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Frame scaling function sets and frame wavelet sets in Rd
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Chaos, Solitons and Fractals 40 (2009) 2483–2490 www.elsevier.com/locate/chaos
Frame scaling function sets and frame wavelet sets in Rd Zhanwei Liu a,*, Guoen Hu a,1, Guochang Wu b a
Department of Applied Mathematics, University of Information Engineering, Zhengzhou, Henan 450002, China b School of Science, Xi’an Jiaotong University, Xi’an, Shanxi 710049, China Accepted 29 October 2007
Abstract In this paper, we classify frame wavelet sets and frame scaling function sets in higher dimensions. Firstly, we obtain a necessary condition for a set to be the frame wavelet sets. Then, we present a necessary and sufficient condition for a set to be a frame scaling function set. We give a property of frame scaling function sets, too. Some corresponding examples are given to prove our theory in each section. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Because nature is not continuous, not periodic, but self-similar, Mohamed El Naschie introduced the E-infinity ðeð1Þ Þ view to describe and understand it. In [1–4], a mathematical formulation was introduced to describe phenomena that is resolution dependent by Mohamed El Naschie with eð1Þ . In his view, space-time is an infinite dimensional fractal that happens to have D = 4 as the expectation value for topological dimension [1–3]. In our low energy resolution, the topological value 3 + 1 means that the world appears to us if it were four-dimensional. The Fouriers transform is a mathematical tool to consider the motion either in the frequency domain or in the time domain. But it cannot takes account into the resolutions and the frequencies simultaneously. Fortunately, the wavelet transform can make up for the disadvantage, which permits a multiresolution analysis of data with different behavior on different scales. Today, Orthonormal bases of wavelets are an ubiquitous and eminently powerful tool that pervades all areas of mathematics. A multiresolution analysis (MRA) was introduced by Mallat [5] and Meyer [6] primarily as a tool to construct and analyze the orthonormal wavelets. When dealing with over complete spanning systems one is naturally lead to the concept of frame [7]. The frame theory plays an important role in the modern time-frequency analysis. It has been developed very fast over the last 10 years, especially in the context of wavelets. The generalization of MRA, a frame multiresolution analysis (FMRA), was considered and applied in the analysis of narrow band signals with more freedom in the constructions of wavelets with fast iterative structures by Benedetto and Li [8]. ffi v for some Lebesgue measurable set B of finite measure, the In the special case that the Fourier transform of w is p1ffiffiffi 2p E study of the wavelet is reduced to the study of the set B, which is simpler in general. If we could characterize the set B that defines a wavelet in such a manner, then we would be able to find ways to construct wavelets and have a better *
1
Corresponding author. E-mail address: [email protected] (Z. Liu). Research was supported by the NSF of China (Grant No. 10671210).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.10.042
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understanding of wavelets in general. In [9], the authors introduced the notion of wavelet set which turned out to be one of the building blocks of their approach to wavelet analysis from an operator theory point of view. From then on, many results are found [10,11]. For example, the important result of the existence of wavelets for unitary systems having an expansive dilation matrix was based on the existence of wavelet sets. In [11], Dai and Larson discussed the characterization of wavelet sets in Rn . In our paper, we release their conditions to the case of the band-limited frame wavelet, and obtain a general result. In [14], Zhang characterized the support G of the Fourier transform of the band-limited scaling function and give an approach to the construction of the scaling function. The second aim of the present article is to generalize their results to the case of frame scaling function associated to FMRA and present a necessary and sufficient condition for a set G to be a frame scaling function set. Based on the relations of four point set G; sðGÞ; 12 G and G n 12 G, we reveal an essential difference of scaling functions corresponding to MRA and FMRA. The paper is organized as follows: Section 2 contains some necessary notations and known results. In Section 3, we discuss the property of the frame wavelet sets. In Section 4, we present some necessary and sufficient conditions for a set G to be a frame scaling function set associated to FMRA. In some conditions, we give a property of frame scaling function sets, too. Some corresponding examples are given in each section.
2. Preliminary A collection of elements f/j : j 2 J g in a Hilbert space H is called a frame if there exist constants a and b, 0 < a 6 b < 1, such that X akf k2 6 jhf ; /j ij2 6 bkf k2 ; 8f 2 H : ð2:1Þ j2J
Let a0 be the supremum of all such numbers a and b0 be the infimum of all such numbers b, then a0 and b0 are called the frame bounds of the frame f/j : j 2 J g. When a0 ¼ b0 we say that the frame is tight. When a0 ¼ b0 ¼ 1 we say the frame is a normalized tight frame. Let A be a d d real invertible matrix. It induces a unitary operator DA acting on L2 ðRd Þ defined by 1
ðDA f ÞðtÞ ¼ jdetAj2 f ðAtÞ;
8f 2 L2 ðRd Þ;
t 2 Rd :
ð2:2Þ
The matrix A is called expansive if all its eigenvalues have modulus greater than one. The operator DA corresponding to a real expansive matrix A is called an A-dilation operator. A vector s in Rd induces a unitary translation operator T s defined by ðT s f ÞðtÞ ¼ f ðt sÞ;
8f 2 L2 ðRd Þ;
t 2 Rd :
ð2:3Þ d
In this article, we will only deal with translation operators T l with l 2 Z . If f 2 L2 ðRd Þ \ L1 ðRd Þ, then we define its Fourier transform: Z 1 ei2phs;ti f ðtÞdt; ðf^ ÞðsÞ ¼ d d 2 R ð2pÞ
ð2:4Þ
where hs; ti denotes the standard inner product in Rd . Let Ed denote the set of all expanding matrices A. Now we fix an arbitrary matrix A 2 Ed . For a function w 2 L2 ðRd Þ, we will consider the affine system W defined by n o j ð2:5Þ W ¼ wj;k ðxÞjwj;k ðxÞ ¼ 22 wðAj x kÞ; j 2 Z; k 2 Z d : In 1988, Mallat introduced a concept of multiresolution analysis (MRA) which is a fundament concept in wavelet analysis. Definition 2.1. Let fV m gm2Z be a sequence of closed subspaces of L2 ðRd Þ satisfying: (1) (2) (3) (4) (5)
V Sm V mþ1 ; 2 d Tm2Z V m ¼ L ðR Þ; m2Z V m ¼ f0g; f 2 V m () f ð2Þ 2 V mþ1 ; m 2 Z; There exists a function / 2 V 0 such that fT k / : k 2 Zg is an orthonormal basis of V 0 .
Then fV j gj2Z is called an MRA and the function / from (5) is called a scaling function.
Z. Liu et al. / Chaos, Solitons and Fractals 40 (2009) 2483–2490
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Notice that fV m : m 2 Zg is said to be an FMRA if condition (5) is replaced by the condition that there exists a / 2 V 0 such that fT k / : k 2 Zg is a normalized tight frame of V 0 . The function / is called a frame scaling function. According to [7], there exists a function w such that the system W defined by (2.5) is a orthogonal base in L2 ðRd Þ. We call w a wavelet. Of course, we can say that w is a normalized tight frame wavelet if the system (2.5) is a normalized tight frame for L2 ðRd Þ. Definition 2.2. For a normalized tight frame wavelet w, let ^ B :¼ closfx 2 Rd : wðxÞ–0g;
ð2:6Þ
^ then B is called the frame wavelet set and write B ¼ suppw. ^ is bounded, then w is called a band-limited frame wavelet. If suppw ^ In the same way, we can define a frame scaling function set G and write G ¼ supp/. Let G be a Lebesgue measurable set of finite measure. For any l 2 Z d , let I l denote the d-cube ½0; 1Þd þ l. For any subset G of Rd , define: [ ðG \ 2pI l 2plÞ: ð2:7Þ sðGÞ ¼ l2Z d
If the above is a disjoint union, we say that G is translation equivalent to sðGÞ, which is a subset of 2pI 0 , where I 0 is the unit d-cube ½0; 1Þd þ l. If G and F are translation equivalent to the same subset in 2pI 0 , then we say G and F are trans: lation equivalent. This defines an equivalent relation and is denoted by ¼. Let lðÞ be the Lebesgue measure. It is clear : : that lðGÞ P lðsðGÞÞ. The equality holds if and only if G¼sðGÞ. If G¼F , then lðGÞ ¼ lðF Þ. Two points x; y 2 G are said to be translation equivalent if x y ¼ 2pl for some l 2 Z d . The translation redundancy index of a point x in G is the number of elements in its equivalent class. We write Gðs; kÞ for the set of all points in G with translation redundancy index k. In general, Gðs; kÞ could be an empty set, proper subset of G, or the set G itself. For k–m; Gðs; kÞ \ Gðs; mÞ ¼ ;, so ! [ [ G ¼ Gðs; 1Þ Gðs; nÞ : ð2:8Þ n2N
Let E be a point set in Rd . Throughout this article, we will use vE to denote the characteristic function of E.
3. Frame wavelet sets In [11], Dai and Larson discussed some characterizations of the wavelet sets. In this section, we will generalize their results to the band-limited frame wavelet, and obtain an important property about frame wavelets sets. At first, we give a Lemma. Lemma 3.1. Let M, N be Lebesgue measurable sets of finite measure in Rd such that sðMÞ ¼ sðN Þ ½0; 2pÞd . If M ¼ Mðs; 1Þ; N ¼ N ðs; 1Þ, then for any 2p periodical function f 2 L2 ðRd Þ, we have d
d
hf ðsÞvM ; ð2pÞ 2 eihl;si i ¼ hf ðsÞvN ; ð2pÞ 2 eihl;si i:
ð3:1Þ
Proof. Since sðMÞ ¼ sðN Þ ½0; 2pÞd , by definition of sðÞ; M ¼ Mðs; 1Þ and N ¼d N ðs; 1Þ, there must exist r; q 2 Z n such that M ¼ sðMðs; 1ÞÞ þ 2pr and N ¼ sðN ðs; 1ÞÞ þ 2pq. Because the system fð2pÞ 2 eihl;si gl2Z d is an orthonormal basis for L2 ½0; 2pÞd , we have Z Z d d d hf ðsÞvM ; ð2pÞ 2 eihl;si i ¼ f ðsÞð2pÞ 2 eihl;si ds ¼ f ðs0 þ 2prÞð2pÞ 2 eihl;s0 þ2pri dðs0 þ 2prÞ ¼
Z
sðMðs;1ÞÞ
M 0
d 2
ihl;s0 i
f ðs Þð2pÞ e sðN Þ d
h
ds ¼
Z
d
f ðs0 þ 2pqÞð2pÞ 2 eihl;s0 þ2pqi dðs0 þ 2pqÞ sðN ðs;1ÞÞ
¼ hf ðsÞvN ; ð2pÞ 2 eihl;si i: Then we get the desired result.
0
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Theorem 3.2. Let B be a Lebesgue measurable set with finite measure in Rd . If B is a frame wavelet set, then B ¼ Bðs; 1Þ. Proof. Assume that set B is a frame wavelet set and lðBðs; m0 ÞÞ > 0 for some m0 > 1. By Lemma 1 [15], we can define gk–0; k^hk–0. Let f ¼ g h, then kf^ k2 –0. g^ ¼ vBð1Þ ðs;m0 Þ and ^h ¼ vBð2Þ ðs;m0 Þ . So k^ Again by Lemma 3.1, we have d
d
h^ gðsÞ; ð2pÞ 2 eihl;si i ¼ h^hðsÞ; ð2pÞ 2 eihl;si i:
ð3:2Þ
^ j;k . Since B be bounded, then Bj;k is bounded also. We can choose many finitely By Definition 2.2, let Bj;k ¼ suppw and define a point set V ¼ fv0 ; v1 ; . . . ; vt g Z d such that distinct points v0 ; v1 ; . . . ; vt 0 2 fvv gt0 P ^ p p ^ ^ ^ Bj;k ¼ [p2V ðBj;k \ 2ppÞ. Letting wðsÞ j;k ¼ wðsÞj;k vBj;k \2pp , we have wðsÞj;k ¼ p2V wðsÞj;k . Then E X XD ^ ^ p ; ð2pÞd2 eihl;si ð2pÞd2 eihl;si : wðsÞ wðsÞ ð3:3Þ j;k j;k ¼ p2V l2Z d
By (3.2) and (3.3), we obtain XX ^ j;k ij2 jh^ g ^h; w kf^ k2 ¼ j2Z k2Z d
2 X X X X d d ^ p ; ð2pÞd2 eihl;si iðh^ ¼ hwðsÞ gðsÞ; ð2pÞ 2 eihl;si i h^ hðsÞ; ð2pÞ 2 eihl;si iÞ ¼ 0 j;k d p2V d j2Z k2Z
l2Z
It is a contradiction with kf^ j2 –0, hence B ¼ Bðs; 1Þ holds. h Then, we will construct an example to prove our result. Example 3.3. Let B ¼ E1 [ E2 ; E1 ¼ p2 ; p ½0; p; E2 ¼ 5p 2 ; 3p ½0; p. Since B ¼ Bðs; 2Þ, by Theorem 3.2, all the ^ ¼ B are not the frame wavelets. functions w 2 L2 ðRd Þ defined by suppw As a result, we get a method to judge band-limited frame wavelets just by their frame wavelet sets.
4. Frame scaling function sets Zhang in [14] characterized the support G of the Fourier transform of the band-limited scaling function and showed ^ ¼ G if and only if that there exists a scaling function / with supp/ 1 1 m d d d G þ 2pk ’ ;ðk 2 Z d Þ ðaÞ G 2G; ðbÞ [m 2 G ’ R ; ðcÞ G þ 2pZ ’ R ; ðdÞ G n G \ 2 2 hold. But all his results just based on the scaling functions associated to MRA. In this section, we will generalize their results to the case of frame scaling function associated to FMRA and present a necessary and sufficient condition for a set G to be a frame scaling function set. We firstly introduce our work. ^ ¼ G if and only Theorem 4.1. Let G be a bounded closed set in Rd . Then there exists a frame scaling function / with supp/ if \ [ m 1 1 ðaÞ G 2G; ðbÞ 2 G ’ Rd ; ðcÞ sðGÞ T d ; ðdÞ G n G ð4:1Þ G þ 2pk ’ ;ðk 2 Z d Þ 2 2 m2Z hold. In order to prove this theorem, we introduce some notations: for m ¼ 0; 1; . . ., 1 G0 ¼ G n G; 2
Gm ¼ 2m Gm0 ;
Gm ¼ 2m G;
Em ¼ Gm þ 2pZ d ;
Em ¼ Gm þ 2pZ d :
ð4:2Þ
Then, we give several Lemmas. Lemma 4.2. [14, Lemma 2.3]. If G satisfies (a) and (d) in (4.1), then we have ð1Þ ðEm n Gm Þ \ G ’ ;ðm P 0Þ;
ð2Þ G \ ðGk þ 2pkÞ ’ ;ðk–0Þ:
ð4:3Þ
Z. Liu et al. / Chaos, Solitons and Fractals 40 (2009) 2483–2490
Lemma 4.3. Let G satisfy (a) in ðN þ 2pZ d Þ [ ðG0 þ 2pZ d Þ [ 12 G þ 2pZ d ¼ Rd .
(4.2).
If
sðGÞ T d
and
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N ¼ T d n sðGÞ,
then
Proof. Assume ðN þ 2pZ d Þ [ G0 þ 2pZ d [ 12 G þ 2pZ d –Rd and a point x 2 Rd satisfying 1 ð4:4Þ x R ðN þ 2pZ d Þ [ ðG0 þ 2pZ d Þ [ G þ 2pZ d : 2 S By k2Z d 2pk ¼ Rd , we can find two points y 2 ½0; 2pd and s 2 Z d such that x ¼ y þ 2ps. Since N [ sðGÞ ¼ T , then y 2 N or y 2 sðGÞ. If y 2 N, we have x ¼ y þ 2ps 2 N þ 2pZ d . If y 2 sðGÞ, by the definition of sðGÞ, there must exist two points z 2 G; l 2 Z d satisfying y ¼ z 2pl. Notice G ¼ G0 [ 12 G, so z 2 G0 or z 2 12 G. If z 2 G0 , then x ¼ y þ 2pk ¼ z 2pl þ 2pk 2 G0 þ 2pZ d . On the other hand, if z 2 12 G, we derive x ¼ y þ 2pk ¼ z 2pl þ 2pk 2 12 G þ 2pZ d . All the results contradict (4.4). Hence, we get the desired result. h Lemma 4.4. Let / 2 L2 ðRd Þ and a closed set G satisfy the following: ^ ¼ G; ð1Þ supp/ ^ ^ ð2Þ /ð2xÞ ¼ H ðxÞ/ðxÞa:e:x 2 Rd ðH 2 L1 ðT d ÞÞ; X ^ þ 2pnÞj2 ¼ 1a:e:x 2 T d n N ; ð3Þ j/ðx
ð4:5Þ ð4:6Þ ð4:7Þ
n2Z d
ð4Þ
[
2m G ’ Rd ;
ð4:8Þ
m2Z
then / is a frame scaling function. Proof. Let V m :¼ spanf/ð2m t nÞ; n 2 Z d gðm 2 ZÞ:
ð4:9Þ
By (4.6), V m V mþ1 . Again by (4.5) and (4.8), we get [ [ [ [ m ^ m Þ ¼ suppf^ ¼ supp/ð2 2 G ¼ Rd : m2Z f 2V m
m2Z
ð4:10Þ
m2Z
Applying a known result [12, Proposition 2.3.4], we obtain that we conclude that / is a frame scaling function. h
S
m2Z V m
¼ L2 ðRd Þ. Again by (4.7) [13, Theorem 2.2.7],
Proof of Theorem 4.1. Sufficiency: Assume that (a), (b), (c), (d) in (4.1) hold. By Lemma 4.4, it is enough that we directly construct / 2 L2 ðRd Þ and H 2 L1 ðT d Þ satisfying (4.5)–(4.8). From the argument of [14, Theorem 2.1], we can define 2p-periodic function H ðxÞ on point set 12 G þ 2pZ d and S m ^ ^ ^ supp/ ¼ G with m2Z 2 G ’ Rd such that the equation /ð2xÞ ¼ H ðxÞ/ðxÞ holds on 12 G þ 2pZ d . ^ þ 2pnÞj2 is a 2p periodical function, we easily conclude by N ¼ T d n sðGÞ that (4.7) ^ ¼ G and P d j/ðx Since supp/ n2Z holds on T d n N . Define H ðxÞ ¼ 0ðx 2 G0 þ 2pZ d [ ðN þ 2pZ d ÞÞ. From Lemma 4.3, H ðxÞ has been well defined on Rd and H 2 L1 ðT d Þ. Again by Lemma 4.3, (d) and (a) in (4.1), we have \ 1 ð1Þ ðN þ 2pZ d Þ \ ðG þ 2pZ d Þ ’ ;; ð2Þ G0 þ 2pZ d ð4:11Þ G þ 2pZ d ’ ;: 2 d For x 2 G0 þ 2pZ d [ ðN þ 2pZ d Þ, we have 2x 2 2G0 þ 4pZ d [ ð2Nþ 4pZ d Þ. By (4.11), then ð2G0 þ 4pZ Þ \ G ’ ; d and ð2N þ 4pZ Þ \ G ’ ;. Hence /ð2xÞ ¼ 0. Again by H ðxÞ ¼ 0ðx 2 G0 þ 2pZ d [ ðN þ 2pZ d ÞÞ, we know that (4.6) holds also on G0 þ 2pZ d [ ðN þ 2pZ d Þ. Combining above results, (4.6) holds on Rd . Up to now, we have constructed / 2 L2 ðRd Þ and H 2 L1 ðT d Þ satisfying (4.5)–(4.8). Sufficiency is proved. Necessity: 1 Suppose that fT n / : n 2 Z d g is the normalized tight frame of V 0 . By Definition 2.1, f22 T n /ð2Þ : n 2 Z d g is the normalized tight frame of V 1 . Since / 2 V 0 V 1 , then X 1 1 h/ðxÞ; 22 T n /ð2xÞi22 T n /ð2xÞ: ð4:12Þ /ðxÞ ¼ n2Z
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Multiplying both side of (4.12) with Fourier transform, there exists m 2 L2 ðT d Þ such that ^ ^ /ð2xÞ ¼ mðxÞ/ðxÞ for a:e:x 2 Rd :
ð4:13Þ
^ ^ ^ By (4.13), if /ðxÞ ¼ 0, then /ð2xÞ ¼ 0. Notice supp/ðxÞ ¼ G, so 12 G G. Multiplying both side of 12 G G with 2, we have (a)Sin (4.1). For j2Z d V j ¼ Rd , by [12, Proposition 2.3.4], we get (b) in (4.1). Since that fT n / : n 2 Z d g is the normalized tight frame of V 0 , by [13, Theorem 2.2.7], we derive (c) in (4.1). Similar to the argument of [14, Theorem 2.1], we have (d) in (4.1). Necessity is proved. h Remark 1. If sðGÞ ¼ T d , we have sðGÞ ¼ T d () G þ 2pZ d ’ Rd . Then [14, Theorem 2.1] is a special case of Theorem 4.1. Lemma 4.5. If G is a bounded closed set in Rd such that G 2G, then the following statements are equivalent: 1 1 ð1Þ G’ G þ 2kp \ G ðk 2 Z d Þ: 2 2 1 1 ð2Þ Gn G \ G þ 2pk ’ ;ðk 2 Z d Þ: 2 2
ð4:14Þ ð4:15Þ
d Proof. ð4:14Þ ) ð4:15Þ. Since 12 G ’ 12 G þ 2kp that (4.15) holds. ) ð4:14Þ. \ 1 Gðk 2 1 Z Þ, it iseasy to know ð4:15Þ 1 If (4.15) holding, then G \ G þ 2pk n G \ G þ 2pk ’ ;ðk 2 Z d Þ. So we get G \ 12 G þ 2pk ’ 2 2 2 1 1 G \ 2 G þ 2pk ðk 2 Z d Þ. Therefore by the formula 12 G \ 12 G þ 2pk ¼ 12 Gðk 2 Z d Þ, we conclude that (4.14) 2 holds. h According to Theorem 4.1 and Lemma 4.5, we deduce easily: ^ ¼ G if and Corollary 4.6. Let G be a bounded closed set in Rd . Then there exists a frame scaling function / with supp/ only if \ [ 1 1 ð1Þ G 2G; ð2Þ 2m G ’ Rd ; ð3Þ sðGÞ T d ; ð4Þ ð4:16Þ G’ G þ 2pk Gðk 2 Z d Þ 2 2 m2Z hold. For a frame scaling set G, if we consider some additional conditions, we can get a more stronger result than Theorem 4.1. ^ ¼ G. If the Theorem 4.7. Let G be a bounded closed set and fT k / : k 2 Z d g be the normalized tight frame of V 0 with supp/ following statements R 2 ^ dx ¼ lðBÞ; (1) 8B G; B /ðxÞ (2) (3)
^ lÞ \ supp/ð ^ kÞ ¼ ;ðl; k 2 Z d ; l–kÞ; supp/ð S d ^ n2Z d supp/ð nÞ ¼ R
hold, we have ðaÞ G 2G;
ðbÞ
[ m2Z
m
d
2 G’R ;
ðcÞ G ¼ Gðs; 1Þ;
ðdÞ
\ 1 1 Gn G G þ 2kp ’ ; ðk 2 Z d Þ: 2 2
ð4:17Þ
Proof. Since (a), (b), (d) are similar to the argument of Theorem 4.1, we need only to prove (c). Suppose lðGðs; m0 Þ > 0Þ for some m0 > 1. Since G is bounded, by [15, Lemma 1], we can choose many finitely distinct points c0 ; c1 ; . . . ; ch 0 2 fcc gh0 and define a point set C ¼ fc0 ; c1 ; . . . ; ch g Z d such that Gc ¼ G \ 2pc. Let ^ c ¼ /ðsÞv ^ /ðsÞ Gc . So we have XX ^ c ; ð2pÞd2 eihl;si ið2pÞd2 eihl;si : ^ h/ðsÞ ð4:18Þ /ðsÞ ¼ c2C l2Z d
hk2 –0; k^ gk2 –0. Define g; h 2 L2 ðRd Þ with g^ ¼ vGð1Þ ðs;m0 Þ ; ^h ¼ vGð2Þ ðs;m0 Þ . So k^
Z. Liu et al. / Chaos, Solitons and Fractals 40 (2009) 2483–2490
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By [15, Lemma 1], and the condition (3) in Theorem 4.7, there exist V 1 ¼ ft1 ; t2 ; . . . ; tm g Z d ; such that Gð1Þ ðs; m0 Þ ¼
[
V 2 ¼ ft1 ; t2 ; . . . ; tn g Z d
^ mÞ \ Gð1Þ ðs; m0 Þ; supp/ð
ð4:19Þ [
Gð2Þ ðs; m0 Þ ¼
m2V l
^ nÞ \ Gð2Þ ðs; m0 Þ: supp/ð
ð4:20Þ
n2V 2
Notice the definition of functions g; h, by the conditions (1), (2) in Theorem 4.7, we have 2 X X X Z 2 2 ^ ^ /ðx mÞ dx ¼ lðGð1Þ ðs; m0 ÞÞ: ¼ k^ jhg; /k ij ¼ jh^ g; /k ij ¼ g k2 : ð1Þ ^ supp/ðmÞ\G ðs;m0 Þ d d m2V k2Z
k2Z
1
Similar to the argument of above formula, we have
P
k2Z d jhh; /k ij
2
¼ k^ hk2 .
2
Let f ¼ g h, it is easy to know that g; h; f 2 V 0 and kf^ k –0. Since the system fT k / : k 2 Z d g is a normalized tight frame of V 0 , by Lemma 3.1, similar to the argument of Theorem 3.2, we have kf^ k2 ¼ 0. It contradicts kf^ k2 –0. Then G ¼ Gðs; 1Þ, and we obtain the desired result. h In the following, we give several examples. 2 2 and G ¼ E1 [ E2 , where E1 ¼ S 14 p; 14 p ; E2 ¼ 38 p; 12 p . Since Example 4.8. Let R2 1 1G 2 2 2 m 2 3 1 1 2G ¼ 2E [ 2E ¼ p; p [ p; p , then G p; p 2G. Notice 2 E ¼ R and E G, we have 1 1 m 2 2 4 2 2 S m 1 2 2 d 1 ’ ;. Hence, we have proved that G satisfies 2 G ¼ R . By the definition of G, we can get easily G \ G þ 2pZ 0 m 2 ^ ¼ G. Eq. (4.1) in Theorem 4.1 and there exists a frame scaling function / with supp/ We define / as follows: 1 ðx 2 GÞ ^ /ðxÞ ¼ : ð4:21Þ 0 ðx R GÞ Next we will testify that / is a frame scaling function. Suppose ( 1 x 2 12 G þ 2pk ðk 2 Z d Þ: H ðxÞ ¼ 0 x R 12 G þ 2pk
ð4:22Þ
^ ^ ^ If x 2 12 G, then /ð2xÞ ¼ 1. By (4.22), we have /ð2xÞ ¼ /ðxÞH ðxÞ x 2 12 G . For x 2 12 G þ 2pkðk–0Þ, we get x R G ^ ^ ^ ^ ¼ /ðxÞH ðxÞ is defined on 12 G þ 2pZ d . and 2x R G. Hence /ð2xÞ ¼ /ðxÞ ¼ 0 holds on 12 G þ 2pkðk–0Þ, then /ð2xÞ 1 d d d Let N ¼ T n G. By the definition of G, we have R ¼ 2 G þ 2pZ [ G0 þ 2pZ d [ ðN þ 2pZ d Þ. If ^ ¼ H ðxÞ ¼ 0. So the formula x 2 G0 þ 2pZ d [ ðN þ 2pZ d Þ, again by the definition of G and (4.22), /ð2xÞ d d ^ ^ ^ ^ ¼ /ðxÞH ðxÞ holds on Rd . On /ð2xÞ ¼ /ðxÞH ðxÞ holds on G0 þ 2pZ [ ðN þ 2pZ Þ. From above results, /ð2xÞ 1 1 d P 2 ^ þ 2pkÞj ¼ 1 for a:e:x 2 T d n N . So, by Lemma the other hand, noticing (4.21) and G 2 p; 2 p , we have k2Z d j/ðx 4.4, the function / defined by (4.21) is a frame scaling function. Example 4.9. Let G R2 and G ¼ E1 [ E2 [ E3 ; where E1 ¼ ½p; p2 ; E1 ¼
2 3 7 1 1 p; p p; p ; E3 ¼ 2E2 : 2 4 8 8
ð4:23Þ
^ ¼ G as follows It is easy to check directly that G satisfies Corollary 4.6. We can construct the scaling function / with / 8 > < 1; x 2 G n ðB [ 2BÞ; ^ ð4:24Þ x 2 B [ 2B; /ðxÞ ¼ p1ffiffi2 ; > : 0; x R G; where B ¼ E2 [ ðE2 2pe1 Þ; e1 ¼ ð1; 0Þ. The proof is trivial. 2 2 Example 4.10. Let G R2 ; G ¼ 12 ; 12 and 2G ¼ ½1; 1 . 1 ðx 2 GÞ ^ /ðxÞ ¼ ; 0 ðx R GÞ
ð4:25Þ
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Similar to the argument of Example 4.8, we can prove that the function defined by (4.25) is a frame scaling function. Furthermore, it is easy to check that the function / satisfies the conditions (1), (2) and (3) in Theorem 4.7. So, the set G satisfies Eq. (4.17).
5. Conclusion In this paper, we discuss firstly the property of the frame wavelet sets. Secondly, we present a necessary and sufficient condition for a set to be a frame scaling function set. Based on the relations of four point sets G; sðGÞ; 12 G and G n 12 G, we reveal an essential difference of scaling functions between FMRA and MRA. By discussing the frequency domain of frame scaling function, the frame scaling functions are well understood. Thirdly, in some conditions, we give a property of frame scaling function sets. In each section, some concrete examples are constructed to prove our theory.
References [1] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19(1):209–36. [2] El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64. [3] El Naschie MS. Hilbert space, the number of Higgs particles and the quantum two-slit experiment. Chaos, Solitons & Fractals 2006;27(1):9–13. [4] El Naschie MS. Hilbert Fock and Cantorian spaces in the quantum two-slit Gedanken experiment. Chaos, Solitons & Fractals 2006;27(1):39–42. [5] Mallat S. Multiresolution approximations and wavelet orthonormal basis of L2 ðRÞ. Trans Am Math Soc 1989;315(1):69–87. [6] Meyer Y. Wavelets and Operators. Cambridge, UK: Cambridge University Press; 1992. [7] Daubechies I. Ten Lectures on Wavelets. In: CBMS-NSFReg Conf Series Appl Math. Philadelphia: SIAM; 1992. [8] Benedetto JJ, Li S. The theory of multiresolution analysis frames and applications to filter banks. Appl Comput Harmon Anal 1998;5(4):398–427. [9] Dai X, Larson D. Wandering Vectors for Unitary Systems and Orthogonal Wavelets. Memoirs of AMS 1998;134(640). [10] Speegle D. The s-elementary wavelets are path-connected. Proc Am Math Soc 1999;127(1):223–33. [11] Dai X, Larson D, Speegle D. Wavelet sets in Rn . J Fourier Anal Appl 1997;3(4):451–6. [12] Long RL. Multidimensional Wavelet Analysis. Beijing: International Publishing Co; 1995. [13] Ron A, Shen Z. Frames and stable bases for shift-invariant subspaces of L2 ðRd Þ. Can J Math 1995;47(5):1051–94. [14] Zhang ZS. Supports of Fourier transform of scaling functions. Appl Comput Harmon Anal 2007;22(2):141–56. [15] Dai X, Diao Q, Gu Y, Han D. Frame wavelet sets in Rd . J Comput Appl Math 2003;155(1):69–82.
Chaos, Solitons and Fractals 40 (2009) 2483–2490 www.elsevier.com/locate/chaos
Frame scaling function sets and frame wavelet sets in Rd Zhanwei Liu a,*, Guoen Hu a,1, Guochang Wu b a
Department of Applied Mathematics, University of Information Engineering, Zhengzhou, Henan 450002, China b School of Science, Xi’an Jiaotong University, Xi’an, Shanxi 710049, China Accepted 29 October 2007
Abstract In this paper, we classify frame wavelet sets and frame scaling function sets in higher dimensions. Firstly, we obtain a necessary condition for a set to be the frame wavelet sets. Then, we present a necessary and sufficient condition for a set to be a frame scaling function set. We give a property of frame scaling function sets, too. Some corresponding examples are given to prove our theory in each section. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Because nature is not continuous, not periodic, but self-similar, Mohamed El Naschie introduced the E-infinity ðeð1Þ Þ view to describe and understand it. In [1–4], a mathematical formulation was introduced to describe phenomena that is resolution dependent by Mohamed El Naschie with eð1Þ . In his view, space-time is an infinite dimensional fractal that happens to have D = 4 as the expectation value for topological dimension [1–3]. In our low energy resolution, the topological value 3 + 1 means that the world appears to us if it were four-dimensional. The Fouriers transform is a mathematical tool to consider the motion either in the frequency domain or in the time domain. But it cannot takes account into the resolutions and the frequencies simultaneously. Fortunately, the wavelet transform can make up for the disadvantage, which permits a multiresolution analysis of data with different behavior on different scales. Today, Orthonormal bases of wavelets are an ubiquitous and eminently powerful tool that pervades all areas of mathematics. A multiresolution analysis (MRA) was introduced by Mallat [5] and Meyer [6] primarily as a tool to construct and analyze the orthonormal wavelets. When dealing with over complete spanning systems one is naturally lead to the concept of frame [7]. The frame theory plays an important role in the modern time-frequency analysis. It has been developed very fast over the last 10 years, especially in the context of wavelets. The generalization of MRA, a frame multiresolution analysis (FMRA), was considered and applied in the analysis of narrow band signals with more freedom in the constructions of wavelets with fast iterative structures by Benedetto and Li [8]. ffi v for some Lebesgue measurable set B of finite measure, the In the special case that the Fourier transform of w is p1ffiffiffi 2p E study of the wavelet is reduced to the study of the set B, which is simpler in general. If we could characterize the set B that defines a wavelet in such a manner, then we would be able to find ways to construct wavelets and have a better *
1
Corresponding author. E-mail address: [email protected] (Z. Liu). Research was supported by the NSF of China (Grant No. 10671210).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.10.042
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understanding of wavelets in general. In [9], the authors introduced the notion of wavelet set which turned out to be one of the building blocks of their approach to wavelet analysis from an operator theory point of view. From then on, many results are found [10,11]. For example, the important result of the existence of wavelets for unitary systems having an expansive dilation matrix was based on the existence of wavelet sets. In [11], Dai and Larson discussed the characterization of wavelet sets in Rn . In our paper, we release their conditions to the case of the band-limited frame wavelet, and obtain a general result. In [14], Zhang characterized the support G of the Fourier transform of the band-limited scaling function and give an approach to the construction of the scaling function. The second aim of the present article is to generalize their results to the case of frame scaling function associated to FMRA and present a necessary and sufficient condition for a set G to be a frame scaling function set. Based on the relations of four point set G; sðGÞ; 12 G and G n 12 G, we reveal an essential difference of scaling functions corresponding to MRA and FMRA. The paper is organized as follows: Section 2 contains some necessary notations and known results. In Section 3, we discuss the property of the frame wavelet sets. In Section 4, we present some necessary and sufficient conditions for a set G to be a frame scaling function set associated to FMRA. In some conditions, we give a property of frame scaling function sets, too. Some corresponding examples are given in each section.
2. Preliminary A collection of elements f/j : j 2 J g in a Hilbert space H is called a frame if there exist constants a and b, 0 < a 6 b < 1, such that X akf k2 6 jhf ; /j ij2 6 bkf k2 ; 8f 2 H : ð2:1Þ j2J
Let a0 be the supremum of all such numbers a and b0 be the infimum of all such numbers b, then a0 and b0 are called the frame bounds of the frame f/j : j 2 J g. When a0 ¼ b0 we say that the frame is tight. When a0 ¼ b0 ¼ 1 we say the frame is a normalized tight frame. Let A be a d d real invertible matrix. It induces a unitary operator DA acting on L2 ðRd Þ defined by 1
ðDA f ÞðtÞ ¼ jdetAj2 f ðAtÞ;
8f 2 L2 ðRd Þ;
t 2 Rd :
ð2:2Þ
The matrix A is called expansive if all its eigenvalues have modulus greater than one. The operator DA corresponding to a real expansive matrix A is called an A-dilation operator. A vector s in Rd induces a unitary translation operator T s defined by ðT s f ÞðtÞ ¼ f ðt sÞ;
8f 2 L2 ðRd Þ;
t 2 Rd :
ð2:3Þ d
In this article, we will only deal with translation operators T l with l 2 Z . If f 2 L2 ðRd Þ \ L1 ðRd Þ, then we define its Fourier transform: Z 1 ei2phs;ti f ðtÞdt; ðf^ ÞðsÞ ¼ d d 2 R ð2pÞ
ð2:4Þ
where hs; ti denotes the standard inner product in Rd . Let Ed denote the set of all expanding matrices A. Now we fix an arbitrary matrix A 2 Ed . For a function w 2 L2 ðRd Þ, we will consider the affine system W defined by n o j ð2:5Þ W ¼ wj;k ðxÞjwj;k ðxÞ ¼ 22 wðAj x kÞ; j 2 Z; k 2 Z d : In 1988, Mallat introduced a concept of multiresolution analysis (MRA) which is a fundament concept in wavelet analysis. Definition 2.1. Let fV m gm2Z be a sequence of closed subspaces of L2 ðRd Þ satisfying: (1) (2) (3) (4) (5)
V Sm V mþ1 ; 2 d Tm2Z V m ¼ L ðR Þ; m2Z V m ¼ f0g; f 2 V m () f ð2Þ 2 V mþ1 ; m 2 Z; There exists a function / 2 V 0 such that fT k / : k 2 Zg is an orthonormal basis of V 0 .
Then fV j gj2Z is called an MRA and the function / from (5) is called a scaling function.
Z. Liu et al. / Chaos, Solitons and Fractals 40 (2009) 2483–2490
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Notice that fV m : m 2 Zg is said to be an FMRA if condition (5) is replaced by the condition that there exists a / 2 V 0 such that fT k / : k 2 Zg is a normalized tight frame of V 0 . The function / is called a frame scaling function. According to [7], there exists a function w such that the system W defined by (2.5) is a orthogonal base in L2 ðRd Þ. We call w a wavelet. Of course, we can say that w is a normalized tight frame wavelet if the system (2.5) is a normalized tight frame for L2 ðRd Þ. Definition 2.2. For a normalized tight frame wavelet w, let ^ B :¼ closfx 2 Rd : wðxÞ–0g;
ð2:6Þ
^ then B is called the frame wavelet set and write B ¼ suppw. ^ is bounded, then w is called a band-limited frame wavelet. If suppw ^ In the same way, we can define a frame scaling function set G and write G ¼ supp/. Let G be a Lebesgue measurable set of finite measure. For any l 2 Z d , let I l denote the d-cube ½0; 1Þd þ l. For any subset G of Rd , define: [ ðG \ 2pI l 2plÞ: ð2:7Þ sðGÞ ¼ l2Z d
If the above is a disjoint union, we say that G is translation equivalent to sðGÞ, which is a subset of 2pI 0 , where I 0 is the unit d-cube ½0; 1Þd þ l. If G and F are translation equivalent to the same subset in 2pI 0 , then we say G and F are trans: lation equivalent. This defines an equivalent relation and is denoted by ¼. Let lðÞ be the Lebesgue measure. It is clear : : that lðGÞ P lðsðGÞÞ. The equality holds if and only if G¼sðGÞ. If G¼F , then lðGÞ ¼ lðF Þ. Two points x; y 2 G are said to be translation equivalent if x y ¼ 2pl for some l 2 Z d . The translation redundancy index of a point x in G is the number of elements in its equivalent class. We write Gðs; kÞ for the set of all points in G with translation redundancy index k. In general, Gðs; kÞ could be an empty set, proper subset of G, or the set G itself. For k–m; Gðs; kÞ \ Gðs; mÞ ¼ ;, so ! [ [ G ¼ Gðs; 1Þ Gðs; nÞ : ð2:8Þ n2N
Let E be a point set in Rd . Throughout this article, we will use vE to denote the characteristic function of E.
3. Frame wavelet sets In [11], Dai and Larson discussed some characterizations of the wavelet sets. In this section, we will generalize their results to the band-limited frame wavelet, and obtain an important property about frame wavelets sets. At first, we give a Lemma. Lemma 3.1. Let M, N be Lebesgue measurable sets of finite measure in Rd such that sðMÞ ¼ sðN Þ ½0; 2pÞd . If M ¼ Mðs; 1Þ; N ¼ N ðs; 1Þ, then for any 2p periodical function f 2 L2 ðRd Þ, we have d
d
hf ðsÞvM ; ð2pÞ 2 eihl;si i ¼ hf ðsÞvN ; ð2pÞ 2 eihl;si i:
ð3:1Þ
Proof. Since sðMÞ ¼ sðN Þ ½0; 2pÞd , by definition of sðÞ; M ¼ Mðs; 1Þ and N ¼d N ðs; 1Þ, there must exist r; q 2 Z n such that M ¼ sðMðs; 1ÞÞ þ 2pr and N ¼ sðN ðs; 1ÞÞ þ 2pq. Because the system fð2pÞ 2 eihl;si gl2Z d is an orthonormal basis for L2 ½0; 2pÞd , we have Z Z d d d hf ðsÞvM ; ð2pÞ 2 eihl;si i ¼ f ðsÞð2pÞ 2 eihl;si ds ¼ f ðs0 þ 2prÞð2pÞ 2 eihl;s0 þ2pri dðs0 þ 2prÞ ¼
Z
sðMðs;1ÞÞ
M 0
d 2
ihl;s0 i
f ðs Þð2pÞ e sðN Þ d
h
ds ¼
Z
d
f ðs0 þ 2pqÞð2pÞ 2 eihl;s0 þ2pqi dðs0 þ 2pqÞ sðN ðs;1ÞÞ
¼ hf ðsÞvN ; ð2pÞ 2 eihl;si i: Then we get the desired result.
0
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Theorem 3.2. Let B be a Lebesgue measurable set with finite measure in Rd . If B is a frame wavelet set, then B ¼ Bðs; 1Þ. Proof. Assume that set B is a frame wavelet set and lðBðs; m0 ÞÞ > 0 for some m0 > 1. By Lemma 1 [15], we can define gk–0; k^hk–0. Let f ¼ g h, then kf^ k2 –0. g^ ¼ vBð1Þ ðs;m0 Þ and ^h ¼ vBð2Þ ðs;m0 Þ . So k^ Again by Lemma 3.1, we have d
d
h^ gðsÞ; ð2pÞ 2 eihl;si i ¼ h^hðsÞ; ð2pÞ 2 eihl;si i:
ð3:2Þ
^ j;k . Since B be bounded, then Bj;k is bounded also. We can choose many finitely By Definition 2.2, let Bj;k ¼ suppw and define a point set V ¼ fv0 ; v1 ; . . . ; vt g Z d such that distinct points v0 ; v1 ; . . . ; vt 0 2 fvv gt0 P ^ p p ^ ^ ^ Bj;k ¼ [p2V ðBj;k \ 2ppÞ. Letting wðsÞ j;k ¼ wðsÞj;k vBj;k \2pp , we have wðsÞj;k ¼ p2V wðsÞj;k . Then E X XD ^ ^ p ; ð2pÞd2 eihl;si ð2pÞd2 eihl;si : wðsÞ wðsÞ ð3:3Þ j;k j;k ¼ p2V l2Z d
By (3.2) and (3.3), we obtain XX ^ j;k ij2 jh^ g ^h; w kf^ k2 ¼ j2Z k2Z d
2 X X X X d d ^ p ; ð2pÞd2 eihl;si iðh^ ¼ hwðsÞ gðsÞ; ð2pÞ 2 eihl;si i h^ hðsÞ; ð2pÞ 2 eihl;si iÞ ¼ 0 j;k d p2V d j2Z k2Z
l2Z
It is a contradiction with kf^ j2 –0, hence B ¼ Bðs; 1Þ holds. h Then, we will construct an example to prove our result. Example 3.3. Let B ¼ E1 [ E2 ; E1 ¼ p2 ; p ½0; p; E2 ¼ 5p 2 ; 3p ½0; p. Since B ¼ Bðs; 2Þ, by Theorem 3.2, all the ^ ¼ B are not the frame wavelets. functions w 2 L2 ðRd Þ defined by suppw As a result, we get a method to judge band-limited frame wavelets just by their frame wavelet sets.
4. Frame scaling function sets Zhang in [14] characterized the support G of the Fourier transform of the band-limited scaling function and showed ^ ¼ G if and only if that there exists a scaling function / with supp/ 1 1 m d d d G þ 2pk ’ ;ðk 2 Z d Þ ðaÞ G 2G; ðbÞ [m 2 G ’ R ; ðcÞ G þ 2pZ ’ R ; ðdÞ G n G \ 2 2 hold. But all his results just based on the scaling functions associated to MRA. In this section, we will generalize their results to the case of frame scaling function associated to FMRA and present a necessary and sufficient condition for a set G to be a frame scaling function set. We firstly introduce our work. ^ ¼ G if and only Theorem 4.1. Let G be a bounded closed set in Rd . Then there exists a frame scaling function / with supp/ if \ [ m 1 1 ðaÞ G 2G; ðbÞ 2 G ’ Rd ; ðcÞ sðGÞ T d ; ðdÞ G n G ð4:1Þ G þ 2pk ’ ;ðk 2 Z d Þ 2 2 m2Z hold. In order to prove this theorem, we introduce some notations: for m ¼ 0; 1; . . ., 1 G0 ¼ G n G; 2
Gm ¼ 2m Gm0 ;
Gm ¼ 2m G;
Em ¼ Gm þ 2pZ d ;
Em ¼ Gm þ 2pZ d :
ð4:2Þ
Then, we give several Lemmas. Lemma 4.2. [14, Lemma 2.3]. If G satisfies (a) and (d) in (4.1), then we have ð1Þ ðEm n Gm Þ \ G ’ ;ðm P 0Þ;
ð2Þ G \ ðGk þ 2pkÞ ’ ;ðk–0Þ:
ð4:3Þ
Z. Liu et al. / Chaos, Solitons and Fractals 40 (2009) 2483–2490
Lemma 4.3. Let G satisfy (a) in ðN þ 2pZ d Þ [ ðG0 þ 2pZ d Þ [ 12 G þ 2pZ d ¼ Rd .
(4.2).
If
sðGÞ T d
and
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N ¼ T d n sðGÞ,
then
Proof. Assume ðN þ 2pZ d Þ [ G0 þ 2pZ d [ 12 G þ 2pZ d –Rd and a point x 2 Rd satisfying 1 ð4:4Þ x R ðN þ 2pZ d Þ [ ðG0 þ 2pZ d Þ [ G þ 2pZ d : 2 S By k2Z d 2pk ¼ Rd , we can find two points y 2 ½0; 2pd and s 2 Z d such that x ¼ y þ 2ps. Since N [ sðGÞ ¼ T , then y 2 N or y 2 sðGÞ. If y 2 N, we have x ¼ y þ 2ps 2 N þ 2pZ d . If y 2 sðGÞ, by the definition of sðGÞ, there must exist two points z 2 G; l 2 Z d satisfying y ¼ z 2pl. Notice G ¼ G0 [ 12 G, so z 2 G0 or z 2 12 G. If z 2 G0 , then x ¼ y þ 2pk ¼ z 2pl þ 2pk 2 G0 þ 2pZ d . On the other hand, if z 2 12 G, we derive x ¼ y þ 2pk ¼ z 2pl þ 2pk 2 12 G þ 2pZ d . All the results contradict (4.4). Hence, we get the desired result. h Lemma 4.4. Let / 2 L2 ðRd Þ and a closed set G satisfy the following: ^ ¼ G; ð1Þ supp/ ^ ^ ð2Þ /ð2xÞ ¼ H ðxÞ/ðxÞa:e:x 2 Rd ðH 2 L1 ðT d ÞÞ; X ^ þ 2pnÞj2 ¼ 1a:e:x 2 T d n N ; ð3Þ j/ðx
ð4:5Þ ð4:6Þ ð4:7Þ
n2Z d
ð4Þ
[
2m G ’ Rd ;
ð4:8Þ
m2Z
then / is a frame scaling function. Proof. Let V m :¼ spanf/ð2m t nÞ; n 2 Z d gðm 2 ZÞ:
ð4:9Þ
By (4.6), V m V mþ1 . Again by (4.5) and (4.8), we get [ [ [ [ m ^ m Þ ¼ suppf^ ¼ supp/ð2 2 G ¼ Rd : m2Z f 2V m
m2Z
ð4:10Þ
m2Z
Applying a known result [12, Proposition 2.3.4], we obtain that we conclude that / is a frame scaling function. h
S
m2Z V m
¼ L2 ðRd Þ. Again by (4.7) [13, Theorem 2.2.7],
Proof of Theorem 4.1. Sufficiency: Assume that (a), (b), (c), (d) in (4.1) hold. By Lemma 4.4, it is enough that we directly construct / 2 L2 ðRd Þ and H 2 L1 ðT d Þ satisfying (4.5)–(4.8). From the argument of [14, Theorem 2.1], we can define 2p-periodic function H ðxÞ on point set 12 G þ 2pZ d and S m ^ ^ ^ supp/ ¼ G with m2Z 2 G ’ Rd such that the equation /ð2xÞ ¼ H ðxÞ/ðxÞ holds on 12 G þ 2pZ d . ^ þ 2pnÞj2 is a 2p periodical function, we easily conclude by N ¼ T d n sðGÞ that (4.7) ^ ¼ G and P d j/ðx Since supp/ n2Z holds on T d n N . Define H ðxÞ ¼ 0ðx 2 G0 þ 2pZ d [ ðN þ 2pZ d ÞÞ. From Lemma 4.3, H ðxÞ has been well defined on Rd and H 2 L1 ðT d Þ. Again by Lemma 4.3, (d) and (a) in (4.1), we have \ 1 ð1Þ ðN þ 2pZ d Þ \ ðG þ 2pZ d Þ ’ ;; ð2Þ G0 þ 2pZ d ð4:11Þ G þ 2pZ d ’ ;: 2 d For x 2 G0 þ 2pZ d [ ðN þ 2pZ d Þ, we have 2x 2 2G0 þ 4pZ d [ ð2Nþ 4pZ d Þ. By (4.11), then ð2G0 þ 4pZ Þ \ G ’ ; d and ð2N þ 4pZ Þ \ G ’ ;. Hence /ð2xÞ ¼ 0. Again by H ðxÞ ¼ 0ðx 2 G0 þ 2pZ d [ ðN þ 2pZ d ÞÞ, we know that (4.6) holds also on G0 þ 2pZ d [ ðN þ 2pZ d Þ. Combining above results, (4.6) holds on Rd . Up to now, we have constructed / 2 L2 ðRd Þ and H 2 L1 ðT d Þ satisfying (4.5)–(4.8). Sufficiency is proved. Necessity: 1 Suppose that fT n / : n 2 Z d g is the normalized tight frame of V 0 . By Definition 2.1, f22 T n /ð2Þ : n 2 Z d g is the normalized tight frame of V 1 . Since / 2 V 0 V 1 , then X 1 1 h/ðxÞ; 22 T n /ð2xÞi22 T n /ð2xÞ: ð4:12Þ /ðxÞ ¼ n2Z
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Multiplying both side of (4.12) with Fourier transform, there exists m 2 L2 ðT d Þ such that ^ ^ /ð2xÞ ¼ mðxÞ/ðxÞ for a:e:x 2 Rd :
ð4:13Þ
^ ^ ^ By (4.13), if /ðxÞ ¼ 0, then /ð2xÞ ¼ 0. Notice supp/ðxÞ ¼ G, so 12 G G. Multiplying both side of 12 G G with 2, we have (a)Sin (4.1). For j2Z d V j ¼ Rd , by [12, Proposition 2.3.4], we get (b) in (4.1). Since that fT n / : n 2 Z d g is the normalized tight frame of V 0 , by [13, Theorem 2.2.7], we derive (c) in (4.1). Similar to the argument of [14, Theorem 2.1], we have (d) in (4.1). Necessity is proved. h Remark 1. If sðGÞ ¼ T d , we have sðGÞ ¼ T d () G þ 2pZ d ’ Rd . Then [14, Theorem 2.1] is a special case of Theorem 4.1. Lemma 4.5. If G is a bounded closed set in Rd such that G 2G, then the following statements are equivalent: 1 1 ð1Þ G’ G þ 2kp \ G ðk 2 Z d Þ: 2 2 1 1 ð2Þ Gn G \ G þ 2pk ’ ;ðk 2 Z d Þ: 2 2
ð4:14Þ ð4:15Þ
d Proof. ð4:14Þ ) ð4:15Þ. Since 12 G ’ 12 G þ 2kp that (4.15) holds. ) ð4:14Þ. \ 1 Gðk 2 1 Z Þ, it iseasy to know ð4:15Þ 1 If (4.15) holding, then G \ G þ 2pk n G \ G þ 2pk ’ ;ðk 2 Z d Þ. So we get G \ 12 G þ 2pk ’ 2 2 2 1 1 G \ 2 G þ 2pk ðk 2 Z d Þ. Therefore by the formula 12 G \ 12 G þ 2pk ¼ 12 Gðk 2 Z d Þ, we conclude that (4.14) 2 holds. h According to Theorem 4.1 and Lemma 4.5, we deduce easily: ^ ¼ G if and Corollary 4.6. Let G be a bounded closed set in Rd . Then there exists a frame scaling function / with supp/ only if \ [ 1 1 ð1Þ G 2G; ð2Þ 2m G ’ Rd ; ð3Þ sðGÞ T d ; ð4Þ ð4:16Þ G’ G þ 2pk Gðk 2 Z d Þ 2 2 m2Z hold. For a frame scaling set G, if we consider some additional conditions, we can get a more stronger result than Theorem 4.1. ^ ¼ G. If the Theorem 4.7. Let G be a bounded closed set and fT k / : k 2 Z d g be the normalized tight frame of V 0 with supp/ following statements R 2 ^ dx ¼ lðBÞ; (1) 8B G; B /ðxÞ (2) (3)
^ lÞ \ supp/ð ^ kÞ ¼ ;ðl; k 2 Z d ; l–kÞ; supp/ð S d ^ n2Z d supp/ð nÞ ¼ R
hold, we have ðaÞ G 2G;
ðbÞ
[ m2Z
m
d
2 G’R ;
ðcÞ G ¼ Gðs; 1Þ;
ðdÞ
\ 1 1 Gn G G þ 2kp ’ ; ðk 2 Z d Þ: 2 2
ð4:17Þ
Proof. Since (a), (b), (d) are similar to the argument of Theorem 4.1, we need only to prove (c). Suppose lðGðs; m0 Þ > 0Þ for some m0 > 1. Since G is bounded, by [15, Lemma 1], we can choose many finitely distinct points c0 ; c1 ; . . . ; ch 0 2 fcc gh0 and define a point set C ¼ fc0 ; c1 ; . . . ; ch g Z d such that Gc ¼ G \ 2pc. Let ^ c ¼ /ðsÞv ^ /ðsÞ Gc . So we have XX ^ c ; ð2pÞd2 eihl;si ið2pÞd2 eihl;si : ^ h/ðsÞ ð4:18Þ /ðsÞ ¼ c2C l2Z d
hk2 –0; k^ gk2 –0. Define g; h 2 L2 ðRd Þ with g^ ¼ vGð1Þ ðs;m0 Þ ; ^h ¼ vGð2Þ ðs;m0 Þ . So k^
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2489
By [15, Lemma 1], and the condition (3) in Theorem 4.7, there exist V 1 ¼ ft1 ; t2 ; . . . ; tm g Z d ; such that Gð1Þ ðs; m0 Þ ¼
[
V 2 ¼ ft1 ; t2 ; . . . ; tn g Z d
^ mÞ \ Gð1Þ ðs; m0 Þ; supp/ð
ð4:19Þ [
Gð2Þ ðs; m0 Þ ¼
m2V l
^ nÞ \ Gð2Þ ðs; m0 Þ: supp/ð
ð4:20Þ
n2V 2
Notice the definition of functions g; h, by the conditions (1), (2) in Theorem 4.7, we have 2 X X X Z 2 2 ^ ^ /ðx mÞ dx ¼ lðGð1Þ ðs; m0 ÞÞ: ¼ k^ jhg; /k ij ¼ jh^ g; /k ij ¼ g k2 : ð1Þ ^ supp/ðmÞ\G ðs;m0 Þ d d m2V k2Z
k2Z
1
Similar to the argument of above formula, we have
P
k2Z d jhh; /k ij
2
¼ k^ hk2 .
2
Let f ¼ g h, it is easy to know that g; h; f 2 V 0 and kf^ k –0. Since the system fT k / : k 2 Z d g is a normalized tight frame of V 0 , by Lemma 3.1, similar to the argument of Theorem 3.2, we have kf^ k2 ¼ 0. It contradicts kf^ k2 –0. Then G ¼ Gðs; 1Þ, and we obtain the desired result. h In the following, we give several examples. 2 2 and G ¼ E1 [ E2 , where E1 ¼ S 14 p; 14 p ; E2 ¼ 38 p; 12 p . Since Example 4.8. Let R2 1 1G 2 2 2 m 2 3 1 1 2G ¼ 2E [ 2E ¼ p; p [ p; p , then G p; p 2G. Notice 2 E ¼ R and E G, we have 1 1 m 2 2 4 2 2 S m 1 2 2 d 1 ’ ;. Hence, we have proved that G satisfies 2 G ¼ R . By the definition of G, we can get easily G \ G þ 2pZ 0 m 2 ^ ¼ G. Eq. (4.1) in Theorem 4.1 and there exists a frame scaling function / with supp/ We define / as follows: 1 ðx 2 GÞ ^ /ðxÞ ¼ : ð4:21Þ 0 ðx R GÞ Next we will testify that / is a frame scaling function. Suppose ( 1 x 2 12 G þ 2pk ðk 2 Z d Þ: H ðxÞ ¼ 0 x R 12 G þ 2pk
ð4:22Þ
^ ^ ^ If x 2 12 G, then /ð2xÞ ¼ 1. By (4.22), we have /ð2xÞ ¼ /ðxÞH ðxÞ x 2 12 G . For x 2 12 G þ 2pkðk–0Þ, we get x R G ^ ^ ^ ^ ¼ /ðxÞH ðxÞ is defined on 12 G þ 2pZ d . and 2x R G. Hence /ð2xÞ ¼ /ðxÞ ¼ 0 holds on 12 G þ 2pkðk–0Þ, then /ð2xÞ 1 d d d Let N ¼ T n G. By the definition of G, we have R ¼ 2 G þ 2pZ [ G0 þ 2pZ d [ ðN þ 2pZ d Þ. If ^ ¼ H ðxÞ ¼ 0. So the formula x 2 G0 þ 2pZ d [ ðN þ 2pZ d Þ, again by the definition of G and (4.22), /ð2xÞ d d ^ ^ ^ ^ ¼ /ðxÞH ðxÞ holds on Rd . On /ð2xÞ ¼ /ðxÞH ðxÞ holds on G0 þ 2pZ [ ðN þ 2pZ Þ. From above results, /ð2xÞ 1 1 d P 2 ^ þ 2pkÞj ¼ 1 for a:e:x 2 T d n N . So, by Lemma the other hand, noticing (4.21) and G 2 p; 2 p , we have k2Z d j/ðx 4.4, the function / defined by (4.21) is a frame scaling function. Example 4.9. Let G R2 and G ¼ E1 [ E2 [ E3 ; where E1 ¼ ½p; p2 ; E1 ¼
2 3 7 1 1 p; p p; p ; E3 ¼ 2E2 : 2 4 8 8
ð4:23Þ
^ ¼ G as follows It is easy to check directly that G satisfies Corollary 4.6. We can construct the scaling function / with / 8 > < 1; x 2 G n ðB [ 2BÞ; ^ ð4:24Þ x 2 B [ 2B; /ðxÞ ¼ p1ffiffi2 ; > : 0; x R G; where B ¼ E2 [ ðE2 2pe1 Þ; e1 ¼ ð1; 0Þ. The proof is trivial. 2 2 Example 4.10. Let G R2 ; G ¼ 12 ; 12 and 2G ¼ ½1; 1 . 1 ðx 2 GÞ ^ /ðxÞ ¼ ; 0 ðx R GÞ
ð4:25Þ
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Similar to the argument of Example 4.8, we can prove that the function defined by (4.25) is a frame scaling function. Furthermore, it is easy to check that the function / satisfies the conditions (1), (2) and (3) in Theorem 4.7. So, the set G satisfies Eq. (4.17).
5. Conclusion In this paper, we discuss firstly the property of the frame wavelet sets. Secondly, we present a necessary and sufficient condition for a set to be a frame scaling function set. Based on the relations of four point sets G; sðGÞ; 12 G and G n 12 G, we reveal an essential difference of scaling functions between FMRA and MRA. By discussing the frequency domain of frame scaling function, the frame scaling functions are well understood. Thirdly, in some conditions, we give a property of frame scaling function sets. In each section, some concrete examples are constructed to prove our theory.
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