e. on R, where E ¼ ð½1; 12Þ [ ð12 ; 1Þ \ X. Then w is a semi-orthogonal MRA PFW for FL2(X). Example 6. Let A ¼
E¼
0 2
S 1 1 ; 6 a; b 6 1, and let FL2(X) be a reducing subspace of L2 ðR2 Þ with X ¼ j2Z ðA Þj E, where 0 2
b n 2 ½0; 1Þ2 : 0 6 n1 6 a; ðn1 aÞ 6 n2 6 1 [ fn 2 ½0; 1Þ2 : a 6 n1 6 1; 0 6 n2 6 1g:
a
^ ¼ v on R2 . Then w is a semi-orthogonal PFW, but not an MRA PFW for FL2(X). Define w via its Fourier transform by w E Proof. By Example 1 in [10], w is a PFW for FL2(X). A simple computation shows that
1 1 ðA Þ1 E ½0; 1Þ ½1; 0Þ and ðA Þj E ½ ; Þ2 2 2 for 1 < j 2 N, which implies that
Dw ðÞ ¼
1 X X j¼1 k2Z2
vððA Þj EþkÞ ðÞ ¼
1 X j¼1
v S ððA Þj EþkÞ ðÞ:
ð37Þ
k2Z2
However, Dw() < 1 a.e. on R2 by (11). So we have Dw 2 Zþ a.e. on R2 , and consequently, w is a semi-orthogonal PFW for 1 is a set with nonzero measure. By (37), we have Dw() P 2 FL2(X) by Theorem 1. Note that ððA Þ1 EÞ \ ðA Þ2 E þ 0 1 . Combined with Theorem 3, it follows that w is not an MRA PFW. The proof is a.e. on ððA Þ1 EÞ \ ðA Þ2 E þ 0 completed. h
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Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
Now we conclude this paper with the following remark: Remark 2. It is unresolved whether w in Example 1 is always an MRA wavelet for L2 ðRd Þ when L = 1, and is always a nonsemi-orthogonal MRA PFW for L2 ðRd Þ when L is an odd number greater than 1. Acknowledgments The authors thank the referees for their valuable comments, which greatly improved the readability of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
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Li, Reducing subspace frame multiresolution analysis and frame wavelets, Commun. Pure Appl. Analy. 6 (2007) 741–756. Y.-Z. Li, F.-Y. Zhou, GMRA-based construction of framelets in reducing subspaces of L2 ðRd Þ, Int. J. Wavelets Multires. Inf. Process. 9 (2011) 237–268. F.-Y. Zhou, Y.-Z. Li, Multivariate FMRAs and FMRA frame wavelets for reducing subspaces of L2 ðRd Þ, Kyoto J. Math. 50 (2010) 83–99. H. Volkmer, Frames of wavelets in Hardy space, Analysis 15 (1995) 405–421. B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. 4 (1997) 380–413. X. Dai, Y. Diao, Q. Gu, Subspaces with normalized tight frame wavelets in R, Proc. Amer. Math. Soc. 130 (2001) 1661–1667. X. Dai, Y. Diao, Q. Gu, D. Han, The existence of subspace wavelet sets, J. Comput. Appl. Math. 155 (2003) 83–90. X. Zhu, G. Wu, A characteristic description of orthonormal wavelet on subspace L2E ðRÞ of L2 ðRÞ, Chaos Solitons Fract. 41 (2009) 2484–2490. Y.-Z. Li, F.-Y. Zhou, Affine and quasi-affine dual wavelet frames in reducing subspaces of L2 ðRd Þ, Acta. Math. Sinica(Chin. Ser.) 53 (2010) 551–562. M. Bownik, Intersection of dilates of shift-invariant spaces, Proc. Amer. Math. Soc. 137 (2009) 563–572. C. de Boor, R.A. DeVore, A. Ron, On the construction of multivariate (pre)wavelets, Constr. Approx. 9 (1993) 123–166. H. Eugenio, G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL, 1996. J. Kovacˇevic´, M. Vetterli, Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for Rn , IEEE Trans. Inform. Theory 38 (1992) 533–555. A. Cohen, I. Daubechies, Nonseparable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993) 51–137. L.F. Villemoes, Continuity of nonseparable quincunx wavelets, Appl. Comput. Harmon. Anal. 1 (1994) 180–187. E. Belogay, Y. Wang, Arbitrarily smooth orthogonal nonseparable wavelets in R2 , SIAM J. Math. Anal. 30 (1999) 678–697. A. Ayache, Construction of non-separable dyadic compactly supported orthonormal wavelet bases for L2 ðR2 Þ of arbitrarily high regularity, Rev. Mat. Iberoamericana 15 (1999) 37–58. G. Gripenberg, A necessary and sufficient condition for the existence of a father wavelet, Studia Math. 114 (1995) 207–226. H.O. Kim, R.Y. Kim, J.K. Lim, Characterizations of biorthogonal wavelets which are associated with biorthogonal multiresolution analyses, Appl. Comput. Harmon. Anal. 11 (2001) 263–272. M. Paluszyn´ski, H. S´ikic´, G. Weiss, S. Xiao, Generalized low pass filters and MRA frame wavelets, J. Geom. Anal. 11 (2001) 311–342. M. Paluszyn´ski, H. S˘ikic´, G. Weiss, S. Xiao, Tight frame wavelets, their dimension functions, MRA tight frame wavelets and connectivity properties, Adv. Comput. Math. 18 (2003) 297–327. H. S˘ikic´, D. Speegle, G. Weiss, Structure of the set of dyadic PFW’s, Frames and operator theory in analysis and signal processing, Contemporary Mathematics, 451, American Mathematics Society Providence, RI, 2008. 263-291. M. Ehler, On multivariate compactly supported bi-frames, J. Fourier Anal. Appl. 13 (2007) 511–532. M. Ehler, Compactly supported multivariate pairs of dual wavelet frames obtained by convolution, Int. J. Wavelets Multires. Inf. Process. 6 (2008) 183– 208. M. Ehler, B. Han, Wavelet bi-frames with few generators from multivariate refinable functions, Appl. Comput. Harmon. Anal. 25 (2008) 407–414. M. Ehler, Nonlinear approximation schemes associated with nonseparable wavelet bi-frames, J. Approx. Theory 161 (2009) 292–313. M. Ehler, K. Koch, The construction of multiwavelet bi-frames and applications to variational image denoising, Int. J. Wavelets Multires. Inf. Process. 8 (2010) 431–455. M. 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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The characterization of a class of multivariate MRA and semi-orthogonal parseval frame wavelets q Yun-Zhang Li ⇑, Feng-Ying Zhou College of Applied Sciences, Beijing University of Technology, Beijing 100124, PR China
a r t i c l e
i n f o
Keywords: Parseval frame wavelet Semi-orthogonal Parseval frame wavelet MRA Parseval frame wavelet Semi-orthogonal MRA Parseval frame wavelet
a b s t r a c t Let A be a d d expansive matrix with jdetAj = 2. This paper addresses Parseval frame wavelets (PFWs) in the setting of reducing subspaces of L2 ðRd Þ. We prove that all semiorthogonal PFWs (semi-orthogonal MRA PFWs) are precisely the ones with their dimension functions being non-negative integer-valued (0 or 1). We also characterize all MRA PFWs. Some examples are provided. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Let H be a separable Hilbert space. An at most countable sequence {fn:n 2 I} in H is called a frame for H if there exist 0 < C1 6 C2 < 1 such that
C 1 kf k2 6
X
jhf ; fn ij2 6 C 2 kf k2
ð1Þ
n2I
for f 2 H, where C1, C2 are called lower frame bound and upper frame bound, respectively; is called a tight frame for H with frame bound C1 if C1 = C2 in (1); is called a Parseval frame for H if C1 = C2 = 1 in (1), which is, in particular, an orthonormal basis for H if kfnk = 1 for n 2 I in addition. {fn:n 2 I} is called a Bessel sequence in H if the right inequality in (1) holds for f 2 H, where C2 is called the Bessel bound; is called a frame sequence if it is a frame for its closed linear span in H. The fundamentals of frames can be found in [1] and [2]. We denote by Z; Zþ and N the set of integers, the set of nonnegative integers and the set of positive integers, respectively. For d 2 N, we denote by Id the d d identity matrix, by M⁄ the transpose of M for a d d real matrix M, by xk the k-th component of x for x 2 Rd , by Td ¼ ½ 12 ; 12 Þd the d-dimensional torus in Rd , and by L2 ðTd Þ R the Hilbert space of Zd -periodic square-integrable functions with the inner product hf ; gi ¼ Td f ðxÞgðxÞdx for f ; g 2 L2 ðTd Þ. The Fourier transform is defined by
^f ðÞ ¼
Z
f ðxÞe2pihx;i dx for f 2 L1 ðRd Þ \ L2 ðRd Þ
Rd
and extended to L2 ðRd Þ by Plancheral’s theorem. Let S be a measurable set in Rd . A collection {Si: i 2 I} of countably many S measurable sets is called a partition of S if S = i2ISi, and Si \ Sj = ; up to a set of measure zero for i, j 2 I with i – j. A d d matrix A is called an expansive matrix if it is an integer matrix with all its eigenvalues being greater than 1 in module. Let A be a d d expansive matrix. Define the dilation operator D and the shift operator Tk with k 2 Zd on L2 ðRd Þ respectively by q Supported by the National Natural Science Foundation of China, Beijing Natural Science Foundation (Grant No. 1092001), the Scientific Research Common Program of Beijing Municipal Commission of Education, the Project-sponsored by SRF for ROCS, SEM of China. ⇑ Corresponding author. E-mail address: [email protected] (Y.-Z. Li).
0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.150
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Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164 1
Df ðÞ :¼ j det Aj2 f ðAÞ; T k f ðÞ :¼ f ð kÞ for f 2 L2 ðRd Þ. Obviously, they are both unitary operators on L2 ðRd Þ. For an arbitrary measurable set X in Rd , we denote by FL2(X) the closed subspace of L2 ðRd Þ of the form
n o FL2 ðXÞ :¼ f 2 L2 ðRd Þ : ^f ðÞ ¼ 0 a:e: on Rd n X : A closed subspace X of L2 ðRd Þ is called a reducing subspace if DX = X and TkX = X for each k 2 Zd . The following proposition provides us with a characterization of reducing subspaces: Proposition 1 ([3], Theorem 1). Let A be a d d expansive matrix. A closed subspace X of L2 ðRd Þ is a reducing subspace if and only if X = FL2(X) for some measurable set X in Rd with the property X = A⁄X. Let A be a d d expansive matrix, and let X be a nonzero reducing subspace of L2 ðRd Þ. By Proposition 1, X corresponds a set X Rd with nonzero measure for which
A X ¼ X and X ¼ FL2 ðXÞ: So, to be specific, we denote a reducing subspace by FL2(X) instead of X. In particular, L2 ðRd Þ is a reducing subspace of L2 ðRd Þ for an arbitrary expansive matrix A, and FL2([0, 1)) (Hardy space) is also a reducing subspace of L2 ðRÞ for an arbitrary 2 6 A 2 N. Throughout this paper, we always assume that A is a d d expansive matrix with jdetAj = 2. For such A, we have the following proposition: Proposition 2. Let A be a d d expansive matrix with jdetAj = 2. Then (i) f( + (A⁄)1e) = f( + (A⁄)1d) for an arbitrary Zd -periodic function f, e and d with {0, e} and {0, d} being both the sets of representatives of distinct cosets in Zd =A Zd ; (ii) there exists 1 6 k0 6 d such that 2ððA Þ1 eÞk0 is odd for each e with {0, e} being a set of representatives of distinct cosets in Zd =A Zd . Proof (i) Since ðA Þ1 ðe dÞ 2 Zd , we have (i) by Zd -periodicity of f. (ii) Fix e with {0, e} being a set of representatives of distinct cosets in Zd =A Zd . Since jdetAj = 2, we have 2ðA Þ1 e 2 Zd . Also observing that ðA Þ1 e R Zd leads to 2ðA Þ1 e 2 Zd n 2Zd . This implies that 2ððA Þ1 eÞk0 is odd for some 1 6 k0 6 d. Let {0, d} be another set of representatives of distinct cosets in Zd =A Zd . We always have ðA Þ1 e ðA Þ1 d 2 Zd , which implies that 2ððA Þ1 e ðA Þ1 dÞ 2 2Zd , and thus 2ððA Þ1 dÞk0 is odd. So (ii) holds. The proof is completed. h Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. A function w 2 FL2(X) is called a Parseval frame wavelet (PFW) for FL2(X) if fDj T k w : j 2 Z; k 2 Zd g is a Parseval frame for FL2(X); is called a semi0 0 0 orthogonal PFW for FL2(X) if it is a PFW for FL2(X) and hDjTkw, Dj Tk0 wi = 0 for (j, k), ðj ; k Þ 2 Z Zd with j – j0 ; is called a wavej 2 2 d let for FL (X) if fD T k w : j 2 Z; k 2 Z g is an orthonormal basis for FL (X), which is obviously a semi-orthogonal PFW. Definition 1. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. A sequence fV j gj2Z of closed subspaces of FL2(X) is called a multiresolution analysis (an MRA) for FL2(X) associated with A if the following conditions are satisfied: (i) (ii) (iii) (iv)
V Vj+1 for j 2 Z; T Sj 2 j2Z V j ¼ FL ðXÞ and j2Z V j ¼ f0g; f 2 V0 if and only if Djf 2 Vj for j 2 Z; there exists / 2 FL2(X) such that fT k / : k 2 Zd g is an orthonormal basis for V0.
Some other MRA structures were introduced in [4–7] for the construction of wavelet frames in L2 ðRd Þ. This definition is a natural generalization of the usual MRA for L2 ðRd Þ. Some more general notions can be found in [8–10]. We can refer to [11–16] for some related results on wavelets in reducing subspaces. In Definition 1, we call / a scaling function of the T MRA. From Theorem 1.1 in [17], it follows that the condition j2Z V j ¼ f0g in (ii) is trivial, a special case of which can be obtained by Corollary 4.14 in [18]. By the definition, V j ¼ spanfDj T k / : k 2 Zd g (so we say / generates the MRA), and there ^ Þ ¼ mðÞ/ðÞ ^ a.e. on Rd . It is easy to check that exists a unique m 2 L2 ðTd Þ such that /ðA
jmðÞj2 þ jmð þ ðA Þ1 eÞj2 ¼ 1
ð2Þ
Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
9153
a.e. on Rd , where {0, e} is a set of representatives of distinct cosets in Zd =A Zd . Let k0 be as in Proposition 2. Define m1 2 L2 ðTd Þ by
m1 ðnÞ ¼ e2pink0 lðA nÞmðn þ ðA Þ1 eÞ 2
ð3Þ
d
and w 2 L ðR Þ via its Fourier transform by
^ ^ Þ1 nÞ; wðnÞ ¼ m1 ððA Þ1 nÞ/ððA
ð4Þ
d
where l is an arbitrary Z -periodic, unimodular and measurable function. Note that
mðnÞ
mðn þ ðA Þ1 eÞ
!
m1 ðnÞ m1 ðn þ ðA Þ1 eÞ is a unitary matrix. By the same procedure as in [[19],Proposition 2.3], we can prove that fT k w : k 2 Zd g is an orthonormal basis for W0 = V1 V0 (W0 is the orthogonal complement of V0 in V1), and thus w is a wavelet for FL2(X), which is independent of the choice of e by Proposition 2. Such w is called an MRA wavelet since it is associated with an MRA. In this paper, we focus on expansive matrices A satisfying jdet Aj = 2. Many applications, such as image compression, employ wavelet bases in R2 . Tensor-product wavelets impose an unnecessary product structure on the plane, which is artificial for natural images. For a 2 2 expansive matrix A with jdet Aj = 2, wavelets are nonseparable, and thus offer the hope of a more isotropic analysis. For a general d d expansive matrix A, a wavelet frame for L2 ðRd Þ induced by a refinable function requires at least jdetAj 1 wavelets. However, wavelet frame systems with few generators can reduce the complexity of the associated wavelet transform. The restriction jdetAj = 2 may admit few wavelets. In addition, by the above argument, MRA wavelets have an explicit expression when jdetAj = 2. See [20–24] for details. MRA wavelets have many desirable features, but they impose some restrictions. Even if for the one-dimensional case 6pin A = 2 in the setting of L2 ðRÞ, there are useful ‘‘filters’’, such as mðnÞ ¼ 1þe2 , that do not produce orthonormal bases, nevertheless, they do produce systems that have the reconstruction property, as well as many useful features. Therefore, it is natural to develop a theory involving more general filters that produce systems having these properties. A natural setting for such a theory is provided by frames (see [[19], Chapter 8]). For the one-dimensional case A = 2, this problem was studied in [4,5,25] and [26] under the setting of L2 ðRÞ. However, the results there do involve certain restrictions and technical 6pin assumptions such as semi-orthogonality, and they exclude the filter mðnÞ ¼ 1þe2 . Fortunately, the authors in [27] and [28] introduced the notions of generalized filter, pseudo-scaling function and MRA PFW (which includes the filter mentioned above), and characterized PFWs being respectively semi-orthogonal PFWs, MRA PFWs and semi-orthogonal MRA PFWs for L2 ðRÞ. Some related results were also obtained in [29]. For d d expansive matrices A satisfying Ad = 2Id, some results on the construction of wavelet frames, wavelet bi-frames and sibling frames for L2 ðRd Þ were obtained in [30–35]. It is obvious that jdetAj = 2 when Ad = 2Id. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Li and Zhou jointly in [9] and [10] obtained an explicit expression of wavelet frames and semi-orthogonal wavelet frames for FL2(X) induced by refinable functions. This paper extends the results in [28] to the case of a general d d expansive matrix A with jdetAj = 2 under the setting of FL2(X). Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. A function m defined on d R is called a generalized filter if it is a Zd -periodic measurable function satisfying (2) a.e. on Rd . We denote by e F the set of all generalized filters. A function / 2 L2 ðRd Þ is called a pseudo-scaling function if there exists m 2 e F (not necessarily unique) such that
^ Þ ¼ mðÞ/ðÞ ^ /ðA
ð5Þ
a.e. on Rd . A PFW w for FL2(X) is called an MRA PFW for FL2(X) if there exist m 2 e F, a Zd -periodic unimodular measurable function l and a pseudo-scaling function / such that both (3) and (4) hold a.e. on Rd . These notions are a natural generalization of those in [27], where the one-dimensional case A = 2 was treated under the setting of L2 ðRÞ. Let us first see some examples. Example 1. Let A be a d d expansive matrix with Ad = 2Id, let L be a positive odd number, and let {0, e} be a set of representatives of distinct cosets in Zd =A Zd with ð2ðA Þ1 eÞk0 being odd. Define w 2 L2 ðRd Þ via its Fourier transform by
^ wðnÞ ¼ ie
0 1 l1 d1 piLððA Þ nÞk0 Y e sinðpLððA Þl1 nÞk0 Þ 1 A 0 sinðpLððA Þ nÞk0 Þ @ pLððA Þl1 nÞk0 l¼0
ð2þLÞpiððA Þ1 nÞk
for n 2 Rd . Then w is an MRA PFW for L2 ðRd Þ. Proof. Define
mðnÞ ¼
1 þ e2piLnk0 ; 2
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Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
and / 2 L2 ðRd Þ via its Fourier Transform by d1 Y
^ /ðnÞ ¼
epiLððA
l
Þ nÞk
0
sinðpLððA Þl nÞk0 Þ
l¼0
pLððA Þl nÞk0
for n 2 Rd . Then it is easy to check that m 2 e F,
^ ^ Þ1 nÞ /ðnÞ ¼ mððA Þ1 nÞ/ððA and
^ wðnÞ ¼ e2piððA
1
Þ
nÞk
0
^ Þ1 nÞ mððA Þ1 n þ ðA Þ1 eÞ/ððA
ð6Þ
for n 2 Rd . This implies that / is a pseudo-scaling function, and that w is a PFW for L2 ðRd Þ by Theorem 4.1 in [9]. Therefore w is an MRA PFW for L2 ðRd Þ by the definition of MRA PFW. The proof is completed. h 6pin
Remark 1. When d = 1, A = 2 and L = 3 in Example 1, mðnÞ ¼ 1þe2 , which is mentioned in the above. There are many expan0 1 1 1 0 1 1 d sive matrices satisfying A = 2Id. For example, A1 ¼ and A2 ¼ @ 0 1 1 A satisfy A21 ¼ 2I2 and A32 ¼ 2I3 , respec1 1 1 1 0 tively. Ehler in [30] provided such a class of examples satisfying Ad = 2Id: for d = 2, 3, let
A¼
0
1 1 ; 1 1
0
2
1
1
1
1
1
B C A ¼ @ 1 1 0 A
and for d > 3, let
1 0 2 1 1 C B . .. B .. . 1 0 0C C B B . .. .. .. .. C C B .. . . . . C: A¼B C B B 0 0 1 0C C B C B @ 1 1 0 A 1 1 0
Example 2. Let A ¼
1 , and let L be a positive odd number. Define w 2 L2 ðR2 Þ by 1
1 1
81 x 2 D1 ; > < L2 ; wðxÞ ¼ 12 ; x 2 D2 ; > : L 0; otherwise for x 2 R2 , where
1 L1 ; 0 6 x1 x 2 < L ; x 2 R2 : 6 x2 < 2 2 Lþ1 1 2 6 x2 < ; 0 6 x 1 x 2 < L : D2 ¼ x 2 R : 2 2
D1 ¼
Then w is an MRA wavelet for L2 ðR2 Þ when L = 1, and is a non-semi-orthogonal MRA PFW for L2 ðR2 Þ when L P 3. Proof. Take
e ¼ 10
and k0 = 1 in Example 1. A simple computation shows that
ð2þLÞpiððA ^ wðnÞ ¼ ie
1
Þ
nÞ1
sinðpLððA Þ1 nÞ1 Þ epiLn1
sinðpLn1 Þ piLððA Þ1 nÞ1 sinðpLððA Þ1 nÞ1 Þ e pLn1 pLððA Þ1 nÞ1
for n 2 R2 . So, by Example 1, w is an MRA PFW for L2 ðR2 Þ. When L = 1, by the same procedure as in Example 1, we can prove that w is associated with / as in (6) with
mðnÞ ¼
1 þ e2pin1 2
and /ðxÞ ¼ v½0;1Þ ðx1 x2 Þv½0;1Þ ðx2 Þ:
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It is easy to check that k/k = 1. This implies that / generates an MRA for L2 ðR2 Þ by Theorem 1 and Theorem 2 in [36], and thus w is an MRA wavelet for L2 ðR2 Þ. When L P 3, it is easy to check that
hw; wðAÞi ¼
2 L1 2L2
– 0:
Therefore w is a non-semi-orthogonal MRA PFW for L2 ðR2 Þ. The proof is completed. h 0 0 1 0 1 0 1 1 0 1 B 0 ⁄ 1 ⁄ 1 @ A @ A B Note that (2(A ) e)1 = 1 for A ¼ 0 1 1 and e ¼ 0 , and that (2(A ) e)1 = 1 for A ¼ @ 1 1 1 0 0 0 1 1 1 B0C C e¼B @ 0 A . By the same procedure as in Example 2, we have the following two examples: 0 0 1 1 1 0 Example 3. Let A ¼ @ 0 1 1 A, and let L be a positive odd number. Define w 2 L2 ðR3 Þ by 1 1 0
2 0 1 1
1 1 1 1
1 1 0C C and 0A 1
81 x 2 D1 ; > < L3 ; 1 wðxÞ ¼ 3 ; x 2 D2 ; > : L 0; otherwise
for x 2 R3 , where
L1 1 < x2 6 ; 0 6 x1 x3 < L; 0 6 x2 þ x3 < L ; 2 2 1 L þ 1 ; 0 6 x1 x3 < L; 0 6 x2 þ x3 < L : D2 ¼ x 2 R3 : < x2 6 2 2
D1 ¼
x 2 R3 :
Then w is an MRA wavelet 0 0 B 0 Example 4. Let A ¼ B @ 1 1
for L2 ðR3 Þ when L = 1, and is a non-semi-orthogonal MRA PFW for L2 ðR3 Þ when L P 3. 1 2 1 1 0 1 0C C, and let L be a positive odd number. Define w 2 L2 ðR4 Þ by 1 1 0 A 1 1 1
81 x 2 D1 ; > < L4 ; 1 wðxÞ ¼ 4 ; x 2 D2 ; > : L 0; otherwise
for x 2 R4 , where
D1 ¼ fx 2 R4 : 0 6 x4 < L; 0 6 x3 þ x4 < L; 0 6 x1 þ x2 þ x3 þ x4 < L; 1 6 2ðx2 þ x3 þ x4 Þ < L 1g; D2 ¼ fx 2 R4 : 0 6 x4 < L; 0 6 x3 þ x4 < L; 0 6 x1 þ x2 þ x3 þ x4 < L; 1 L 6 2ðx2 þ x3 þ x4 Þ < 1g: Then w is an MRA wavelet for L2 ðR4 Þ when L = 1, and is a non-semi-orthogonal MRA PFW for L2 ðR4 Þ when L P 3. For examples in the setting of reducing subspaces, see Example 5 and Example 6 in Section 3. This paper focuses on the characterization of PFWs being respectively semi-orthogonal PFWs, MRA PFWs and semi-orthogonal MRA PFWs in the setting of reducing subspaces. Let w 2 L2 ðRd Þ, and let A be a d d expansive matrix with jdetAj = 2. Define its periodization function rw by
rw ðÞ ¼
X
^ þ kÞj2 jwð
ð7Þ
k2Zd
a.e on Rd , its dimension function Dw by
Dw ðÞ ¼
2 1 X X ^ j wððA Þ ð þ kÞÞ
ð8Þ
j¼1 k2Zd
a.e. on Rd , and define Wj for each j 2 N by
^ Þj ð þ kÞÞ : k 2 Zd g Wj ðÞ ¼ fwððA
ð9Þ
a.e. on Rd , and Fw() by
F w ðÞ ¼ spanfWj ðÞ : j 2 Ng
ð10Þ
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a.e. on Rd , where the closure is taken in l ðZd Þ. Note that
Z Td
Dw ðnÞdn ¼
1 Z X j¼1
^ Þj nÞj2 dn ¼ jwððA
1 X
Rd
^ 2 ¼ kwk ^ 2¼ 2j kwk
j¼1
Z Td
rw ðnÞdn:
2
ð11Þ 2
We have rw(), Dw() < 1 a.e. on Rd by their Zd -periodicity, Wj ðÞ 2 l ðZd Þ a.e. on Rd , and Fw() is a closed subspace of l ðZd Þ a.e. on Rd . Define
tk ðÞ ¼
1 X
^ Þj ÞwððA ^ Þj ð þ kÞÞ wððA
ð12Þ
j¼0
a.e. on Rd for k 2 Zd . Note that
1 X ^ j ^ j ^ 2 þ Dw ðÞÞ12 ðrw ðÞ þ Dw ðÞÞ12 < 1 wððA Þ ÞwððA Þ ð þ kÞÞ 6 ðjwðÞj j¼0
a.e. on Rd by Cauchy–Schwartz inequality. It follows that tk is well defined a.e. on Rd for k 2 Zd . In this paper, we obtain the following results: Theorem 1. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X). Then w is a semi-orthogonal PFW for FL2(X) if and only if Dw ðÞ 2 Zþ a.e. on Rd . Theorem 2. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X). Then w is an MRA PFW for FL2(X) if and only if dimFw() 2 {0, 1} a.e. on Rd . Theorem 3. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X). Then w is a semi-orthogonal MRA PFW for FL2(X) if and only if Dw() 2 {0, 1} a.e. on Rd . This paper is organized as follows. Section 2 is devoted to proof of Theorem 1; Section 3 is devoted to proof of Theorem 2 and Theorem 3. 2. Proof of Theorem 1 The following lemma is borrowed from [16], the L2 ðRd Þ version (i.e. X ¼ Rd ) of which was obtained independently by Bownik in [37] and Calogero in [38]. Lemma 1 ([16], Theorem 3). Let A be a d d expansive matrix, let FL2(X) be a reducing subspace of L2 ðRd Þ, and let w 2 FL2(X). Then w is a PFW for FL2(X) if and only if
X
^ Þj Þj2 ¼ v ðÞ and tk ðÞ ¼ 0 jwððA X
j2Z
a.e. on Rd for k 2 Zd n A Zd . Lemma 2 ([12], Lemma 3.1). Let A be a d d expansive matrix. Then, for an arbitrary f 2 L1 ðRd Þ, limj?1f(Aj) = 0 a.e. on Rd . The following lemma is a well-known result, some variations of which can be found in [2,4,39] and [40]. Lemma 3. For w 2 L2 ðRd Þ, the following hold: (i) fT k w : k 2 Zd g is a Bessel sequence with Bessel bound C if and only if rw() 6 C a.e. on Rd ; (ii) fT k w : k 2 Zd g is a frame sequence with frame bound C1 and C2 if and only if C1vU() 6 rw() 6 C2vU() a.e. on Rd , where U ¼ fn 2 Rd : rw ðnÞ > 0g. Lemma 4. Let w 2 L2 ðRd Þ, let A be a d d expansive matrix with jdetAj = 2, and let {0, e} be a set of representatives of distinct cosets in Zd =A Zd . Then
Dw ðÞ þ Dw ð þ ðA Þ1 eÞ ¼ Dw ðA Þ þ rw ðA Þ a.e. on Rd . Proof. Since Zd ¼ A Zd þ f0; eg, we have
Dw ðA Þ ¼
1 X X j¼1 k2Zd
^ Þjþ1 ð þ kÞÞj2 þ jwððA
1 X X j¼1 k2Zd
^ Þjþ1 ð þ k þ ðA Þ1 eÞÞj2 jwððA
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a.e. on Rd , and consequently, by the definition of Dw and rw,
Dw ðA Þ ¼ Dw ðÞ
X
^ ð þ kÞÞj2 þ Dw ð þ ðA Þ1 eÞ jwðA
k2Zd
X
^ ð þ k þ ðA Þ1 eÞÞj2 ¼ Dw ðÞ þ Dw ð þ ðA Þ1 eÞ rw ðA Þ jwðA
k2Zd
a.e. on Rd . The proof is completed. h Lemma 5. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X). Then the following are equivalent: (i) w is a semi-orthogonal PFW for FL2(X); (ii) rw() = vU() a.e. on Rd , where U ¼ fn 2 Rd : rw ðnÞ > 0g; P (iii) kwk2 ¼ k2Zd jhw; T k wij2 ; P d ^ j ^ for j 2 N. (iv) d wððA Þ ð þ kÞÞwð þ kÞ ¼ 0 a.e. on R k2Z
Proof. Write W j ¼ spanfDj T k w : k 2 Zd g for j 2 Z. Since w is a PFW for FL2(X), we have
f ¼
XX
hf ; Dj T k wiDj T k w for f 2 W 0 ;
ð13Þ
j2Z k2Zd
kwk2 ¼
2 X X j hw; D T k wi :
ð14Þ
j2Z k2Zd
Note that w being a semi-orthogonal PFW for FL2(X) is equivalent to W0\Wj for 0 – j 2 Z. Combined with (13), it follows that (i) is equivalent to fT k w : k 2 Zd g being a Parseval frame for W0, which implies the equivalence between (i) and (ii) by Lemma 3. By Lemma 3, we also have (ii) implies (iii). Next we prove that ðiiiÞ ) ðiÞ () ðiv Þ. Since 0 0 0 0 hDj T k w; Dj T k0 wi ¼ hw; Dj j T k0 Aj0 j k wi for k, k 2 Zd and j, j 2 Z with j0 > j, (i) is equivalent to hw, DjTkwi = 0 for j > 0 and j d k 2 Z . Suppose (iii) holds. Then hw, D Tkwi = 0 for 0 – j 2 Z by (14), and thus (i) holds. Now we turn to the equivalence between (i) and (iv). Note that
Z j 2pihk;ni ^ wððA ^ Þj nÞe2pihk;ðA Þj ni dn ¼ 22 ^ Þj nÞwðnÞe ^ wðnÞ wððA dn Rd Rd ! Z X j ^ Þj ðn þ lÞÞwðn ^ þ lÞ e2pihk;ni dn wððA ¼ 22 j
hw; Dj T k wi ¼ 22
Z
Td
l2Zd
for j > 0 and k 2 Zd . By the equivalence between (i) and hw, DjTkwi = 0 for j > 0 and k 2 Zd , we obtain the equivalence between (i) and (iv). The proof is completed. h Lemma 6. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X). Define
Hn ðÞ ¼
1 X X
^ Þj ÞwððA ^ Þj ð þ kÞÞ ^ Þn ð þ kÞÞwððA wððA
j¼0 k2Zd
a.e. on Rd for each n 2 N. Then
^ Hn ðÞ ¼ Hn1 ðA Þ þ wðÞ
X
^ þ kÞwððA ^ Þn ð þ kÞÞ wð
k2Zd
a.e. on Rd for 1 < n 2 N. Proof. We first prove that Hn() is well-defined a.e. on Rd . Note that, for n 2 N and a.e. n 2 Rd , !1 !12 2 2 X 1 1 X X X X ^ Þj nÞj ^ Þn ðn þ kÞÞjjwððA ^ Þj ðn þ kÞÞj 6 ^ Þj nÞj ^ Þn ðn þ kÞÞj2 ^ Þj ðn þ kÞÞ jwððA jwððA jwððA jwððA wððA j¼0
j¼0
k2Zd
6
1 X j¼0
k2Zd
^ Þj nÞj jwððA
X
k2Zd
!12 ^ Þj ðn þ kÞÞj2 jwððA
n
1 2
ðrw ððA Þ nÞÞ 6
j¼0
k2Zd 1
1 X
n
1
1
ðrw ðnÞ þ Dw ðnÞÞ2 ðrw ððA Þ nÞÞ2 6 ð1 þ Dw ðnÞÞ2 < 1
!12 ^ Þj nÞj2 jwððA
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by Lemma 1 and Lemma 3. So 1 X X
^ Þj ÞwððA ^ Þj ð þ kÞÞ ^ Þn ð þ kÞÞwððA wððA
j¼0 k2Zd
converges absolutely a.e. on Rd . It follows that Hn() is well-defined, and that
Hn ðÞ ¼
X
^ Þn ð þ kÞÞtk ðÞ wððA
ð15Þ
k2Zd
a.e. on Rd for n 2 N, where tk is defined as in (12). Also observing that
^ Þ1 þkÞ þ t A k ðÞ ^ Þ1 ÞwððA tk ððA Þ1 Þ ¼ wððA leads to
X
Hnþ1 ððA Þ1 Þ ¼
^ Þnþ1 ððA Þ1 þkÞÞt k ððA Þ1 Þ wððA
k2Zd
^ Þ1 Þ ¼ wððA
X nþ1 1 ^ ðA Þ ððA Þ þkÞ wððA ^ Þ1 þkÞ w k2Zd
þ
X n
^ ðA Þ ð þ A kÞ t A k ðÞ w k2Zd
X nþ1 1 ^ ðA Þ ððA Þ þkÞ wððA ^ Þ1 þkÞ þ Hn ðÞ ^ w Þ Þ ¼ wððA 1
k2Zd d
a.e. on R for n 2 N by Lemma 1. The proof is completed.
h
Proof of Theorem 1. Sufficiency. Suppose Dw ðÞ 2 Zþ a.e. on Rd . Then rw ðÞ 2 Zþ a.e. on Rd by Lemma 4. Also observing that w is a PFW for FL2(X) leads to
X
jhf ; T k wij2 6
XX
jhf ; Dj T k wij2 6 kf k2
j2Z k2Zd
k2Zd
for f 2 spanfT k w : k 2 Zd g, which implies that rw() 6 1 a.e. on Rd by Lemma 3. It follows that rw() = vU() a.e. on Rd , where U ¼ fn 2 Rd : rw ðnÞ > 0g. So w is a semi-orthogonal PFW for FL2(X) by Lemma 5. Necessity. Suppose w is a semi-orthogonal PFW for FL2(X). Then
X
^ Þj ð þ kÞÞwð ^ þ kÞ ¼ 0 wððA
k2Zd
a.e. on Rd for j 2 N by Lemma 5. Write
Gn ðÞ ¼
1 X X
^ Þj ÞwððA ^ Þj ð þ kÞÞ ^ Þn ð þ kÞÞwððA wððA
j¼1 k2Zd
a.e. on Rd for n 2 N. Then Gn() = Hn(), and Gn() = Gn1(A⁄) a.e. on Rd for 1 < n 2 N by Lemma 6. It follows that
Gn ðÞ ¼ G1 ððA Þn1 Þ a:e: on Rd for n 2 N:
ð16Þ
Note that
G1 ðÞ ¼
X
^ þA kÞtA k ðA Þ wðA
k2Zd
a.e. on Rd . By Lemma 1, we have
G1 ðÞ ¼
X
^ þkÞt k ðA Þ ¼ wðA
k2Zd
1 X
^ Þjþ1 Þ wððA
j¼0
which together with Lemma 5 implies that
^ Þrw ðA Þ G1 ðÞ ¼ wðA a.e. on Rd . Therefore,
^ Þn Þrw ððA Þn Þ Gn ðÞ ¼ G1 ððA Þn1 Þ ¼ wððA
X k2Zd
^ þkÞwððA ^ Þj ðA þkÞÞ; wðA
Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
9159
^ Þn Þ ¼ 0 when a.e. on Rd for n 2 N. Also observing that rw((A⁄)n) is either 0 or 1 a.e. on Rd by Lemma 5, and that wððA ^ Þn Þ a.e. on Rd for n 2 N. So rw((A⁄)n) = 0, we obtain that Gn ðÞ ¼ wððA
Wn ðÞ ¼
1 X hWn ðÞ; Wj ðÞil2 ðZd Þ Wj ðÞ j¼1
P 2 d a.e. on R for n 2 N, where Wj is defined as in (9) for j 2 N. Note that 1 n¼1 kWn ðÞkl2 ðZd Þ ¼ Dw ðÞ < 1 a.e. on R , where Dw is defined as in (8). Applying [[19], Chapter 7, Lemma 3.7], we obtain that d
Dw ðÞ ¼ dimF w ðÞ a.e. on Rd , where Fw is defined as in (10). Also observing Dw() < 1 a.e. on Rd leads to Dw ðÞ 2 Zþ a.e. on Rd . The proof is completed. h
3. Proof of Theorem 2 and Theorem 3 In this section, we focus on the proof of Theorem 2 and Theorem 3. For this purpose, we first establish two lemmas. Let w 2 L2 ðRd Þ, let A be a d d expansive matrix with jdetAj = 2, and let Dw and Wj with j 2 N be defined as in (8) and (9), respectively. Define
Z :¼ fn 2 Rd : Dw ðnÞ ¼ 0g;
ð17Þ
d
ð18Þ
P 1 :¼ fn 2 R : kW1 ðnÞkl2 ðZd Þ – 0g; d
P j :¼ fn 2 R : kWj ðnÞkl2 ðZd Þ – 0 and kWl ðnÞkl2 ðZd Þ ¼ 0;
for 1 6 l 6 j 1g for j > 1:
ð19Þ
Obviously, the sets Z and P j ’s are Zd -periodic and measurable, and they form a partition of Rd . Lemma 7. Let A be a d d expansive matrix with jdetAj = 2, let FL2(X) be a reducing subspace of L2 ðRd Þ, and let w 2 FL2(X). Assume that dim Fw() 2 {0, 1} a.e. on Rd . Define /0 by
8 if n 2 Z; < 0; pffiffiffiffiffiffiffiffiffi ^ 0 ðnÞ :¼ / Dw ðnÞ j ^ : kW ðnÞk wððA Þ nÞ; if n 2 P j for some j 2 N; 2 j
ð20Þ
l ðZd Þ
where Z and P j are defined as in (17)–(19), respectively. Then (i) /0 2 FL2(X); ^ 0 ðÞj2 ¼ P1 jwððA ^ Þj Þj2 a.e. on Rd ; (ii) j/ j¼1 d ^ 0 ðA Þ ¼ m0 ðÞ/ ^ 0 ðÞ (iii) there exists a Z -periodic measurable function m0 : Rd ! C such that jm0()j 6 1 a.e. on Rd , and that / a.e. on Rd .
Proof (i) It is easy to check that
X
^ 0 ð þ kÞj2 ¼ Dw ðÞ j/
ð21Þ
k2Zd
a.e. on Rd , which implies that /0 2 L2 ðRd Þ by (11). Also observing (20) gives (i). (ii) Note that (ii) is trivial a.e. on Z. Next we prove that it holds a.e. on each P l with l 2 N. Since dimFw() = 1 a.e. on P l , to each j 2 N there corresponds a Zd -periodic measurable function klj : P l ! C such that
Wj ðÞ ¼ klj ðÞWl ðÞ a.e. on P l . It follows that 1 X
^ Þj Þj2 ¼ jwððA
j¼1
Dw ðÞ ¼
1 X
!
jklj ðÞj2
j¼1 1 X j¼1
kWj ðÞk2l2 ðZd Þ ¼
1 X j¼1
which leads to (ii) a.e. on P l .
^ Þl Þj2 ; jwððA ! jklj ðÞj2 kWl ðÞk2l2 ðZd Þ ;
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P ^ Þl ð þ ðA Þ1 kÞÞj2 P kWl ðÞk22 d for 2 6 l 2 N, to each 2 6 l 2 N there corresponds a (iii) Since kWl1 ðA Þk2l2 ðZd Þ ¼ k2Zd jwððA l ðZ Þ d Z -periodic measurable function cl : P l ! f1; 2; . . . ; l 1g such that A P l P cl ðÞ . Also observing that dimFw(A⁄) = 1 a.e. on P l , we conclude that, for each 2 6 l 2 N, there exists a Zd -periodic measurable function kl : P l ! C such that kl() – 0 a.e. on P l and
Wl1 ðA Þ ¼ kl ðÞWcl ðÞ ðA Þ
ð22Þ
a.e. on P l . Similarly, for each l 2 N, there exists a Zd -periodic measurable function ~ kl : P 1 \ ðA Þ1 P l ! C such that
Wlþ1 ðÞ ¼ ~kl ðÞW1 ðÞ
ð23Þ
1
d
a.e. on P 1 \ ðA Þ P l . Define m0 : R ! C by
m0 ðnÞ ¼
8 pffiffiffiffiffiffiffiffiffiffiffi ffi Dw ðA nÞkWl ðnÞk 2 d > l ðZ Þ > p ffiffiffiffiffiffiffiffi ffi > ; for n 2 P l with l P 2; > Dw ðnÞkWc ðnÞ ðA nÞk 2 d kl ðnÞ > > l ðZ Þ l > > p ffiffiffiffiffiffiffiffiffiffiffi ffi > D ðA nÞkW ðnÞk ~k ðnÞ > w 1 > l2 ðZd Þ l > ffi > ; for n 2 P 1 \ ðA Þ1 P l with l 2 N; > pffiffiffiffiffiffiffiffi Dw ðnÞkWl ðA nÞk 2 d > l ðZ Þ > < e
2pink 0
kW1 ðnþðA Þ1 eÞk 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > > Dw ðnþðA Þ1 eÞ > > > > > p1ffiffi ; > > 2 > > > > > > 0; :
l ðZd Þ
;
for n 2 Z \ ðP 1 ðA Þ1 eÞ;
ð24Þ
for n 2 Z \ ðZ ðA Þ1 eÞ; 1 S T S for n 2 ðP 1 \ ðA Þ1 ZÞ ðZ ð P l ðA Þ1 eÞÞ; l¼2 d
where {0, e} is a set of representatives of distinct cosets in Z =A Z , and k0 is as in Proposition 2. Then m0 is Zd -periodic ^ 0 ðA Þ ¼ m0 ðÞ/ ^ 0 ðÞ a.e. on Z c by (20), (22) and (23), / ^ 0 ðA Þ ¼ 0 ¼ m0 ðÞ/ ^ 0 ðÞ a.e. on Z by (ii) and (20). and measurable, / c d ^ Also observing that there exists a mapping s : Z ! Z such that /0 ð þ sðÞÞ – 0 a.e. on Z c by (21), we have
jm0 ðÞj ¼ jm0 ð þ sðÞÞj ¼
d
^ 0 ðA ð þ sðÞÞÞj j/ ^ 0 ð þ sðÞÞj j/
^ 0 ðA ð þ sðÞÞÞj2 6 j/ ^ 0 ð þ sðÞÞj2 a.e. on Rd by (ii). Therefore we have jm0()j 6 1 a.e. on Z c . It is a.e. on Z c . However j/ obvious that jm0()j 6 1 a.e. on Z by (24). The proof is completed. h Lemma 8. Let A be a d d expansive matrix with jdetAj = 2, and let FL2(X) be a reducing subspace of L2 ðRd Þ. Assume that w is a PFW for FL2(X), and that dimFw() 2 {0, 1} a.e. on Rd . Define the Zd -periodic measurable function m1 : Rd ! C by
m1 ðnÞ ¼
8 1 pffiffi ; for n 2 Z \ ðZ ðA Þ1 eÞ; > > > 2 > > > < m0 ðn þ ðA Þ1 eÞ; for n 2 Z n ðZ ðA Þ1 eÞ; 2pink
0 kW1 ðnÞk 2 e > > pffiffiffiffiffiffiffiffiffi l ðZd Þ ; > > Dw ðnÞ > > : 0;
ð25Þ
for n 2 P 1 ; for n 2 P l with l P 2;
where {0, e} is a set of representatives of distinct cosets in Zd =A Zd , k0 is as in Proposition 2, Z and P j are respectively defined as in (17)–(19), and m0 is defined as in (24). Then (i) (ii) (iii) (iv)
^ nÞ ¼ e2pink0 m1 ðnÞ/ ^ 0 ðnÞ for a.e. n 2 Rd ; jm1(n)j 6 1, and wðA 2 2 d jm0()j + jm1()j = 1 a.e. on R ; m0 ðÞm0 ð þ ðA Þ1 eÞ ¼ m1 ð þ ðA Þ1 eÞm1 ðÞ a.e. on Rd ; jm0()j = jm1( + (A⁄)1e))j, and jm1()j2 + jm1( + (A⁄)1e)j2 = 1 a.e. on Rd .
Proof ^ Þ ¼ 0 a.e. on P c . We only need to prove that m1 ðÞ/ ^ 0 ðÞ ¼ 0 a.e. on (i) By (25) and (20), (i) holds a.e. on P 1 . Note that wðA 1 c c ^ 0 ðÞ ¼ 0 a.e. on Z. So (i) holds. P 1 . It is true since m1() = 0 a.e. on ðP 1 [ ZÞ , and / (ii) By (24) and (25), m0 ðÞ ¼ m1 ðÞ ¼ p1ffiffi2 a.e. on Z \ ðZ ðA Þ1 eÞ, which implies (ii) holds a.e. on Z \ ðZ ðA Þ1 eÞ. Note that
jm0 ðÞj2 þ jm1 ðÞj2 ¼ jm0 ð þ ðA Þ1 eÞj2 þ jm1 ð þ ðA Þ1 eÞj2 a.e. on Z n ðZ ðA Þ1 eÞ by (24) and (25). So, to prove (ii) a.e. on Rd , we only need to prove that
Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
jm0 ðÞj2 þ jm1 ðÞj2 ¼ 1 a:e: on Z c :
9161
ð26Þ
By (ii) and (iii) in Lemma 7 and (i) in Lemma 8, we have
^ 0 ðÞj2 ¼ jwðA ^ Þj2 þ j/ ^ 0 ðA Þj2 ¼ ðjm0 ðÞj2 þ jm1 ðÞj2 Þj/ ^ 0 ðÞj2 j/ a.e. on Rd , which implies that
X
^ 0 ð þ kÞj2 ¼ ðjm0 ðÞj2 þ jm1 ðÞj2 Þ j/
k2Zd
X
^ 0 ð þ kÞj2 j/
k2Zd
a.e. on Rd . Also applying Lemma 7 (ii) leads to
Dw ðÞ ¼ jm0 ðÞj2 þ jm1 ðÞj2 Dw ðÞ a.e. on Rd . It gives (26) due to the fact that Dw() – 0 a.e. on Z c . (iii) By (24) and (25), we have
m0 ðÞ ¼ m1 ð þ ðA Þ1 eÞ;
m0 ð þ ðA Þ1 eÞ ¼ m1 ðÞ
ð27Þ 1
a.e. on Z, which gives (iii) a.e. on Z. Observing n þ ðA Þ leads to
1
⁄ 1
e 2 Z for n 2 Z ðA Þ e, and applying (27) to n + (A ) e
m0 ðn þ ðA Þ1 eÞ ¼ m1 ðn þ 2ðA Þ1 eÞ ¼ m1 ðnÞ; m0 ðnÞ ¼ m0 ðn þ ðA Þ1 e þ ðA Þ1 eÞ ¼ m1 ðn þ ðA Þ1 eÞ: Therefore (iii) holds a.e. on Z [ ðZ ðA Þ1 eÞ. Next we prove that it holds a.e. on ðZ [ ðZ ðA Þ1 eÞÞc . By (i) and Zd periodicity of m1,
^ Þj ÞwððA ^ Þj ð þ kÞÞ ¼ jm1 ððA Þj1 Þj2 / ^ 0 ððA Þj1 Þ/ ^ 0 ððA Þj1 ð þ kÞÞ wððA a.e. on Rd for j 2 N and k 2 Zd . Combined with (ii) and Lemma 7 (iii), it follows that
^ Þj ÞwððA ^ Þj ð þ kÞÞ ¼ ð1 jm0 ððA Þj1 Þj2 Þ/ ^ 0 ððA Þj1 ð þ kÞÞ ^ 0 ððA Þj1 Þ/ wððA ^ 0 ððA Þj1 ð þ kÞÞ m0 ððA Þj1 Þ/ ^ 0 ððA Þj1 Þm0 ððA Þj1 ð þ kÞÞ/ ^ 0 ððA Þj1 ð þ kÞÞ ^ 0 ððA Þj1 Þ/ ¼/ ^ 0 ððA Þj1 Þ/ ^ 0 ððA Þj1 ð þ kÞÞ / ^ 0 ððA Þj ð þ kÞÞ ^ 0 ððA Þj Þ/ ¼/ a.e. on Rd for j 2 N and k 2 Zd , which implies that 1 X
^ Þj ÞwððA ^ Þj ð þ kÞÞ ¼ / ^ 0 ð þ kÞ ^ 0 ðÞ/ wððA
j¼1
a.e. on Rd for k 2 Zd by Lemma 2. So, by Lemma 1,
0¼
1 X
^ Þj ÞwððA ^ Þj ð þ qÞÞ ¼ wðÞ ^ þ qÞ þ / ^ 0 ðÞ/ ^ 0 ð þ qÞ ^ wð wððA
ð28Þ
j¼0
a.e. on Rd for q 2 Zd n A Zd . Since ðA Þ1 q ðA Þ1 e 2 Zd , we have
e2piððA
1
Þ
qÞk
0
¼ e2piððA
1
Þ
eÞk0
¼ 1;
and thus
^ wð ^ þ qÞ ¼ m1 ððA Þ1 Þm1 ððA Þ1 ð þ qÞÞ/ ^ 0 ððA Þ1 ð þ qÞÞ ^ 0 ððA Þ1 Þ/ wðÞ a.e. on Rd for q 2 Zd n A Zd by (i). Also from (28) and Lemma 7, it follows that
^ 0 ððA Þ1 Þ/ ^ 0 ððA Þ1 ð þ qÞÞ ¼ 0 m0 ððA Þ1 Þm0 ððA Þ1 ð þ qÞÞ m1 ððA Þ1 Þm1 ððA Þ1 ð þ qÞÞ / a.e. on Rd for q 2 Zd n A Zd , equivalently,
^ 0 ðÞ/ ^ 0 ð þ ðA Þ1 qÞ ¼ 0 m0 ðÞm0 ð þ ðA Þ1 qÞ m1 ðÞm1 ð þ ðA Þ1 qÞ /
ð29Þ
a.e. on Rd for q 2 Zd n A Zd . Since Dw()Dw( + (A⁄)1e) – 0 a.e. on ðZ [ ðZ ðA Þ1 eÞÞc , there exist two mappings s1, s2: ðZ [ ðZ ðA Þ1 eÞÞc ! Zd such that
^ 0 ð þ s1 ðÞÞ/ ^ 0 ð þ s2 ðÞ þ ðA Þ1 eÞ – 0 /
ð30Þ
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Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
a.e. on ðZ [ ðZ ðA Þ1 eÞÞc by (21). Combined with (29), it follows that
^ 0 ð þ s1 ðÞÞ/ ^ 0 ð þ s1 ðÞ þ ðA Þ1 ððA Þðs2 ðÞ s1 ðÞÞ þ eÞÞ ¼ 0; m0 ðÞm0 ð þ ðA Þ1 eÞ m1 ðÞm1 ð þ ðA Þ1 eÞ / namely,
^ 0 ð þ s1 ðÞÞ/ ^ 0 ð þ s2 ðÞ þ ðA Þ1 eÞ ¼ 0 m0 ðÞm0 ð þ ðA Þ1 eÞ m1 ðÞm1 ð þ ðA Þ1 eÞ / a.e. on ðZ [ ðZ ðA Þ1 eÞÞc . Combined with (30), it follows that
m0 ðÞm0 ð þ ðA Þ1 eÞ ¼ m1 ðÞm1 ð þ ðA Þ1 eÞ a.e. on ðZ [ ðZ ðA Þ1 eÞÞc . So (iii) holds. (iv) By (iii) and (ii), we have
jm0 ðÞj2 ¼ jm0 ðÞj2 jm0 ð þ ðA Þ1 eÞj2 þ jm0 ðÞj2 jm1 ð þ ðA Þ1 eÞj2 ¼ jm1 ðÞj2 jm1 ð þ ðA Þ1 eÞj2 þ jm0 ðÞj2 jm1 ð þ ðA Þ1 eÞj2 ¼ jm1 ð þ ðA Þ1 eÞj2 a.e. on Rd , and thus
jm0 ðÞj ¼ jm1 ð þ ðA Þ1 eÞj
ð31Þ
d
a.e. on R . Combined with (ii), it follows that
jm1 ðÞj2 þ jm1 ð þ ðA Þ1 eÞj2 ¼ jm1 ðÞj2 þ jm0 ðÞj2 ¼ 1 a.e. on Rd . The proof is completed.
h
Proof of Theorem 2. Sufficiency. Define C ¼ fn 2 Rd : m0 ððA Þ1 n þ ðA Þ1 eÞ ¼ 0g, and
lðÞ ¼
8 > > <
m1 ððA Þ1 Þ m0 ððA Þ1 þðA Þ1 eÞ
on Cc ; ð32Þ
1 1 > Þ eÞ > : m1 ððA Þ þðA on C; 1
m0 ððA Þ
Þ
where m0 and m1 are defined as in (24) and (25), respectively. Then l is well-defined, unimodular and measurable by Lemma 8 (ii) and (iv). We also claim that l is Zd -periodic. By Zd -periodicity of m0 and m1, l is A Zd -periodic. Also observing that Zd ¼ A Zd þ f0; eg, to prove l being Zd -periodic, we only need to prove that
lðÞ ¼ lð þ eÞ a:e: on Rd :
ð33Þ
By Lemma 8 (ii) and (iv), we have
jm0 ðÞj2 þ jm0 ð þ ðA Þ1 eÞj2 ¼ 1 a:e: on Rd :
ð34Þ
It follows that C \ (C e) = ;, which implies that (33) holds a.e. on C [ (C e) by (32). Now we turn to (C [ (C e))c. By Lemma 8 (iii) and (32), we have
0 – m0 ððA Þ1 Þm0 ððA Þ1 þðA Þ1 eÞ ¼ m1 ððA Þ1 þðA Þ1 eÞm1 ððA Þ1 Þ ¼ lð þ eÞm0 ððA Þ1 ÞlðÞm0 ððA Þ1 þðA Þ1 eÞ ¼ lð þ eÞlðÞm0 ððA Þ1 Þm0 ððA Þ1 þðA Þ1 eÞ a.e. on (C [ (C e))c. It follows that lð þ eÞlðÞ ¼ 1 a.e. on (C [ (C e))c, and thus l() = l( + e) a.e. on (C [ (C e))c due to the fact that l is unimodular. Therefore (33) holds, and thus l is a unimodular, Zd -periodic and measurable function. Note ^ 0 ðA Þ ¼ m0 ðÞ/ ^ 0 ðÞ a.e. on Rd by Lemma 7 that jm0()j2 + jm0( + (A⁄)1e)j2 = 1 a.e. on Rd by Lemma 8 (ii) and (iv), and that / (iii). To prove the sufficiency, we only need to prove that
^ nÞ ¼ e2pink0 lðA nÞm0 ðn þ ðA Þ1 eÞ/ ^ 0 ðnÞ wðA for a.e. n 2 Rd . Also observing that
^ nÞ ¼ e2pink0 m1 ðnÞ/ ^ 0 ðnÞ wðA for a.e. n 2 Rd , it suffices to prove that
m1 ðÞ ¼ lðA Þm0 ð þ ðA Þ1 eÞ
ð35Þ
Y.-Z. Li, F.-Y. Zhou / Applied Mathematics and Computation 217 (2011) 9151–9164
9163
a.e. on Rd . By Lemma 8 (iv),
jm1 ðÞj ¼ jm0 ð þ ðA Þ1 eÞj a.e. on Rd , which implies that (35) holds a.e. on (A⁄)1C. However, (35) also holds a.e. on ((A⁄)1C)c by (32). Therefore (35) holds a.e. on Rd , and the sufficiency follows. Necessity. Suppose w is an MRA PFW for FL2(X). Then there exist / 2 FL2(X), m 2 L2 ðTd Þ, and a unimodular Zd -periodic function l such that m 2 e F and 2piððA ^ wðnÞ ¼e
1
Þ
nÞ
^ Þ1 nÞ lðnÞmððA Þ1 n þ ðA Þ1 eÞ/ððA
k0
for a.e. n 2 Rd . It follows that
^ Þj ðn þ kÞÞ ¼ e2piððA wððA ¼e
j1
Þ
nÞ
k0
2piððA Þj1 nÞ
k0
^ Þj1 ðn þ kÞÞ lððA Þj nÞmððA Þj1 n þ ðA Þ1 eÞ/ððA j
j1
1
lððA Þ nÞmððA Þ n þ ðA Þ eÞ
j2 Y
!
n
^ þ kÞ mððA Þ nÞ /ðn
n¼0
for a.e. n 2 Rd , j 2 N and k 2 Zd , where Zd -periodicity of l and m is used in the first equality, and we make the convention that Q1 n d n¼0 mððA Þ nÞ ¼ 1 for a.e. n 2 R . Therefore, we have
Wj ðnÞ ¼ e
2piððA Þj1 nÞk
0
j
j1
1
lððA Þ nÞmððA Þ n þ ðA Þ eÞ
j2 Y
! n
mððA Þ nÞ U0 ðnÞ
ð36Þ
n¼0
^ þ kÞ : k 2 Zd g. By (36), dimFw() 2 {0, 1} a.e. on Rd . The proof is completed. h for a.e. n 2 Rd and j 2 N, where U0 ðÞ ¼ f/ð Proof of Theorem 3. By Theorem 1 and its proof, w is a semi-orthogonal PFW for FL2(X) if and only if Dw ðÞ ¼ dimF w ðÞ 2 Zþ a.e. on Rd . Combined with Theorem 2, it follows that w is a semi-orthogonal MRA PFW for FL2(X) if and only if Dw() = dimFw() 2 {0, 1} a.e. on Rd . Note that Dw() 2 {0, 1} a.e. on Rd implies that w is a semi-orthogonal PFW for FL2(X) by Theorem 1, and thus implies that Dw() = dimFw() 2 {0, 1} a.e. on Rd by the proof of the necessity of Theorem 1. Therefore, w is a semi-orthogonal MRA PFW for FL2(X) if and only if Dw() 2 {0, 1} a.e. on Rd . The proof is completed. h Applying Lemma 1 and Theorem 3, we have the following example: Example 5. Let A = 2, and let FL2(X) be a reducing subspace of L2 ðRÞ. Define w via its Fourier transform by
^ ¼ epi v ðÞ wðÞ E a.e. on R, where E ¼ ð½1; 12Þ [ ð12 ; 1Þ \ X. Then w is a semi-orthogonal MRA PFW for FL2(X). Example 6. Let A ¼
E¼
0 2
S 1 1 ; 6 a; b 6 1, and let FL2(X) be a reducing subspace of L2 ðR2 Þ with X ¼ j2Z ðA Þj E, where 0 2
b n 2 ½0; 1Þ2 : 0 6 n1 6 a; ðn1 aÞ 6 n2 6 1 [ fn 2 ½0; 1Þ2 : a 6 n1 6 1; 0 6 n2 6 1g:
a
^ ¼ v on R2 . Then w is a semi-orthogonal PFW, but not an MRA PFW for FL2(X). Define w via its Fourier transform by w E Proof. By Example 1 in [10], w is a PFW for FL2(X). A simple computation shows that
1 1 ðA Þ1 E ½0; 1Þ ½1; 0Þ and ðA Þj E ½ ; Þ2 2 2 for 1 < j 2 N, which implies that
Dw ðÞ ¼
1 X X j¼1 k2Z2
vððA Þj EþkÞ ðÞ ¼
1 X j¼1
v S ððA Þj EþkÞ ðÞ:
ð37Þ
k2Z2
However, Dw() < 1 a.e. on R2 by (11). So we have Dw 2 Zþ a.e. on R2 , and consequently, w is a semi-orthogonal PFW for 1 is a set with nonzero measure. By (37), we have Dw() P 2 FL2(X) by Theorem 1. Note that ððA Þ1 EÞ \ ðA Þ2 E þ 0 1 . Combined with Theorem 3, it follows that w is not an MRA PFW. The proof is a.e. on ððA Þ1 EÞ \ ðA Þ2 E þ 0 completed. h
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Now we conclude this paper with the following remark: Remark 2. It is unresolved whether w in Example 1 is always an MRA wavelet for L2 ðRd Þ when L = 1, and is always a nonsemi-orthogonal MRA PFW for L2 ðRd Þ when L is an odd number greater than 1. Acknowledgments The authors thank the referees for their valuable comments, which greatly improved the readability of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
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