On Dual Wavelet Tight Frames

Transcription

1 On Dual Wavelet Tight Frames Bin Han Department of Mathematical Sciences University of Alberta Edmonton, AB T6G 2G, Canada Abstract. A characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton s result on wavelet tight frames in L 2 (IR) is generalized to the n-dimensional case. Two ways of constructing certain dual wavelet tight frames in L 2 (IR n ) are suggested. Finally examples of smooth wavelet tight frames in L 2 (IR) and H 2 (IR) are provided. In particular, an example is given to demonstrate that there is a function ψ, whose Fourier transform is positive, compactly supported and infinitely differentiable, which generates a non-mra wavelet tight frame in H 2 (IR). Keywords: Bessel sequence, dual wavelet tight frame, multiresolution analysis, wavelet basis. AMS Subject Classification: 4A30, 4A63, 42C0.

2 . Introduction A dual wavelet tight frame (DWTF) differs from a biorthogonal wavelet basis in that it may be linearly dependent. In many applications such as signal analysis, this freedom of redundancy is useful (cf.[5, 9]), due to its ability to reduce the presence of uncorrelated noise in the signal. This paper is concerned with the characterization and construction of dual wavelet tight frames (DWTFs), in certain subspaces of L 2 := L 2 (IR n ), for any general dilation matrix. In particular, we construct many wavelet tight frames whose Fourier transforms are positive, compactly supported and infinitely differentiable. An example of a non-mra smooth wavelet tight frame in H 2 (IR) is given in this paper. An n n integer matrix M is said to be a dilation matrix if all the eigenvalues, λ i, of M satisfy λ i >. In applications, M is often chosen to be 2I n n. But some other dilation matrices such as ( ) and ( ) are also useful in applications such as image compression (cf.[9]). In this paper, we always let n denote the dimension of the Euclidean space IR n. Without further mention, M denotes an n n dilation matrix with m := detm, and M denotes the transpose of the matrix M. Throughout this paper, we let S denote a measurable subset of IR n such that χ S (x) = χ M S(x) a.e. and S\intS = 0 where ints denotes the interior of S, and S\intS := {x IR n : x S and x ints} and S\intS denotes the Lebesgue measure of S\intS. We consider the following special subspace F L 2 (S) of L 2 (IR n ) defined as and let F L 2 (S) := {f L 2 (IR n ) : supp f S} L 2 00(S) := {f : f L and supp f {ξ S : C ξ C}for some C > }. As an example, F L 2 ([0, )) = H 2 (IR). Note that L 2 00(S) is a dense subset of F L 2 (S). The Fourier transform of f L (IR n ) is given by f(ξ) := f(t)e iξ t dt. () n/2 IR n The Plancherel theorem then provides an extension of this to L 2 (IR n ) such that f, g = f, ĝ and f = f with, and denoting the standard inner product and norm in the Hilbert space L 2 (IR n ), respectively. To state the definition of a DWTF in F L 2 (S), we recall that a family {ψ,, ψ d } in F L 2 (S) is said to generate a wavelet frame in F L 2 (S) if there exist a lower frame bound and an upper frame bound for the family {ψ i;j,γ } i d,j ZZ,γ ZZ n. By this we mean that there exist some positive constants A and B such that d (.) A f 2 f, ψ i;j,γ 2 B f 2, γ ZZ n f F L 2 (S), 2

3 where (.2) ψ i;j,γ (x) := m j/2 ψ i (M j x γ), i d, j ZZ, γ ZZ n. We say that a function ψ L 2 forms a Bessel sequence with Bessel bound B if it satisfies (.3) f, ψ j,γ 2 B f 2, f L 2. j ZZ γ ZZ n where ψ j,γ (x) := m j/2 ψ(m j x γ). Hence, if a family {ψ,, ψ d } generates a wavelet frame in F L 2 (S), then observing that for any f L 2, we have f, ψ i;j,γ = g, ψ i;j,γ where ĝ = χ S ˆf. We see that each function in {ψ,, ψ d } forms a Bessel sequence. Now we give the definition of a DWTF in F L 2 (S). Let Ψ := {ψ,, ψ d } be a family of functions in F L 2 (S) with, Ψ := { ψ,, ψ d } denoting its dual family of functions in F L 2 (S). This notation for the dual family will be used throughout. Here the word dual is used literally and just means that the number of functions in Ψ is equal to the number of functions in Ψ. We say that Ψ with Ψ generates a DWTF in F L 2 (S) if each function in both Ψ and Ψ forms a Bessel sequence and for some non-zero constant B, d (.4) B f, g = f, ψ i;j,γ ψ i;j,γ, g, γ ZZ n f, g F L 2 (S). Moreover, when Ψ is the dual family of itself, we say that Ψ generates a wavelet tight frame in F L 2 (S). If Ψ with Ψ generates a DWTF in F L 2 (S) and they satisfy the following biorthogonal conditions: (.5) ψ i;j,γ, ψ i ;j,γ = δ i,i δ j,j δ γ,γ, i, i d, j, j ZZ, γ, γ ZZ n, we say that Ψ with Ψ generates a biorthogonal wavelet basis in F L 2 (S) and when Ψ is the dual family of itself, we say that Ψ generates a wavelet basis in F L 2 (S). From the above definition of a DWTF, and by using the Cauchy-Schwarz inequality, it is easily seen that if Ψ with Ψ generates a DWTF in F L 2 (S), each of Ψ and Ψ generates a wavelet frame in F L 2 (S). Moreover it gives rise to the series representation in L 2 of any function f F L 2 (S), namely, (.6) d f = B = B d f, ψ i;j,γ ψ i;j,γ γ ZZ n f, ψ i;j,γ ψ i;j,γ. γ ZZ n The above equality means that any signal having finite energy can be decomposed into a sequence of numbers, i.e., { f, ψ i;j,γ : i d, j ZZ, γ ZZ n }, and the original signal can be reconstructed from this sequence. Such an exact reconstruction property is desirable in applications. Even without biorthogonality, a DWTF shares most of the advantages, such as good time and frequency localization, exact reconstruction etc., of a biorthogonal wavelet basis. 3

4 Although some results on DWTFs appeared elsewhere by other authors in various forms, in this paper we present a more complete and general approach to this problem. The following is an outline of the paper. In Section 2, we characterize DWTFs in F L 2 (S) for any dilation matrix M by using two classes of equations which appeared in Lemarié [5] and were known to Meyer, Bonami, Soria and Weiss [3], Chui and Shi [4, 5]. In Section 3, we generalize Lawton s result on tight frames in L 2 (IR) to DWTFs in F L 2 (S) and this extension can be used to construct compactly supported DWTFs. Two ways of constructing certain band-limited DWTFs in L 2 (IR n ) are suggested (a function is said to be band-limited if its Fourier transform is compactly supported). Examples are provided to illustrate the general theory. Finally, in Section 4, for the case n = and M = 2, we can construct special kinds of wavelets and wavelet tight frames in L 2 (IR) and H 2 (IR). We shall discuss how to modify them to obtain smooth band-limited C wavelet tight frames. It is well-known that under certain decay conditions, there is no function ψ in H 2 (IR) such that ψ is continuous and ψ generates a wavelet basis in H 2 (IR) (cf. []). In Han [0] (also cf.[,3,6,7]), it is proved that under certain decay conditions, if ψ is continuous and ψ generates a wavelet basis in H 2 (IR), then ψ can be obtained from a Multiresolution Analysis(MRA). For the concept of a multiresolution analysis, the reader is referred to [7,8,9,2,3,8,2]. In this paper we will give an example to show that there is a function ψ, whose Fourier transform is positive, compactly supported and infinitely differentiable, which generates a non-mra wavelet tight frame in H 2 (IR). The definition of a non-multiresolution Analysis (non-mra) wavelet tight frame will be given in Section Characterization of Dual Wavelet Tight Frames Concerning the characterization and construction of wavelet bases, biorthogonal wavelet bases and wavelet frames, a lot of results have been obtained in the current literature. For example, Daubechies constructed many compactly supported wavelets in L 2 (IR) in [8]. Bonami, Soria and Weiss characterized band-limited wavelets in [3], and many properties of wavelet frames are obtained by Chui and Shi in [4, 5]. In [7], many examples of compactly supported biorthogonal wavelet bases in L 2 (IR) are given. After reading these and other papers, we give a relatively unified approach to DWTFs. In this section, we shall characterize DWTFs and discuss the relation between DWTFs and biorthogonal wavelet bases. We remind the reader again that, throughout this paper, n refers to the dimension of the Euclidean space IR n. M denotes an n n dilation matrix with m := detm, and M, the transpose of the matrix M. S denotes a measurable subset of IR n such that χ S (x) = χ M S(x) a.e. and S\intS = 0. Following Daubechies [9], we say that a function ψ L 2 (IR n ) is admissible provided that ψ(m j ξ) 2 L 2 loc(ir n ) j ZZ where f L 2 loc(ir n ) means that for any compact set K IR n, K f(x) 2 <. In fact, 4

5 when n = and M = 2, the above definition is equivalent to saying that ψ(ξ) 2 dξ = ξ [ 2, ] [,2] ξ ψ(2 j ξ) 2 dξ <. j ZZ Here and throughout this paper, M j means the j th power of M, and M j means the j th power of the inverse matrix of M. A proof similar to that of [Theorem A, 4] gives the following proposition. Proposition 2.. If ψ L 2 (IR n ) forms a Bessel sequence with Bessel bound B, then ψ(m j ξ) 2 B() n and ψ is admissible. j ZZ By using the above proposition, we observe the following result which gives us a necessary condition for a family Ψ with Ψ to generate a DWTF. Later, we shall see that, under some additional decay conditions, this is also a sufficient condition. Theorem 2.2. Suppose a family Ψ := {ψ,, ψ d } with Ψ := { ψ,, ψ d } in F L 2 (S) := {f L 2 (IR n ) : supp f S} generates a DWTF in F L 2 (S) with frame bound B. Then the following equations (2.) and (2.2) d i= j=0 d ψ i (M j ξ) ψ i (M j ξ) = ψ i (M j ξ) ψ i (M j (ξ + γ 0 )) = 0 hold with the series converging absolutely. B () n χ S(ξ) a.e., γ 0 ZZ n \M ZZ n Proof. Let S 0 := ints\{0}. From the assumption S\intS = 0, to prove (2.), it suffices to prove that (2.3) d ψ i (M j ξ) ψ i (M j ξ) = B () n a.e. ξ S 0. By the Parseval equality and the polarization identity, setting T := [0, ) n, we have the following equality (2.4) d i= j Z γ Z n f, ψ i;j,γ ψ i;j,γ, g = d i= j Z where the bracket product is defined as [f, g](ζ) := Thus [ f(m j ), ψ i ](ζ) = γ ZZ n m j T [ f(m j ), ψ i ](ζ)[ ψi, ĝ(m j )](ζ)dζ, f, g L 2 T γ ZZ n f(ζ + γ)g(ζ + γ). For a detailed proof of (2.4), the reader is referred to [4]. f(m j (ζ+γ)) ψ i (ζ+γ) and similarly for [ ψ i, ĝ(m j )](ζ). 5

6 Note that Ψ with Ψ generates a DWTF in F L 2 (S). Hence by definition, Ψ with Ψ satisfies equation (.4). By (2.4), we can rewrite (.4) as (2.5) B f, ĝ = d m j T [ f(m j ), ψ i ](ζ)[ ψ i, ĝ(m j )](ζ)dζ, f, g F L 2 (S). T For any fixed ξ S 0, it is easy to see that for any k ZZ, there is a unique γ k ZZ n such that ξ D k (ξ) := M k (T + γ k ). Since M is an n n dilation matrix, there exists a norm on IR n such that M < where M := sup M x. x = Note that for any ζ D k (ξ) and k < 0, ζ ξ C M k with C = sup x y. x,y T Since ξ 0 and ξ ints, we can choose k ξ < 0 such that D k (ξ) ints, and Then we have ξ ( M ) 2C M k > 0 k k ξ. M j D k (ξ) D k (ξ) = j < 0, k k ξ, j, k ZZ. In fact, for any x, y D k (ξ), note that M < and x ξ + C M k and y ξ C M k. We have for any j < 0 and for any x, y D k (ξ), M j x y y M j x y M x ξ ( M ) 2C M k > 0. Hence we have (2.6) M i D k (ξ) M j D k (ξ) = i j, k k ξ, i, j, k ZZ. For any k k ξ, choose f(ζ) = ĝ(ζ) = D k (ξ) /2 χ Dk (ξ)(ζ). It is clear that f, g F L 2 (S). Thus (2.5) yields B = B f, ĝ = R k + L k where R k and L k are defined as follows (2.7) R k := d m j T j k k M i= T [ f(m j ), ψ i ](ζ)[ ψ i, ĝ(m j )](ζ)dζ, 6

7 and (2.8) L k := d m j T j<k k M i= with the integer k M < 0 depending only on M such that T [ f(m j ), ψ i ](ζ)[ ψ i ( ), ĝ(m j )](ζ)dζ. (2.9) γ M j (, ) n γ ZZ n \{0}, j k M. Now we can calculate and simplify R k. For any j k k M, (2.9) implies that which gives us γ M (k j) (, ) n = M j D k (ξ) M j D k (ξ) γ ZZ n \{0}, (2.0) (M j D k (ξ) + γ) (M j D k (ξ) + γ ) = γ γ, γ, γ ZZ n. Note that f(ζ) = ĝ(ζ) = D k (ξ) /2 χ Dk (ξ)(ζ). Thus for any j k k M, [ f(m j ), ψ i ](ζ)[ ψ i, ĝ(m j )](ζ) = D k (ξ) [ ψ i ψ i, χ Dk (ξ)(m j )](ζ). So (2.) d R k = mj T χ Dk (ξ)(m j ζ) ψ i (ζ) ψ i (ζ)dζ i= j k k M D k (ξ) IR n = T d ψ i (M j ζ) ψ i (M j ζ)dζ. D k (ξ) i= j k M k D k (ξ) In the following, we shall prove that we have (2.2) [ f(m j ), ψ i ](ζ) D k (ξ) /2 D k (ξ) /2 [χ Dk (ξ)(m j ), ψ i 2 ](ζ) 2 lim L k = 0. By the Cauchy-Schwarz inequality, k χ Dk (ξ)(m j (ζ + γ)) ψ i (ζ + γ) γ ZZ n χ Dk (ξ)(m j (ζ + γ)) γ ZZ n Note that by definition, ξ D k (ξ) = M k (T + γ k ) for a unique γ k ZZ n, and χ T (ζ + γ) =. We observe that γ ZZ n 2. (2.3) γ Z n χ Dk (ξ)(m j (ζ + γ)) = γ Z n χ M (k j) T (ζ M k j γ k + γ) { = m k j, j k,, j > k, from which it follows that (2.4) γ ZZ n χ Dk (ξ)(m j (ζ + γ)) m k M m k j = D k(ξ) m km T m j, j < k k M. 7

8 Hence, by the Cauchy-Schwarz inequality again, from (2.2) and (2.4), we derive that L k d m k M i= j<k k T M d ( m k M m k M i= j<k k M d i= j>k M k [χ Dk (ξ)(m j ), ψ i 2 ] 2 (ζ)[χdk (ξ)(m j ), ψ i 2 ] 2 (ζ)dζ χ M j D IR n k (ξ)(ζ) ψ i (ζ) 2 dζ χ M j D IR n k (ξ)(ζ) ψ 2 i (ζ) 2 dζ Let E k := j>k M k M j D k (ξ). (2.6) implies that Observing that L k m k M i= j>k M k ) ( ) 2 χ M j D IR n k (ξ) ψ i (ζ) 2 2 dζ j>k M k χ M j D IR n k (ξ) ψ i (ζ) 2 dζ χ M j D k (ξ)(ζ) = χ Ek (ζ). So d ( ψ ) /2 ( ) /2 i (ζ) 2 dζ ψ i (ζ) 2 dζ. E k E k lim inf ζ =, we obtain k ζ E k lim ψ i (ζ) 2 dζ = lim ψ i (ζ) 2 dζ = 0 k E k k E k which implies lim L k = 0. k By Proposition 2., each function in both Ψ and Ψ is admissible. Hence, d i= j k M k ψ i (M j ζ) ψ i (M j ζ) L 2 loc(ir n ). Note that B = R k + L k. The Lebesgue differentiation Theorem and (2.) yield B = lim k T D k (ξ) D k (ξ) d i= j k M k d = () n ψ i (M j ζ) ψ i (M j ζ) ψ i (M j ζ) ψ i (M j ζ)dζ a.e.. Hence, (2.3) is proved. To see that (2.2) is true, first observe that (.4) also says that (2.5) B χ S f, χ S ĝ = d γ ZZ n f, ψ i;j,γ ψ i;j,γ, g, f, g L 2. As in the proof of (2.), for any fixed ξ IR n \ZZ n and γ 0 ZZ n \M ZZ n, there exist integers k ξ < 0 and k ξ+γ0 < 0 such that (2.6) holds for ξ and ξ + γ 0, respectively. Since ξ IR n \ZZ n, it is easy to see that there is an integer k 0 < min(k ξ, k ξ+γ0, k M ) such that (2.6) D k (ξ) (D k (ξ) + γ) = γ ZZ n \{0}, k k

9 For any k k 0, set f(ζ) = D k (ξ) /2 χ Dk (ξ)(ζ) and ĝ(ζ) = D k (ξ) /2 χ Dk (ξ+γ 0 )(ζ). From (2.4), (2.5) and (2.6), we obtain (2.7) 0 = B χ S f, χs ĝ = d i= j Z m j T [ f(m j ), ψ i ](ζ)[ ψi, ĝ(m j )](ζ)dζ =: R k + L k, T where R k and L k are defined as in (2.7) and (2.8). Note that D k (ξ) = M k (T + γ k ) where γ k ZZ n is the unique multi-integer such that ξ D k (ξ). Since k k 0 < 0, it is easily seen that D k (ξ + γ 0 ) = D k (ξ) + γ 0. Thus by (2.6) and γ 0 ZZ n \M ZZ n, we have for any j > 0, [ f(m j ), ψ i ](ζ)[ ψ i, ĝ(m j )](ζ) = Thus R k = d i= k k M j 0 D k (ξ) [χ D k (ξ)(m j ), ψ i ](ζ) [χ Dk (ξ)+γ 0 (M j ), ψ i ](ζ) = 0. m j T [ f(m j ), ψ i ](ζ)[ ψ i, ĝ(m j )](ζ)dζ. T A similar argument as in the proof of (2.) will give us that for any k k M j 0, Hence [ f(m j ), ψ i ](ζ)[ ψ i, ĝ(m j )](ζ) = (2.8) R k = T D k (ξ) d i= 0 j k M k D k (ξ) [ ψ i ( ) ψ i ( + M j γ 0 ), χ Dk (ξ)(m j )](ζ). D k (ξ) ψ i (M j ζ) ψ i (M j (ζ + γ 0 ))dζ, Similarly, as in the proof of (2.), lim L k = 0. Hence lim R k = 0 by R k + L k = 0. k k Again by the Lebesgue differentiation theorem, we obtain (2.2) from (2.8) which completes the proof. At first glance, it seems that equations (2.) and (2.2) are too complicated to provide any useful information. We will prove that under some decay condition on Ψ(no decay condition on ˆΨ is required), (2.) and (2.2) are sufficient conditions for Ψ with Ψ to generate a DWTF in F L 2 (S) provided that each function in both Ψ and Ψ forms a Bessel sequence. In fact, by using (2.) and (2.2) directly, there is a relatively easy way of constructing certain smooth wavelet tight frames in F L 2 (S). Examples are provided in Sections 3 and 4. The particular function space of interest to us is W L 2 defined by (2.9) W := {ψ L 2 : δ ψ( ) L and ψ L 2 δ 2 for some δ > 0, δ 2 > 0}. For example, if for some positive constants C and ε, ψ(ξ) C( + ξ ) n/2 ε a.e. ξ IR n, where denotes the Euclidean norm in IR n. Then ψ W. However, there are examples, such as Haar wavelets in several dimensions, that satisfy (2.9) while violating the simpler, yet stronger decay assumption. 9

10 The decay condition in (2.9) will be used in the following technical Lemma 2.3. We don t know whether the following Lemma is still true without requiring Ψ W. Lemma 2.3. Let Ψ := {ψ,, ψ d } and Ψ := { ψ,, ψ d } be two families of functions in F L 2 (S) such that Ψ W. Suppose that each function in both Ψ and Ψ is admissible and [ ψ i, ψ i ] L for i d. If (2.) and (2.2) hold, then d (2.20) B f, g = f, ψ i;j,γ ψ i;j,γ, g, γ ZZ n f, g L 2 00(S). where L 2 00(S) := {f : f L and supp f {ξ S : C ξ C}for some C > } as defined before. Proof. To see that (2.20) is true for any f, g L 2 00(S), by (2.4), we can rewrite the right side of (2.20) as d d = = T f, ψ i;j,γ ψ i;j,γ, g γ ZZ n m j T [ f(m j ), ψ i ](ξ)[ ψ i, ĝ(m j )](ξ)dξ d IR n T γ ZZ n ψ i (M j ξ) ψ i (M j ξ + γ)ĝ(ξ) f(ξ + M j γ)dξ, where T := [0, ) n as defined in the proof of Theorem 2.2. Let h(ξ) := d γ ZZ n ψ i (M j ξ) ψ i (M j ξ + γ)ĝ(ξ) f(ξ + M j γ). Suppose h L. Then we can rearrange the order of terms in the above sum to get d i= j Z γ Z n f, ψ i;j,γ ψ i;j,γ, g = () n IR n f(ξ)ĝ(ξ) +() n IR n ĝ(ξ) j 0 Z d ψ i (M j ξ) ψ i (M j ξ)dξ i= j Z γ 0 Z n \M Z n f(ξ + M j0 γ 0) j=0 i= d ψ i (M j (M j0 ξ)) ψi (M j (M j0 ξ + γ0))dξ. Thus by (2.) and (2.2), it is easy to see that (2.20) holds. Hence, to complete the proof, it suffices to prove h L. Since f, g L 2 00(S), then there exists j 0 ZZ such that ĝ(ξ) f(ξ + M j γ) = 0 a.e. γ ZZ n \{0}, j j 0. Hence we can write h(ξ)dξ = I + I 2 where IR n d I := ψ i (M j ξ) ψ i (M j ξ) f(ξ)ĝ(ξ) dξ, IR n 0

11 and I 2 := d IR n i= j>j 0 γ ZZ n \{0} ψ i (M j ξ) ψ i (M j ξ + γ)ĝ(ξ) f(ξ + M j γ) dξ. By the assumption, setting E := suppĝ, we have I f d L ĝ L ψ i (M j ξ) ψ i (M j ξ) dξ i= E j ZZ f d L ĝ L ψ i (M j ξ) 2 dξ /2 ψ /2 i (M j ξ) 2 dξ i= E j ZZ E j ZZ <. On the other hand, the estimation of I 2 is much more complicated. By g L 2 00(S), we have I 2 = Note that d i= d j>j 0 ĝ L i= d i= γ ZZ n j>j 0 ĝ L m j j>j 0 ĝ L m j E γ ZZ n M j E ψ i (M j ξ) ψ i (M j ξ + γ) f(ξ + M j γ) dξ M j E ψ i (ξ) ψ i (ξ + γ) f(m j (ξ + γ)) dξ ψ i (ξ) [ ψ i, f(m j ) ](ξ)dξ. [ ψ i, f(m j ) ](ξ) [ f(m j ), f(m j )](ξ) /2 [ ψ i, ψ i ](ξ) /2. From the above inequalities, we have d I 2 ĝ L m j i= j>j 0 d C C d C 2 d m j E m j i= j>j M j E 0 ( m j i= j>j M j E 0 ( m j i= j>j M j E 0 ψ i (ξ) [ f(m j ), f(m j )] 2 (ξ)[ ψ i, ψ i ] 2 (ξ)dξ ψ i (ξ) δ δ 2 2 [ f(m j ), f(m j )] 2 (ξ)dξ ψ ) ( ) i (ξ) 2 δ 2 2 dξ ψ i (ξ) δ 2 2 [ f(m j ), f(m j )](ξ)dξ M je ψ i (ξ) δ 2 [ ) 2 f(m j ), f(m j )](ξ)dξ, where C = ĝ L sup [ ψ i, ψ ( ) /2 i ] /2 L and C 2 = C sup ψ i (ξ) 2 δ 2 dξ. i d i d IR n Note that g L 2 00(S). Thus E = suppĝ {ξ : C ξ C} for some constant C >. Hence M j ξ C M j ξ E and γ ZZ n χ M j E(ξ + γ) C 3 m j ξ IR n, j > j 0

12 where C 3 is a positive constant depending only on E and j 0. Hence ψ i (ξ) δ 2 [ f(m j ), f(m j )](ξ)dξ M j E = M j E ( ξ δ ψ i (ξ) ) δ 2 ξ δ δ 2 [ f(m j ), f(m j )](ξ)dξ ψ δ 2 δ i ( ) R δ δ 2 M j δ δ 2 [ f(m j ), f(m j )](ξ)dξ L M j E C 4 M j δ δ 2 [ f(m j ), f(m j )](ξ) χ M j E(ξ + γ)dξ T γ ZZ n C 3 C 4 m j M j δ δ 2 [ f(m j ), f(m j )](ξ)dξ = C 3 C 4 m j M j δ δ 2 IR n f(m j ξ) 2 dξ = C 3 C 4 f 2 m 2j M j δ δ 2 with C 4 = C δ δ 2 max ψ δ 2 i d δ i ( ). Let C 5 = C 3 C 4 f 2. We have L T d d I 2 C 2 m j C /2 5 m j M j δ δ 2 /2 C 2 C /2 5 M j δ δ 2/2 <, i= j>j 0 i= j>j 0 by the spectral radius ρ(m ) <. Hence h L which completes the proof. The above Lemma 2.3 states a weak version of the converse theorem of Theorem 2.2. To achieve our main result, we have to prove that (2.20) holds for any f and g in F L 2 (S). To do this, the following lemma related to Bessel sequences is needed. Lemma 2.4. If ψ L 2 (IR n ) forms a Bessel sequence with Bessel bound B. Then [ ψ, ψ] B/() n. Proof. For any finitely supported sequence {a γ } γ ZZ n l 2 (ZZ n ), we have a γ ψ 0,γ = sup a γ ψ 0,γ, g sup {a γ } l 2 { ψ 0,γ, g } l 2 B {a γ } l 2. γ ZZ n g γ ZZ n g Let ã(z) := a γ z γ, z {(z,, z n ) : z = z 2 = = z n = }. We have γ ZZ n ã(e iξ ) 2 [ ψ, ψ](ξ)dξ = a γ ψ 0,γ B ã(e iξ ) 2 dξ, [0,) n γ ZZ n [0,) n from which it follows that [ ψ, ψ] B/() n. Now we are ready to state and prove our main result in this section. The following theorem gives a characterization of DWTFs by using equations (2.) and (2.2). Theorem 2.5. Let Ψ := {ψ,, ψ d } and Ψ := { ψ,, ψ d } be two families in F L 2 (S). Suppose that Ψ W where W is defined in (2.9). Then Ψ with Ψ generates a DWTF 2

13 in F L 2 (S) with frame bound B 0 if and only if each function in both Ψ and Ψ forms a Bessel sequence and equations (2.) and (2.2) hold. Proof. To see the necessity, by the definition of a DWTF, we have that each function in both Ψ and Ψ forms a Bessel sequence. From Theorem 2.2, we see that (2.) and (2.2) hold. To see the sufficiency, since each function in both Ψ and Ψ forms a Bessel sequence, from Proposition 2. and Lemma 2.4, each function in both Ψ and Ψ is admissible and [ ψ i, ψ i ] L for i d. Hence by Lemma 2.3, (2.20) holds. Note that L 2 00(S) is dense in F L 2 (S). For any f, g F L 2 (S), there exist sequences (f l ) and (g l ) in L 2 00(S) such that lim f l f = lim g l g = 0. Thus l l B f, g d γ ZZ n f, ψ i;j,γ ψ i;j,γ, g B f, g B f l, g l + B f l, g l + + d d d γ ZZ n f l, ψ i;j,γ ψ i;j,γ, g l f l f, ψ i;j,γ ψ i;j,γ, g l γ ZZ n f, ψ i;j,γ ψ i;j,γ, g l g. γ ZZ n Note that the second term in the right side of the above inequality is 0 by f l, g l L 2 00(S) and (2.20). Since each function in both Ψ and Ψ forms a Bessel sequence with the common Bessel bound being C, by the Cauchy-Schwarz inequality, we get d and similarly γ ZZ n f l f, ψ i;j,γ ψ i;j,γ, g l d /2 d f l f, ψ i;j,γ 2 γ ZZ n /2 ψ i;j,γ, g l 2 C f l f g l, γ ZZ n d Since lim l f l f = lim l g l g = 0, we have B f, g d γ ZZ n f, ψ i;j,γ ψ i;j,γ, g l g C f g l g. γ ZZ n f, ψ i;j,γ ψ i;j,γ, g lim B f, g B f l, g l l + lim l C( f l f g l + f g l g ) = 0. Hence, (.4) holds. So Ψ with Ψ generates a DWTF in F L 2 (S). To use the above theorem to check whether a family Ψ with Ψ generates a DWTF, we still have to check whether each function in both Ψ and Ψ forms a Bessel sequence. 3

14 Several sufficient conditions to guarantee that a function ψ forms a Bessel sequence can be found in the current literature. For example, Theorem in [5] states that if ψ satisfies ψ(x) C( + x ) ε x IR for some positive numbers ε and C, and ψ is piecewise Lip α for some 0 < α, then ψ forms a Bessel sequence provided that ψ(x)dx = 0. We cite another result here(cf. [6]) which is close to our decay conditions stated in the definition of W in (2.9). Proposition 2.6. Suppose ψ L 2 (IR n ) satisfies ψ(m j ) δ L (IR n ) and ψ( + j ZZ γ ZZ n γ) 2 δ L (IR n ) for some positive constant δ. Then ψ forms a Bessel sequence. Proof. Let T := [0, ) n. By (2.4) and the Cauchy-Schwarz inequality, we have that for any f L 2 (IR n ), f, ψ j,γ 2 = m j T f(m j (ξ + γ)) ψ(ξ + γ) 2 dξ j Z γ Z n j Z T γ Z n m j T ψ(ξ + γ) 2 δ f(m j (ξ + γ)) 2 ψ(ξ + γ) δ dξ j Z T γ Z n γ Z n ψ( + γ) 2 δ L T m j γ Z n j Z R f(m j ξ) 2 ψ(ξ) δ dξ n = ψ( + γ) 2 δ L T f(ξ) 2 ψ(m j ξ) δ dξ γ Z n j Z IR n T ψ( + γ) 2 δ L ψ(m j ) δ L f 2, γ Z n j Z IR which completes the proof. For example, if there exist some positive constants C, δ, ε such that (2.2) ψ(ξ) C ξ δ and ψ(ξ) C( + ξ ) n/2 ε a.e., then it is easy to check that ψ satisfies the assumption in Proposition 2.6. Hence ψ forms a Bessel sequence and obviously ψ is also in W. In passing we mention that the sufficient conditions of (2.2) are given by Weiss in [2]. The result for wavelet tight frames corresponding to Theorem 2.5 is the following: Theorem 2.7. Suppose Ψ := {ψ,, ψ d } F L 2 (S) W. Then Ψ generates a wavelet tight frame in F L 2 (S) with frame bound B > 0 if and only if, [ ψ i, ψ i ] L for i d, and the following equations (2.22) and d ψ i (M j ξ) 2 = B () n χ S(ξ) a.e., (2.23) d i= j=0 ψ i (M j ξ) ψ i (M j (ξ + γ 0 )) = 0 γ 0 ZZ n \M ZZ n hold with the series converging absolutely. 4

15 Proof. By Theorem 2.5, it suffices to show that if [ ψ i, ψ i ] L for i d and (2.22), (2.23) hold, then each ψ i forms a Bessel sequence. Since L 2 00(S) is dense in F L 2 (S), for any fixed f F L 2 (S), there exists a sequence (f l ) in L 2 00(S) such that lim f l f = 0. l Note that equation (2.22) also says that each ψ i is admissible. Thus by (2.20) in Lemma 2.3, we have that for any N IN, So d i= j <N γ <N f, ψ i;j,γ 2 lim d l i= j <N γ <N f l, ψ i;j,γ 2 lim l B f l = B f 2. (2.24) d γ ZZ n f, ψ i;j,γ 2 B f 2, f F L 2 (S). Note that ψ i;j,γ F L 2 (S). Thus f, ψ i;j,γ = fχ S, ψ i;j,γ for any f L 2 (IR n ). Hence, (2.24) yields d γ ZZ n f, ψ i;j,γ 2 = d γ ZZ n fχ S, ψ i;j,γ 2 B fχ S 2 B f 2 which means that each ψ i forms a Bessel sequence. So we are done. From the results we obtained for a DWTF, it is now very easy to characterize a biorthogonal wavelet basis. In fact, there is a very close relation between them. Corollary 2.8. Let Ψ := {ψ,, ψ d } and Ψ := { ψ,, ψ d } be two families in F L 2 (S). Then Ψ with Ψ generates a biorthogonal wavelet basis in F L 2 (S) if and only if (2.25) [ ψ i, ψ j (M k )])ξ) = [ ψ i, ψ j (M k )](ξ) = () n δ i,jδ 0,k k IN {0}, i, j d, and Ψ with Ψ generates a DWTF in F L 2 (S) with frame bound B =. Proof. By the definition of a biorthogonal wavelet basis in F L 2 (S), it is straightforward to see that the claim is true since (.5) is equivalent to (2.25). Corresponding to Corollary 2.8, the characterization of a wavelet basis is as follows: Corollary 2.9. Let Ψ := {ψ,, ψ d } be a family in F L 2 (S). Then Ψ generates a wavelet basis in F L 2 (S) if and only if Ψ generates a wavelet tight frame in F L 2 (S) with tight frame bound B =, and ψ i = for i d. In Section 4, we will give an example(see Example in Section 4) to show that if Ψ generates a wavelet tight frame in F L 2 (S) with tight frame bound B =, it does not necessarily follow that Ψ generates a wavelet basis in F L 2 (S). It is not difficult to see that (2.25) in Corollary 2.8 can be replaced by the following condition: 5

16 (*) For any {a i,j,γ } i d,j ZZ,γ ZZ n satisfying d i= j ZZ γ ZZ a i,j,γ 2 <, if n d then a i,j,γ = 0 i d, j ZZ, γ ZZ n. γ ZZ n a i,j,γ ψ i;j,γ = 0 in L 2, By Corollary 2.8, (*) reveals that the main difference between a DWTF and a biorthogonal wavelet basis is the possibility of redundancy. As we shall see this freedom of redundancy will make it much easier to construct some DWTFs in F L 2 (S). Examples will be given in Sections 3 and 4 to illustrate this fact. 3. Construction of Dual Wavelet Tight Frames in F L 2 (S) In this section, we shall generalize Lawton s result (see [4]) on wavelet tight frames in L 2 (IR) to the n-dimensional case. Two ways of constructing DWTFs in F L 2 (S) are suggested. Before proceeding further, we state the following interesting Lemma which was pointed out to us by Weiss and Wang (cf.[20, 2]) while the author was visiting Washington University. Its proof presented in this paper is given by the author himself. Lemma 3.. If f L (IR n ). Then lim f(m j ξ) = 0 j + Proof. We shall prove this lemma for the special case M = 2I n n. For any general dilation matrix M, the proof is similar. Let E := {ξ IR n : < ξ 2} and define g(ξ) := lim sup j + f(2 j ξ), ξ IR n. It suffices to prove that g(ξ) = 0 a.e.. Suppose not, by g(ξ) = g(2ξ), there exist a constant ε > 0 and a measurable set E 0 E such that E 0 > 0 and g(ξ) > ε ξ E 0. For any N IN and ξ E 0, the definition of g guarantees the existence of j ξ IN such that j ξ N and f(2 j ξ ξ) > ε. That is, ξ E jξ j=ne j where E j := {ξ E 0 : f(2 j ξ) > ε}. Hence E 0 j=ne j which gives us that E j E 0 > 0 N IN. j=n Therefore E j = +. On the other hand, from the fact that 2 i E 2 j E = i j= j, i, j IN and E j E 0 E j IN, we have 2 i E i 2 j E j = i j, i, j IN. Hence f(ξ) dξ f(ξ) dξ ε2 j E j ε E j = + IR n j= 2 j E j j= j= which is a contradiction to the fact that f L. So g(ξ) = 0 a.e. which implies lim j + f(2j ξ) = 0. Throughout this section, {γ i } m i=0 denotes a full collection of representatives of distinct cosets of ZZ n /M ZZ n with γ 0 = 0. For any given collection of ZZ n -periodic functions 6 a.e..

17 p i, 0 i d, here and subsequently, we shall use P (p0,,p d ) to denote the following matrix: p 0 (ξ + M γ 0 ), p 0 (ξ + M γ ),, p 0 (ξ + M γ m ) p (3.) (ξ + M γ 0 ), p (ξ + M γ ),, p (ξ + M γ m )......, p d (ξ + M γ 0 ), p d (ξ + M γ ),, p d (ξ + M γ m ) and P (p0,,p d )(ξ) denotes the complex conjugate transpose of the matrix P (p0,,p d ); that is to say, P (p0,,p d )(ξ) := P (p0,,p d )(ξ) T. We generalize Lawton s result on wavelet tight frames in L 2 (IR) to the n-dimensional case as follows: Theorem 3.2. Suppose φ F L 2 (S) and φ F L 2 (S) W satisfy lim φ(m j ξ) φ(m j ξ) = Bχ S (ξ) j a.e. for some nonzero constant B, and φ(m ξ) = p 0 (ξ) φ(ξ) and φ(m ξ) = p 0 (ξ) φ(ξ) where p 0, p 0 are ZZ n -periodic measurable functions satisfying (3.2) m i=0 p 0 (ξ + M γ i ) p 0 (ξ + M γ i ) =. Also suppose that there exists a collection of ZZ n -periodic measurable functions p i, p i L (IR n ), i m such that P ( p0, p,, p m ) (ξ) P (p0,p,,p m )(ξ) = I m m a.e. where P ( p0, p,, p m ) (ξ) and P (p 0,p,,p m )(ξ) are defined by (3.). Let ψ i (ξ) := p i (M ξ) φ(m ξ) and ψ i (ξ) := p i (M ξ) φ(m ξ). Then {ψ,, ψ m } with { ψ,, ψ m } generates a DWTF in F L 2 (S) with frame bound () n B provided that each function in {ψ,, ψ m, ψ,, ψ m } forms a Bessel sequence. Proof. First of all, it is easy to see that ψ i, ψ i F L 2 (S). To verify the claim, by Theorem 2.5, it suffices to check equations (2.) and (2.2). Note that by Proposition 2., each function in {ψ,, ψ m, ψ,, ψ m } is admissible which implies that the left side of (2.) converges absolutely. Thus by the definition of ψ i and ψ i, m ψ i (M j ξ) ψ i (M j ξ) = j ZZ m i= p i (M j ξ) p i (M j ξ) φ(m j ξ) φ(m j ξ). 7

18 Since P ( p0, p,, p m ) (ξ) P (p0,p,,p m )(ξ) = I m m implies (3.3) we obtain m m i=0 p i (ξ + M γ j )p i (ξ + M γ k ) = δ j,k, 0 j, k m, ψ i (M j ξ) ψ i (M j ξ) = ( p 0 (M j ξ) p 0 (M j ξ)) φ(m j ξ) φ(m j ξ) j ZZ = ( φ(m j ξ) φ(m j ξ) φ(m ) j 2 ξ) φ(m j 2 ξ) j ZZ = lim φ(m j ξ) φ(m j ξ) lim φ(m j ξ) φ(m j ξ) j j + = Bχ S (ξ) where we used lim φ(m j ξ) φ(m j ξ) = Bχ S (ξ) and the fact j lim φ(m j ξ) φ(m j ξ) = 0 j + by Lemma 3.. Hence equation (2.) is verified. To verify equation (2.2), a similar argument as in the above proof shows that m i= j=0 ψ i (M j ξ) ψ i (M j (ξ + γ)) m = p i (M j ξ) φ(m j ξ) p i (M j (ξ + γ)) φ(m j (ξ + γ)) i= j=0 = φ(m m ξ) φ(m (ξ + γ)) p i (M ξ) p i (M (ξ + γ)) i= + φ(m m j ξ) φ(m j (ξ + γ)) p i (M j ξ) p i (M j ξ). j=0 i= Note that for any fixed γ ZZ n \M ZZ n, we have γ γ k modulo M ZZ n for some k m. By (3.3) and p i (M (ξ + γ)) = p i (M (ξ + γ k )), we have m i= and for any j IN {0}, p i (M ξ) p i (M (ξ + γ)) = p 0 (M ξ) p 0 (M (ξ + γ)), m i= p i (M j ξ) p i (M j ξ) = p 0 (M j ξ) p 0 (M j ξ) = p 0 (M j ξ) p 0 (M j (ξ + γ). Hence m ψ i (M j ξ) ψ i (M j (ξ + γ)) i= j=0 = φ(ξ) φ(ξ + γ) + ( φ(m j ξ) φ(m j (ξ + γ)) φ(m j+ ξ) φ(m j+ (ξ + γ)) j=0 = lim φ(m j ξ) φ(m j (ξ + γ)) j + = 0. 8

19 So equation (2.2) is verified which completes the proof. The result for a wavelet tight frame corresponding to Theorem 3.2 is the following: Theorem 3.3. Suppose φ F L 2 (S) W satisfies [ φ, φ] L, lim j φ(m j ξ) 2 = Bχ S (ξ) for some positive constant B, and φ(m ξ) = p 0 (ξ) φ(ξ) for some ZZ n periodic measurable function p 0 such that (3.4) m i=0 p 0 (ξ + M γ i ) 2 =. Also suppose that there exists a collection of ZZ n -periodic measurable functions p i, i m such that P (p0,,p m )(ξ) is a unitary matrix. Let ψ i (ξ) := p i (M ξ) φ(m ξ). Then {ψ,, ψ m } generates a wavelet tight frame in F L 2 (S) with frame bound () n B. We make some remarks here. The method described in the above Theorem 3.2 of constructing a DWTF is the same method used to obtain a biorthogonal wavelet basis from an MRA(cf. [8, 2, 3]). Hence all the conditions in Theorem 3.2 are natural except the requirement φ W. When m = 2, there exists a vector v IR n such that γ, v =. If we choose p 2 (ξ) := e i ξ,v p 0 (ξ + M γ ) and p (ξ) := e i ξ,v p 0 (ξ + M γ ), then it is easy to check that P ( p0, p ) (ξ) P (p0,p )(ξ) = I 2 2. When m > 2, it is more difficult to find a collection of ZZ n -periodic functions p,, p m, p 0,, p m such that P ( p0,, p m ) (ξ) P (p0,,p m )(ξ) = I m m (cf [2, 3]). In a recent paper [9], Ron and Shen made an interesting observation concerning the construction of compactly supported wavelet tight frames. They pointed out that in Theorem 3.3, we in fact don t have to demand that P (p0,,p m )(ξ) be a square matrix. Note that in the proof of Theorem 3.2, we never use the identity (3.2). Hence the conditions in Theorem 3.2 can be relaxed. In fact, without any change of the proof of Theorem 3.2, we have the following more general result: Theorem 3.4. Suppose φ F L 2 (S) and φ F L 2 (S) W satisfy lim φ(m j ξ) φ(m j ξ) = Bχ S (ξ) j a.e. for some nonzero constant B, and φ(m ξ) = p 0 (ξ) φ(ξ), φ(m ξ) = p 0 (ξ) φ(ξ) where p 0, p 0 are ZZ n periodic measurable functions. Also suppose that there exists a collection of ZZ n -periodic measurable functions p i, p i L, i d such that P ( p0, p,, p d ) (ξ) P (p0,p,,p d )(ξ) = I d d. Let ψ i (ξ) := p i (M ξ) φ(m ξ) and ψ i (ξ) := p i (M ξ) φ(m ξ). Then {ψ,, ψ d } with { ψ,, ψ d } generates a DWTF in F L 2 (S) with frame bound () n B provided that each function in {ψ,, ψ d, ψ,, ψ d } forms a Bessel sequence. 9

20 Note that Theorem 3.2 corresponds to the special case of d = m in Theorem 3.4 which can be used to construct DWTFs with compact support. Similarly, a more general version of Theorem 3.3 holds. The following interesting example is given in [9]. Example. have and Let n = and M = 2. Let φ be the centred B-spline of order 2d. The we φ(ξ) = sin2d (ξ/2) (ξ/2) 2d φ(2ξ) = p 0 (ξ) φ(ξ) with p 0 (ξ) = cos 2d (ξ/2). For any i 2d, define p i (ξ) := ( ) 2d sin i i (ξ/2)cos 2d i (ξ/2). Then {ψ,, ψ 2d }, which is defined by ψ i (ξ) = p i (ξ/2) φ(ξ/2), i 2d, generates a wavelet tight frame with frame bound in L 2 (IR) and each ψ i, i 2d has compact support. To apply Theorem 3.2 and Theorem 3.4, we have to check whether each ψ i defined in these theorems forms a Bessel sequence or not. The following provides us a convenient sufficient condition to do this. Proposition 3.5. Suppose φ(ξ) C( + ξ ) n/2 ε for some positive constants C and ε. Also suppose that a ZZ n -periodic function p(ξ) satisfies p(ξ) C ξ δ, ξ [ π, π] n for some positive constants C and δ. Let ψ(ξ) := p(m ξ) φ(m ξ). Then ψ forms a Bessel sequence. Proof. It is easy to check that ψ satisfies (2.2). Hence by Proposition 2.6, ψ forms a Bessel sequence. On one hand, without requiring d = m in Theorem 3.4, it is simpler to construct DWTFs by Theorem 3.4 than by Theorem 3.2. On the other hand, it is of interest to consider Theorem 3.2 which is a special case of d = m in Theorem 3.4 since the resulting DWTF in Theorem 3.2 may actually be a biorthogonal wavelet basis. Without a proof, we mention that if in addition, [ φ, φ] L, [ φ, φ] L and [ φ, φ] = a.e., () n then the resulting DWTF in Theorem 3.2 is actually a biorthogonal wavelet basis. Observe that if φ is continuous at 0, then φ(ξ) = φ(0)π j= p 0 (M j ξ) by φ(m ξ) = p 0 (ξ) φ(ξ). Hence, to use Theorem 3.2, we have to find a pair of ZZ n -periodic functions p 0 and p 0 satisfying (3.2). The following result given by Wellend and Lundberg in [22] leads to a way of constructing a new pair of such functions from any known pair p 0 and p 0 satisfying (3.2). Before proceeding further, we introduce some notations. Let e 0 := (,,, ) ZZ m. For any α, β ZZ m, we say that α β if for any k m, α k β k where α k is the k-th component of α. For any η = (η,, η m ) ZZ m, we define 20

21 m η := η k and for any N IN, we define the multinomial coefficient as follows: k= ( ) N + η := η (N + η )! (N )!η!η 2! η m!. Lemma 3.6. Let X := (X, X 2,, X m ) and for any 0 < η = (η,, η m ) ZZ m, denote X η := X η X η 2 2 X η m m. Define If m i=0 q N (X) = q N (X, X 2,, X m ) := X i =, then m i=0 0 η (N )e 0 ( ) N + η X η η. X N i q N ( X i ) = where X i = (X 0,, X i, X i+,, X m ). By the above Lemma, the following proposition shows how to obtain a new pair of ZZ n -periodic functions satisfying (3.2). Proposition 3.7. Suppose two ZZ n -periodic measurable functions p 0 and p 0 satisfy (3.2). For any N IN, let h N (ξ) := q N (p 0 (ξ + M γ ) p 0 (ξ + M γ ),, p 0 (ξ + M γ m ) p 0 (ξ + M γ m )). Then we have m i=0 p N 0 (ξ + M γ i ) p N 0 (ξ + M γ i )h N (ξ + M γ i ) =. Proof. Take X i = p 0 (ξ + M γ i ) p 0 (ξ + M γ i ), then m i=0 X i = by (3.2). By Lemma 3.6, it suffices to prove that q N ( X i ) = h N (ξ + M γ i ). Note that q N is symmetric about its arguments and γ + γ i, γ 2 + γ i,, γ m + γ i are congruent to γ 0,, γ i, γ i+,, γ m modulo M ZZ n. By the fact that p 0, p 0 are ZZ n -periodic, we have h N (ξ + M γ i ) = q N (p 0 (ξ + M (γ + γ i )) p 0 (ξ + M (γ + γ i )),, p 0 (ξ + M (γ m + γ i )) p 0 (ξ + M (γ m + γ i ))) = q N (p 0 (ξ + M γ 0 ) p 0 (ξ + M γ 0 ),, p 0 (ξ + M γ i ) p 0 (ξ + M γ i ), p 0 (ξ + M γ i+ ) p 0 (ξ + M γ i+ ),, p 0 (ξ + M γ m ) p 0 (ξ + M γ m )) = q N (X 0,, X i, X i+,, X m ) = q N ( X i ) which completes the proof. Now we give a simple example to illustrate the above general theory. 2

22 Example. Let n = and M = 2. Let S(IR) denote the Schwarz class of functions. Suppose ϕ S(IR) is a Lemarié-Meyer scaling function(cf. [5, 8]) such that ϕ(2ξ) = m 0 (ξ) ϕ(ξ) where m 0 C is a periodic function and m 0 (ξ) 2 + m 0 (ξ + π) 2 =. To find an example of such ϕ, please see [8]. Taking N = 2, by Proposition 3.7, we have (3.5) m 0 (ξ) 4 h 2 (ξ) + m 0 (ξ + π) 4 h 2 (ξ + π) = where h 2 (ξ) := + 2 m 0 (ξ + π) 2. Since m 0 (0) = and m 0 C imply h 2 (0) = and h 2 C, Π j=h 2 (ξ/2 j ) converges uniformly in any bounded closed interval. Thus φ(ξ) := ϕ(ξ/2) 2 Π j=h 2 (ξ/2 j ) is well-defined and φ is band-limited. Let p 0 (ξ) := m 0 (ξ) 2, p 0 (ξ) := m 0 (ξ) 2 h 2 (ξ) and φ(ξ) := ϕ(ξ) 2. It is easy to see that φ(2ξ) = p 0 (ξ) φ(ξ) and φ(2ξ) = p 0 (ξ) φ(ξ). Let p (ξ) := e iξ/2 p 0 (ξ + π) and p (ξ) := e iξ/2 p 0 (ξ + π). It is easily seen that P ( p0, p ) (ξ) P (p0,p )(ξ) = I 2 2. Set ψ(ξ) = p (ξ/2) φ(ξ/2) and ψ(ξ) = p i (ξ/2) φ(ξ/2). By Proposition 3.5, we have that each of ψ and ψ forms a Bessel Sequence. Hence by Theorem 3.2 and lim j φ(2 j ξ) φ(2 j ξ) =, ψ with ψ generates a DWTF in L 2 (IR) with frame bound. In fact, ψ with ψ generates a biorthogonal wavelet basis in L 2 (IR). In the rest of this section, we give another way of constructing certain wavelet tight frames in F L 2 (S). It is well known(cf. [, 3, 0, 6, 7]) that if {ψ,, ψ d } generates a wavelet basis in L 2 (IR n ) for a dilation matrix M = 2I n n, and ψ i, i d are smooth enough, then {ψ,, ψ d } can be derived from an MRA which implies d 2 n. The following result shows that this is not the case for wavelet tight frames. For simplicity, in the rest of this section, we assume M = 2I n n. Theorem 3.8. Let f L 2 (IR n ) be a band-limited function. Suppose g(ξ) := f(2 j ξ) 2 <, a.e.. Let S := suppg. For any integer k such that (3.6) sup ξ ξ 2 < inf γ ZZ ξ,ξ 2 supp f n \{0} 2k γ, Define { ψ(ξ) := () n/2 f(2 k ξ)/ g(ξ), when ξ S; 0 when ξ S. Then ψ generates a wavelet tight frame in F L 2 (S) with frame bound. Proof. Since f is band-limited and ψ(ξ) () n/2, it is easily seen that ψ is bandlimited and ψ F L 2 (S). By Theorem 2.7, it suffices to verify equations (2.22) and (2.23). By the definition of ψ, it is obvious that ψ(2 j ξ) 2 = j ZZ () χ S(ξ). To verify n (2.23), it suffices to prove that j ZZ (3.7) ψ(2 j ξ) ψ(2 j (ξ + γ)) = 0 a.e. γ ZZ n \2ZZ n, j IN {0}. 22

23 Note that supp ψ(2 j ) = 2 j k supp f and supp ψ(2 j ( + γ)) = 2 j k supp f γ. (3.6) implies that for any γ ZZ n \2ZZ n, j IN {0}, supp ψ(2 j ) supp ψ(2 j ( + γ)) = 2 (j+k)n supp f (supp f 2 k 2 j γ) = 0. Hence (3.7) holds. Thus the proof is completed. Example. Let η S(IR) with supp η [ 8π, 8 π] be a Lemarié-Meyer wavelet in 3 3 L 2 (IR)(cf. [8]), that is, η generates a wavelet basis in L 2 (IR). Choose f(ξ) := η( ξ ), ξ IR n. It is easy to see that g(ξ) := f(2 j ξ) 2 = and j ZZ 6 3 π = sup ξ,ξ 2 supp f ξ ξ 2 < inf γ ZZ n \{0} 22 γ = 8π. Define ψ(ξ) := () n 2 f(4ξ) = () n 2 η(4 ξ ). Then by Theorem 3.8, ψ generates a wavelet tight frame in L 2 (IR n ) with tight frame bound and clearly ψ S(IR n ). 4. Wavelet Tight Frames in H 2 (IR) and L 2 (IR) With Frame Bound B = A function is said to be band-limited if its Fourier transform is compactly supported. In this section, we shall use the results developed in Section 2 to construct smooth bandlimited wavelet tight frames in H 2 (IR) and L 2 (IR). Without further mention, in this section, n =, M = 2 and S = (, ) or [0, ). Our procedure to construct smooth band-limited wavelet tight frames in H 2 (IR) and L 2 (IR) is described as follows: first we construct a function ψ(ξ) = χ K (ξ) where K is a union of some mutually disjoint closed intervals in S such that ψ generates a wavelet tight frame in F L 2 (S). Then we show that, for some special K, we can obtain a smooth wavelet tight frame by modifying ψ. Finally, the definition of a non-mra wavelet tight frame is given and examples are provided to illustrate the general theory. In this section, unless specified, otherwise the tight frame bound B is assumed to be. Characteristic functions can be used to construct wavelet tight frames and wavelet bases as follows. Theorem 4.. Let {I i } l i= be a family of mutually disjoint closed intervals in S and define ψ(ξ) := χ K (ξ) where K := l i=i i. Suppose the following two conditions are satisfied (i) χ K (2 j ξ) = χ S (ξ) a.e., and j ZZ (ii) χ K (ξ + γ) γ ZZ a.e.. Then ψ generates a wavelet tight frame in F L 2 (S). In particular, if equality holds almost everywhere in (ii), then ψ generates a wavelet basis in F L 2 (S) and ψ can be derived from an MRA with one scaling function if and only if ( j=2 j K) ( γ ZZ (K + 4πγ)) = 0. 23

24 Proof. By Theorem 2.7, it suffices to verify equation (2.23). Note that condition (ii) implies that supp ψ (supp ψ + γ) = 0 for any γ ZZ\{0}. Hence it is easy to see that ψ(2 j ξ) ψ(2 j (ξ + γ)) = 0 for any γ ZZ\2ZZ, j IN {0} which gives us (2.23). When equality holds in (ii), then [ ψ, ψ] = / which implies ψ =. By Corollary 2.9, ψ generates a wavelet basis in F L 2 (S). For the last assertion, a general result is presented in Han [0]. The following are some examples obtained from the above theorem. More examples can be obtained easily by direct computation. Example. Let K := [a, 2a], 0 < a. Then the corresponding function ψ in Theorem 4. generates a wavelet tight frame in H 2 (IR). When 0 < a <, the above ψ can only generate a wavelet tight frame in H 2 (IR)with tight frame bound. It can not generate a wavelet basis in H 2 (IR) since ψ =. Example 2. Let K := [a, b] [2 s b, 2 s+ a] where a = l ε l > 0 and < 2 s+ 2 s b < 2(l ε) with s, l IN and 0 < ε < 2s+ l. Then ψ generates a wavelet tight frame 2 s+ 2 s+ 2 in H 2 (IR). For example, letting s = 2, l = 6, ε = /20, we have a = 5 and 5 < b < Choosing b = , then K = [ π, π] [ π, π] which satisfies the condition (i) and (ii) in Theorem 4.. The following examples provide some wavelet bases in L 2 (IR) and H 2 (IR). Example 3. Let K = [a, 2a] [2a 4π, a ], when 0 < a <. Then ψ generates a wavelet basis in L 2 (IR) and ψ can be derived from an MRA with one scaling function. Example 4. Let K = [a, b] [2 s b, 2 s+ a] [ 2c, c], where a = k, b = 4πk, c = 2 s+ 2 s+ k 2 s b, with < k < 2 s, k, s IN. Then ψ generates a wavelet basis in L 2 (IR). When k is odd, ψ can be generated by an MRA; When k is even, ψ can not be derived from an MRA with one scaling function. Example 5. Let K denote [ 2 s π, 2 s+ π] [ ] 2 s π, 22s+ π 2 s+ and K = K K where s IN. Then ψ generates a wavelet basis in L 2 (IR). When s 2, ψ can not be derived from an MRA with one scaling function. In the case s = 2, this is Journé s counterexample(cf.[9]). Example 6. Let K = [a, b] [2 s b, 2 s+ a], where a = +k, b = k and k < 2 s+ 2 s 2 s+ 2, k, s IN. Then ψ generates a wavelet basis in H 2 (IR). Also we know that when k is odd, ψ can not be derived from an MRA with one scaling function. Now we show how to modify the wavelet tight frames constructed by Theorem 4. to obtain smooth wavelet tight frames. The following lemma will be needed later and has appeared elsewhere in the literature. 24

25 Lemma 4.2. There exists a function θ C (IR) satisfying θ(x) = 0 when x and θ(x) = when x and (4.) θ(x) 2 + θ( x) 2 =, x IR. Proof. Let { 0, when x 0, f(x) := e x 2, when x > 0. Then it is not difficult to see that f C and f(x) > 0 when x > 0. Define g(x) := x f( + t)f( t)dt. It follows that g C, and g(x) = 0 when x < and g(x) > 0 when x >. Note that g(x) = g() for x. If we set θ(x) := g(x) g( x) 2 + g(x) 2, then θ is well-defined and satisfies all the requirements in Lemma 4.2. Given a closed interval I = [a, b] and two positive numbers δ, δ 2 such that δ + δ 2 b a, we define θ( x a δ ), when x < a + δ, (4.2) f (I;δ,δ 2 )(x) :=, when a + δ x b δ 2, θ( b x δ 2 ), when x > b δ 2, where θ is defined in Lemma 4.2. From Lemma 4.2, it is clear that f (I;δ,δ 2 ) C 00 and suppf (I;δ,δ 2 ) [a δ, b + δ 2 ]. Hence f (I;δ,δ 2 ) can be considered as obtained from χ I by smoothly modifying χ I. To prove our main result in this section, the following notation is also needed. Given a closed subset K IR, we define (4.3) d(k) := dist(k, γ ZZ\{0} (K + γ)) where for any subsets A and B, dist(a, B) := inf x A,y B x y. The following result states that if the mutually disjoint intervals I,, I l satisfy the conditions in Theorem 4. and d( l i=i i ) > 0, then we can obtain a new smooth wavelet tight frame. For simplicity, we discuss the case S = [0, ). When S = (, ), a similar result can be easily obtained. Theorem 4.3. Suppose that a family of disjoint closed intervals I i = [a i, b i ], i l in (0, ) is arranged in a decreasing order, i.e., 0 < b l < b l < < b, and satisfies (4.4) l χ Ii (2 j ξ) = χ [0, ) (ξ). If d( l i=i i ) > 0, then for any 0 < δ < 2 min{d( l i=i i ), min {b i a i }, min dist(i i, I j )}, i l i<j l let (4.5) ψ δ (ξ) := f (I ;δ/2,δ)(ξ) + 25 l f (Ii ;2 k i δ,2 k i δ)(ξ), i=2

26 where k i is the unique non-negative integer such that 2 k i I i [ 2 b, b ]. We have ψ δ S(IR) and ψ δ generates a wavelet tight frame with frame bound in H 2 (IR). Proof. Let f (ξ) := f (I ;δ/2,δ)(ξ) and f i (ξ) := f (Ii ;2 k i δ,2 k i δ)(ξ). Since k i 0, by (4.2), we have suppf [a δ/2, b + δ] and for any 2 i l, suppf i [a i 2 k i δ, b i + 2 k i δ] [a i δ, b i + δ]. Hence for any i l, suppf i [a i δ, b i + δ]. By (4.3) and 0 < δ < 2 d({i i} l i=), it is easy to see that (4.6) d( l i=[a i δ, b i + δ]) d( l i=i i ) 2δ > 0, from which it follows that for any γ ZZ\{0}, K (K + γ) = where K := l i=suppf i l i=[a i δ, b i + δ]. Thus (4.7) f i (2 j ξ)f k (2 j (ξ + γ)) = 0 i, k l, j IN {0}, γ ZZ\{0}. Hence for any j IN {0} and γ ZZ\{0}, ψ δ (2 j ξ) ψ δ (2 j (ξ + γ)) = 0 a.e. which implies (2.23) is true. To check (2.22), note that (4.4) is equivalent to (4.8) χ I (ξ) + l i=2 χ 2 k iii (ξ) = χ [ 2 b,b ] (ξ). Observe that by (4.2), for any positive numbers δ, δ 2, δ 3 and 0 < a < b < c, and f (I;δ,δ 2 )(2 k ξ) = f (2 k I,2 k δ,2 k δ 2 )(ξ), f 2 ([a,b];δ,δ 2 )(ξ) + f 2 ([b,c];δ 2,δ 3 )(ξ) = f 2 ([a,c];δ,δ 3 )(ξ). Combining the above facts with (4.8), we obtain l f 2 (ξ) + fi 2 (2 k i ξ) = f 2 ([ 2 b,b ]; δ,δ)(ξ). 2 i=2 l Hence fi 2 (2 j ξ) = χ [0, ) (ξ) a.e.. Note that 0 < δ < min 2 i<j l{dist(i i, I j )} gives us that ψ l δ (ξ) 2 = f i 2 (ξ). Thus i= j ZZ ψ δ (2 j ξ) 2 = l f 2 i (2 j ξ) = χ [0, )(ξ). Hence (2.22) is verified. By Theorem 2.7, ψ δ generates a wavelet tight frame with tight frame bound in H 2 (IR) and clearly ψ δ S(IR) by f i S(IR), i l. Reviewing the examples constructed in Example and 2, by Theorem 4.3, we have the following: 26

27 Example 7. Let I := [a, 2a], 0 < a <. Then d(i) = a > 0. Hence for any 0 < δ < min{ a, a}, let ψ 2 δ (ξ) := f (I; δ,δ)(ξ). Then ψ δ generates a wavelet tight 2 frame in H 2 (IR). Example 8. Let I := [ 5 0 π, π] and I := [ π, π]. It is not difficult to check that d(i I 2 ) = π > 0. Hence for any 0 < δ < π, the corresponding function ψ 0 20 δ defined in Theorem 4.3 generates a wavelet tight frame in H 2 (IR) and ψ δ S(IR). Many examples of smooth wavelet tight frames in H 2 (IR) and L 2 (IR) can be constructed by using Theorem 4.3. We mention that although for any wavelet basis constructed in Theorem 4., d( l i=i i ) = 0, a similar method also works for some wavelet bases in L 2 (IR) constructed in Theorem 4.. Due to the fact d( l i=i i ) = 0, the procedure is much more complicated. For more details on how to carry out this modification method to wavelet bases in L 2 (IR), please see [2, 3, 0, 20, 2]. The following interesting result on wavelet bases in H 2 (IR) due to Auscher in [] says that: Theorem. There does not exist a function ψ H 2 (IR) such that ψ generates a wavelet basis in H 2 (IR) and ˆψ C (IR) and ˆψ(ξ) + ˆψ (ξ) C ξ α for ξ, α > /2. From the above result, the following interesting question was asked by one of the referees: Question. Does there exist any function ψ H 2 (IR) such that ψ is continuous and ψ generates a non-mra wavelet tight frame in H 2 (IR)? The answer is Yes. We can even find a function ψ H 2 (IR) such that ψ S(IR) and ψ generates a wavelet tight frame in H 2 (IR). Here we still have to explain by what we mean a non-mra wavelet tight frame. In the following definition, we still consider the case S = [0, ) or S = (, ). A similar definition can be given for the general case. Let ψ F L 2 (S) be a given function such that ψ generates a wavelet tight frame in F L 2 (S). If there exist two periodic measurable functions p 0, p and a function φ F L 2 (S) such that and ψ(ξ) = p (ξ/2) φ(ξ/2) and φ(ξ) = p 0 (ξ/2) φ(ξ/2), P (ξ) := ( p0 (ξ), ) p 0 (ξ + π) p (ξ), p (ξ + π) is a unitary matrix, i.e., P (ξ) P (ξ) = I 2 2, then we say that ψ generates an MRA wavelet tight frame in F L 2 (S). Otherwise, we say that ψ generates a non-mra wavelet tight frame in F L 2 (S). Suppose ψ generates an MRA wavelet tight frame in F L 2 (S) and is derived from some φ, p 0, p as in the above definition. Then we have φ(2ξ) 2 + ψ(2ξ) 2 = ( p 0 (ξ) 2 + p (ξ) 2 ) φ(ξ) 2 = φ(ξ) 2 a.e.. 27