International Journal of Analysis and Applications ISSN 2291-8639 Volume 9, Number 2 (2015), 68-82 http://www.etamaths.com CHARACTERIZATION OF BIORTHOGONAL MULTIWAVELET PACKETS WITH ARBITRARY DILATION MATRIX FIRDOUS A. SHAH1 AND R. ABASS2,∗ Abstract. In this paper, we investigate the characterization of biorthogonal multiwavelet packets associated with arbitrary matrix dilations and particular- ly of orthonormal multiwavelet packets by means of basic equations in Fourier domain. 1. INTRODUCTION It is well-known that the classical orthonormal wavelet bases have poor fre- quency localization. For example, if the wavelet ψ is band limited, then the measure of the supp of ψ̂j,k is 2 j-times that of supp ψ̂. To overcome this disadvantage, Coif- man etal. [8] constructed univariate orthogonal wavelet packets. The fundamental idea of wavelet packet analysis is to construct a library of orthonormal bases for L2(R), which can be searched in real time for the best expansion with respect to a given application. Well known Daubechies orthogonal wavelets are a special case of wavelet packets. Chui and Li [6] generalized the concept of orthogonal wavelet packets to the case of non-orthogonal wavelet packets so that they can be can be employed to the spline wavelets and so on. The introduction of biorthogonal wavelet packets attributes to Cohen and Daubechies [7]. They have also shown that all the wavelet packets, constructed in this way, are not led to Riesz bases for L2(R). Shen [18] generalized the notion of univariate orthogonal wavelet packets to the case of multivariate wavelet packets. Other notable generalizations are the wavelet pack- ets related to the Walsh polynomials on R+ [13,14,16], higher dimensional wavelet packets with arbitrary dilation matrix [9], the orthogonal version of vector-valued wavelet packets [5] and the M-band framelet packets [17]. On the other hand, multiwavelets are natural extension and generalization of traditional wavelets. They have received considerable attention from the wavelet research communities both in the theory as well as in applications. They can be seen as vector valued-wavelets that satisfy conditions in which matrices are involved rather than scalars as in the wavelet case. Multiwavelets can own symmetry, or- thogonality, short support and high order vanishing moments, however traditional wavelets can not possess all these properties at the same time (see [10]). As far as the characterization of multiwavelets is concerned, Calogero studied the character- ization of all multiwavelets associated with general expanding maps of Rn in [4]. 2010 Mathematics Subject Classification. 42C40; 42C15; 65T60. Key words and phrases. multiwavelet; multiresolution analysis; scaling function; multiwavelet packet; matrix dilation; Fourier transform. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 68 CHARACTERIZATION OF BIORTHOGONAL MULTIWAVELET 69 The Calogero’s work was further extended by Bownik [2], taking into consideration the dilation matrices which preserves the standard lattice Zn in terms of affine systems. In the same year, another characterization of orthonormal multiwavelets was given by Rzeszotnik [11] for expanding dilations that preserves the lattice Zn. However, Bownik [3] has presented a new approach to characterize all orthonormal multiwavelets by means of basic equations in the Fourier domain. Recently, Yang and Cheng [20] have generalized the concept of wavelet packets to the case of multiwavelet packets associated with a dilation factor a which were more flexible in applications. Subsequently, Behera [1] extended the results of Yang and Cheng to the multivariate multiwavelet packets associated with a dilation matrix A. He proved lemmas on the so-called splitting trick and several theorems concerning the Fourier transform of the multiwavelet packets and the construction of multiwavelet packets to show that their translates form an orthonormal basis of L2(Rd). Later on, Sun and Li [19] have given the construction and properties of generalized orthogonal multiwavelet packets based on the results discussed in [20]. Orthogonal wavelet packets have many desired properties such as compact support, good frequency localization and vanishing moments. However, there is no continuous symmetry which is a much desired property in imaging the compres- sion and signal processing. To achieve symmetry, several generalizations of scalar orthogonal wavelet packets have been investigated in literature. The biorthogo- nal wavelet packets achieve symmetry where the orthogonality is replaced by the biorthogonality. The characterization of multiwavelet packets associated with the dilation matrix A on general lattices has been studied by the author in [12, 15]. In this paper, we further investigate the characterization of biorthogonal multiwavelet packets associated with arbitrary matrix dilations and particularly of orthonormal multiwavelet packets by means of basic equations in Fourier domain. We have structured the article as follows. In Section 2, we state some basic preliminaries, notations and definitions including the definition of multiresoltion analysis associated with arbitrary dilation matrix A and the corresponding multi- wavelet packets. In Section 3, we establish our main results concerning with the characterization of biorthogonal multiwavelet packets on Rd. 2. NOTATIONS AND PRELIMINARIES Throughout, this paper, we use the following notations. Let R and C be all real and complex numbers, respectively. Z and Z+ denote all integers and all non-negative integers, respectively. Zd and Rd denote the set of all d-tuples integers and d-tuples of reals, respectively. Assume that we have an expansive dilation matrix A, i.e., all eigenvalues λ of A satisfy |λ| > 1 and preserves the lattice Γ. Let a = |detA|,A∗ = transpose of A and B be a d×d non-singular matrix. Also, if A is expanding so is A∗. Considering Zd as an additive group, we see that AZd is a normal subgroup of Zd so we can form the cosets of AZd in Zd. It is well known fact that the number of distinct cosets of AZd in Zd is equal to a = |detA| (see[21]). With A and B defined as above, we consider 70 SHAH AND ABASS Λ(A,B) = { α ∈ Rd : ∃ (j,m) ∈ Z×B∗−1(Zd) : α = A∗−jm } , (2.1) and IA,B(α) = { (j,m) ∈ Z×B∗−1(Zd) : α = A∗−jm } . (2.2) The set Λ(A,B) is thought of as the set of all A-adic vectors relative to the lattice B∗−1(Zd), i. e., the set of representatives of the equivalence classes of Z×B∗−1(Zd) with respect to the equivalence relation defined by (j,m) ∼ (j′,m′) if and only if α = A∗−jm = A∗−j ′ . Further, the set IA,B(α) is the set of points of Z×B∗−1(Zd) in the equivalence class of α ∈ Λ(A,B). Since it is a well known fact that for every dilation matrix A, there exists a Hermitian norm ‖·‖∗ in Rd, and constants λmax ≥ λmin > 1, such that if B denotes the unit ball in the new norm, centered at the origin, then B ⊂ λminB ⊂ A∗(B) ⊂ λmaxB. For each k ∈ Z, we define Hk as Hk = A ∗k(B), 2H0 ⊂ Hη, |B| = 1. where η be the smallest integer. Then, the quasi-distance ρ on Rd induced by the dilation A∗ is given by ρ(ξ,ζ) = { |detA|j if ξ − ζ ∈ Hj+1 \Hj 0 if ξ = ζ. Furthermore, it is easy to verify that the Hardy-Littlewood maximal operator MHLf(ζ) = sup k∈Z 1 |Hk| ∫ ζ+Hk ∣∣f(ξ)∣∣dξ is bounded from L1 to L1-weak norm and lim k→−∞ 1 |Hk| ∫ ξ+Hk f(ξ) dξ = f(ξ), for a.e. ξ ∈ Rd. (2.3) Definition 2.1. A countable family {fα}α∈A of elements in a separable Hilbert space H is a frame if there exist constants A,B, 0 < A ≤ B < ∞ satisfying A ∥∥f∥∥2 2 ≤ ∑ α∈A ∣∣〈f,fα〉∣∣2 ≤ B∥∥f∥∥22 (2.4) for all f ∈ H. The constants A and B independent of f for which (2.4) holds are called frame bounds. A frame is a tight frame if A and B can be chosen so that A = B and is a normalized tight frame if A = B = 1. If only the right hand side inequality holds in (2.4), we say that {fα}α∈A is a Bessel squence with constant B. CHARACTERIZATION OF BIORTHOGONAL MULTIWAVELET 71 Lemma 2.2 [3]. Two families {fα : α ∈A} and { f̃α : α ∈A } constitute a biorthog- onal pair if and only if they are Bessel sequences and satisfy P(f,g) = ∑ α∈A 〈 f,fα 〉〈 f̃α,g 〉 = 〈 f,g 〉 for all f,g in a dense subset D of H, where P(f,g) is a bi-linear functional on H×H. Using polarization identity along with the Definition 2.1 implies that P(f,f) = ∥∥f∥∥2, for all f ∈ L2(Rd), (2.5) which is equivalent to P(f,g) = 〈f,g〉, for f ∈D. We recall the notion of higher dimensional multiresolution analysis associated with multiplicity L and orthogonal multiwavelets of L2 ( Rd ) (see [1]). Definition 2.3. A sequence {Vj}j∈Z of closed subspaces of L 2 ( Rd ) is called a multiresolution analysis (MRA) of L2 ( Rd ) of multiplicity L associated with the dilation matrix A if the following conditions are satisfied: (i) Vj ⊂ Vj+1 for all j ∈ Z; (ii) ⋃ j∈ZVj is dense in L 2 ( Rd ) and ⋂ j∈ZVj = {0} ; (iii) f ∈ Vj if and only if f(A·) ∈ Vj+1 for all j ∈ Z; (iv) there exist L-functions Φ = {ϕ1,ϕ2, . . . ,ϕL} ∈ V0, such that the system of functions {ϕ`(x−k)} L `=1,k∈Zd, forms an orthonormal basis for subspace V0. The L-functions whose existence is asserted in (iv) are called scaling functions of the given MRA. Given a multiresolution analysis {Vj}j∈Z, we define another sequence {Wj}j∈Z of closed subspaces of L 2 ( Rd ) by Wj = Vj+1 Vj,j ∈ Z. These subspaces inherit the scaling property of {Vj}, namely f ∈ Wj if and only if f(A·) ∈ Wj+1. (2.6) Further, they are mutually orthogonal, and we have the following orthogonal de- compositions: L2 ( Rd ) = ⊕ j∈Z Wj = V0 ⊕ (⊕ j≥0 Wj ) . (2.7) A set of functions {ψr` : 1 ≤ ` ≤ L, 1 ≤ r ≤ a− 1} in L 2 ( Rd ) is said to be a set of basic multiwavelets associated with the MRA of multiplicity L if the collection{ ψr` (.−k) : 1 ≤ r ≤ a− 1, 1 ≤ ` ≤ L, k ∈ Z d } 72 SHAH AND ABASS forms an orthonormal basis for W0. Now, in view of (2.6) and (2.7), it is clear that if {ψr` : 1 ≤ ` ≤ L, 1 ≤ r ≤ a− 1} is a basic set of multiwavelets, then{ aj/2ψr` (A j.−k) : j ∈ Z, k ∈ Zd, 1 ≤ ` ≤ L, 1 ≤ r ≤ a− 1 } forms an orthonormal basis for L2 ( Rd ) (see [1]). For any n ∈ Z+, we define the basic multiwavelet packets ωn` ; 1 ≤ ` ≤ L recursively as follows. We denote ω0` = ϕ`, 1 ≤ ` ≤ L, the scaling functions and ωr` = ψ r ` ,r ∈ Z +, 1 ≤ ` ≤ L as the possible candidates for basic multiwavelets. Then, define ωs+ar` (x) = L∑ j=1 ∑ k∈Zd hs`jk a 1/2 ωr` (Ax−k), 0 ≤ s ≤ a− 1, 1 ≤ ` ≤ L (2.8) where ( hs`jk ) is a unitary matrix (see [1]). Taking Fourier transform on both sides of (2.8), we obtain ( ωs+ar` )∧ (ξ) = L∑ j=1 hs`j(B −1ξ) ( ωr` )∧ (B−1ξ). (2.9) Note that (2.8) defines ωn` for every non-negative integer n and every ` such that 1 ≤ ` ≤ L. The set of functions {ωn` : n ∈ Z +, 1 ≤ ` ≤ L} as defined above are called the basic multiwavelet packets corresponding to the MRA {Vj}j∈Z of L 2(Rd) of multiplicity L associated with matrix A. Definition 2.4. Let {ωn` : n ∈ Z +, 1 ≤ ` ≤ L} be the basic multiwavelet packets. The collection P = { |detA|j/2ωn` (A.− k) : 1 ≤ ` ≤ L,j ∈ Z,k ∈ Z d } is called the general multiwavelet packets associated with MRA {Vj : j ∈ Z} of L2 ( Rd ) of multiplicity L over matrix dilation A. Corresponding to some orthonormal scaling vector Φ = ω0` , the family of multiwavelet packets ωn` defines a family of subspaces of L 2(Rd) as follows: Unj = span { aj/2ωn` (A jx−k) : k ∈ Zd, 1 ≤ ` ≤ L } ; j ∈ Z,n ∈ Z+. (2.10) Observe that U0j = Vj, U 1 j = Wj = a−1⊕ r=1 Urj , j ∈ Z CHARACTERIZATION OF BIORTHOGONAL MULTIWAVELET 73 so that the orthogonal decomposition Vj+1 = Vj ⊕Wj, can be written as U0j+1 = a−1⊕ r=0 Urj . (2.11) A generalization of this result for other values of n = 1, 2, . . . can be written as Unj+1 = a−1⊕ r=0 Uan+rj , j ∈ Z. (2.12) The following proposition is proved in [1]. Proposition 2.5. If j ≥ 0, then Wj = a−1⊕ r=0 Urj = a2−1⊕ r=a Urj−1 = · · · = at+1−1⊕ r=at Urj−t = aj+1−1⊕ r=aj Ur0 where Unj is defined in (2.10). Using this decomposition, we get the multiwavelet packets decomposition of subspaces Wj, j ≥ 0. Similar to the orthogonal multiwavelet packets, the biorthogonal multiwavelet packets associated with the biorthogonal scaling vector Φ̃ are given by ω̃s+ar` (x) = L∑ j=1 ∑ k∈Zd h̃s`jk a 1/2 ω̃r` (Ax−k), 0 ≤ s ≤ a− 1, 1 ≤ ` ≤ L. (2.13) Implementation of Fourier transform of (2.13) yields ( ω̃s+ar` )∧ (ξ) = L∑ j=1 h̃s`j(B −1ξ) ( ω̃r` )∧ (B−1ξ). (2.14) Let ωn` be general multiwavelet packets associated with the dilation matrix A. Then, we consider the system F(A,B) = { ωn`,j,k : j ∈ Z,k ∈ Z d,` = 1, . . . ,L,aj ≤ n < aj+1 } (2.15) where ωn`,j,k(x) = |detA| j/2 ωn` ( Ajx−Bk ) . Similarly, for the biorthogonal multiwavelet packets, we have F̃(A,B) = { ω̃n`,j,k : j ∈ Z,k ∈ Z d,` = 1, . . . ,L,aj ≤ n < aj+1 } (2.16) where ω̃n`,j,k(x) = |detA| j/2 ω̃n` ( Ajx−Bk ) . 74 SHAH AND ABASS The bi-linear functional P(f,g) associated to the multiwavelet packets systems F(A,B) and F̃(A,B) is given by P(f,g) = aj+1−1∑ n=aj L∑ `=1 ∑ j∈Z ∑ k∈Zd 〈 f,ωn`,j,k 〉〈 ω̃n`,j,k,g 〉 . (2.16) We will also consider the set D as a dense subset of L2 ( Rd ) defined by D = { f ∈ L2 ( Rd ) : f̂ ∈ L∞(Rd), f̂ has compact support in Rd \{0} } . 3. CHARACTERIZATION OF BIORTHOGONAL MULTIWAVELET PACKETS In this section, we prove our main results concerning the characterization of biorthog- onal multiwavelet packets associated with arbitrary matrix dilations by means of the Fourier transform. Theorem 3.1. Suppose {ωn` : n ∈ Z +,` = 1, . . . ,L} and {ω̃n` : n ∈ Z +,` = 1, . . . ,L} are the basic multiwavelet packets associated with a pair of biorthogonal scaling func- tions Φ and Φ̃ such that the following functions are locally integrable: aj+1−1∑ n=aj L∑ `=1 ∑ j∈Z ∣∣ω̂n` (A∗jξ)∣∣2, a j+1−1∑ n=aj L∑ `=1 ∑ j∈Z ∣∣∣ˆ̃ωn` (A∗jξ)∣∣∣2 . (3.1) Then, the bi-linear functional P(f,g) converges absolutely for all f,g ∈ D. More- over, the multiwavelet packets ωn` and ω̃ n ` satisfy: 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∑ (j,m)∈IA,B(α) ˆ̃ωn` ( A∗jξ ) ω̂n` ( A∗j(ξ + A∗−jm) ) = δα,0, (3.2) for a.e. ξ ∈ Rd and for all α ∈ Λ(A,B), if and only if P(f,g) = 〈f,g〉, for all f,g ∈ D. Proof. First of all we prove that P(f,g) is absolutely convergent. For this, fix j ∈ Z and let Gj = aj+1−1∑ n=aj L∑ `=1 ∑ k∈Zd 〈f,ωn`,j,k〉〈ω̃ n `,j,k,f〉. (3.3) Implementation of Parseval’s identity gives 〈 f,ωn`,j,k 〉 = |detA|j/2 ∫ Rd f̂ ( A∗jζ ) ω̂n` (ζ) e 2πiB(k)·ζ dζ, CHARACTERIZATION OF BIORTHOGONAL MULTIWAVELET 75 and 〈 ω̃n`,j,k,f 〉 = |detA|j/2 ∫ Rd f̂ ( A∗jζ ) ˆ̃ωn` (ξ) e −2πiB(k)·ξ dξ. Let F n `,j(ξ) = ∑ s∈Zd f̂ ( A∗j(ξ + B∗−1s) ) ω̂n` ( ξ + B∗−1s ) . Then, by virtue of Fourier inversion formula for the function F n `,j ◦B ∗−1, we obtain F n `,j(ξ) = ∑ k∈Zd { |detB| ∫ B∗−1([0,1]d) F n `,j(ζ) e 2πiBk·ζ dζ } e−2πiBk·ξ = |detB| ∑ k∈Zd {∫ Rd f̂ ( A∗jζ ) ω̂n` (ζ) e 2πiBk·ζ dζ } e−2πiBk·ξ. Thus, Gj as defined in (3.3) can be written as Gj = |detA|j |detB| aj+1−1∑ n=aj L∑ `=1 ∫ Rd f̂ ( A∗jξ ) ˆ̃ωn` (ξ) F n `,j(ξ) dξ = 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∫ Rd f̂(ξ) ˆ̃ωn` ( A∗−jξ ) F n `,j ( A∗−jξ ) dξ = 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∫ Rd f̂(ξ) ˆ̃ωn` ( A∗−jξ ) {∑ s∈Zd f̂ ( ξ + A∗jB∗−1s ) ω̂n` ( A∗−jξ + B∗−1s )} dξ. Now, in order to show that the convergence of ∑ j∈Z Gj is absolute and uncondition- al, it is sufficient to prove that the following two series are absolutely convergent: aj+1−1∑ n=aj L∑ `=1 ∑ j∈Z ∫ Rd f̂(ξ) ˆ̃ωn` ( A∗−jξ ) f̂(ξ) ω̂n` ( A∗−jξ ) dξ, and aj+1−1∑ n=aj L∑ `=1 ∑ j∈Z ∫ Rd f̂(ξ) ˆ̃ωn` ( A∗−jξ ) ∑ s∈Zd\{0} f̂ ( ξ + A∗jB∗−1s ) ω̂n` ( A∗−jξ + B∗−1s )dξ. 76 SHAH AND ABASS From our assumptions on the basic multiwavelet packets ωn` and ω̃ n ` , it is clear that the first of these series converges absolutely. Moreover, we have 2 ∣∣∣ˆ̃ωn` (A∗−jξ) ω̂n` (A∗−jξ + B∗−1s)∣∣∣ ≤ ∣∣∣ˆ̃ωn` (A∗−jξ)∣∣∣2 + ∣∣∣ω̂n` (A∗−jξ + B∗−1s)∣∣∣2. Further, it is easy to verify that the convergence of the second series follows from the convergence of: aj+1−1∑ n=aj L∑ `=1 ∑ j∈Z ∑ s∈Zd\{0} ∫ Rd ∣∣∣f̂(ξ)∣∣∣ ∣∣∣f̂(ξ + A∗jB∗−1s)∣∣∣∣∣∣ˆ̃ωn` (A∗−jξ)∣∣∣2 dξ = ∫ Rd aj+1−1∑ n=aj L∑ `=1  ∑ j∈Z ∑ s∈Zd\{0} |detA|j ∣∣∣f̂(A∗jξ)∣∣∣∣∣∣f̂(A∗jξ + A∗jB∗−1s)∣∣∣   ∣∣∣ˆ̃ωn` (ξ)∣∣∣2 dξ, and from the convergence of a similar series, with ω̃n` replaced by ω n ` . But as s 6= 0, therefore there exists J ∈ Z such that f̂ ( A∗jξ ) f̂ ( A∗jξ + A∗jB∗−1s ) = 0, for all j ≥ J. On the other hand, for each fixed j ∈ Z, and ξ ∈ Rd, the number of s ∈ Zd, for which the above product is nonzero, is less than or equal to C|detA|−j for some constant C. Thus, we have ∑ j∈Z ∑ s∈Zd\{0} |detA|j ∣∣∣f̂(A∗jξ)∣∣∣∣∣∣f̂(A∗jξ + A∗jB∗−1s)∣∣∣ ≤ C ∑ j≤J ∥∥∥f̂∥∥∥2 ∞ χF ( A∗jξ ) , where F is compact in Rd\{0} . Observe that if b′ < |A∗jξ| < b, there exists K > 0, which does not depend on ξ, such that the number of j for which this is nonzero is less than K for every ξ. Hence, the above sum can be estimated from above by CK‖f̂‖2∞ and it proves the convergence of second sum. Hence, we can rearrange the series for P(f,g) to obtain P(f,f) = ∑ α∈Λ(A,B) ∫ Rd f̂(ξ)f̂(ξ + α) ×   1|detB| aj+1−1∑ n=aj L∑ `=1 ∑ (j,m)∈IA,B(α) ˆ̃ωn` ( A∗jξ ) ω̂n` ( A∗j(ξ + α) )dξ. Therefore, it is enough to show that if P(f,g) = 〈f,g〉 for all f,g ∈ D, then the second condition follows. For this, we write P(f,g) = M(f,g) + R(f,g), CHARACTERIZATION OF BIORTHOGONAL MULTIWAVELET 77 with M(f,g) = 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∫ Rd ĝ(ξ)f̂(ξ)  ∑ j∈Z ˆ̃ωn` ( A∗jξ ) ω̂n` (A ∗jξ)   dξ, and R(f,g) = 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∑ α∈Λ(A,B)\{0} ∫ Rd ĝ(ξ)f̂(ξ + α) ∑ (j,m)∈IA,B(α) ˆ̃ωn` ( A∗jξ ) ω̂n` ( A∗j(ξ + α) ) dξ. Now, let us fix, ξ0 ∈ Rd \{0} ,k ∈ Z, and consider f = g = f1, where f1 is defined by f̂1(ξ) = 1 |Hk|1/2 χHk(ξ). Then, M(f1,f1) = 1 |detB||Hk| aj+1−1∑ n=aj L∑ `=1 ∫ Hk ∑ j∈Z ˆ̃ωn` ( A∗jξ ) ω̂n` ( A∗jξ ) dξ, and ∣∣R(f1,f1)∣∣ ≤ 1|detB||Hk| aj+1−1∑ n=aj L∑ `=1 ∑ α∈Λ(A,B) α 6=0 ∑ (j,m)∈IA,B(α) × ∫ Hk∩(α+Hk) ∣∣∣ˆ̃ωn` (A∗jξ) ω̂n` (A∗j(ξ + α))∣∣∣dξ ≤ 1 |detB||Hk|   aj+1−1∑ n=aj L∑ `=1 ∑ α∈Λ(A,B) α6=0 ∑ (j,m)∈IA,B(α) ∫ Hk∩(α+Hk) ∣∣∣ˆ̃ωn` (A∗jξ)∣∣∣2 dξ   1/2 ×   aj+1−1∑ n=aj L∑ `=1 ∑ α∈Λ(A,B) α6=0 ∑ (j,m)∈IA,B(α) ∫ Hk∩(α+Hk) ∣∣ω̂n` (A∗j(ξ + α))∣∣2dξ   1/2 . To estimate R(f1,f1), we observe that if α 6∈ Hk+η, then Hk ∩ (α + Hk) = ∅. Therefore, we may assume that α ∈ Hk+η and α 6= 0. Also, if (j,m) ∈ IA,B(α), then m ∈ A∗j(Hk+η) ∩B∗−1(Zd) and j ≥−k + c1, 78 SHAH AND ABASS where c1 is the largest integer such that Hk+η ∩B∗−1(Zd) = {0} . Therefore, under these observations, we have ∣∣R(f1,f1)∣∣ ≤ 1|detB||Hk|   aj+1−1∑ n=aj L∑ `=1 ∑ j≥−k+c1 ∑ m 6=0 m∈A∗j(Hk+η)∩B ∗−1(Zd) ∫ ξ0+Hk ∣∣∣ˆ̃ωn` (A∗jξ)∣∣∣2 dξ   1/2 ×   aj+1−1∑ n=aj L∑ `=1 ∑ j≥−k+c1 ∑ m 6=0 m∈A∗j(Hk+η)∩B ∗−1(Zd) ∫ ξ0+Hk ∣∣ω̂n` (A∗jξ)∣∣2dξ   1/2 . Now, in order to estimate the first factor in the above product, we observe that 1 |Hk| aj+1−1∑ n=aj L∑ `=1 ∑ j≥−k+c1 ∑ m 6=0 m∈A∗j(Hk+η)∩B∗−1(Zd) ∫ ξ0+Hk ∣∣∣ˆ̃ωn` (A∗jξ)∣∣∣2 dξ ≤ |detA|−k aj+1−1∑ n=aj L∑ `=1 ∑ j≥−k+c1 C|detA|j+k|detA|−j ∫ A∗j(ξ0+Hk) ∣∣∣ˆ̃ωn` (ξ)∣∣∣2 dξ ≤ C ∑ j≥−k+c1 ∫ A∗j(ξ0+Hk) ∣∣∣ˆ̃ωn` (ξ)∣∣∣2 dξ. Here, we have used the fact that the number of points of the lattice B∗−1(Zd), different from the origin and contained in the set A∗j(Hk+η) = Hj+k+η, is smaller than a constant multiple of the volume of this set. Similar estimate holds for the second factor. Since the sets A∗j(ξ0+Hk), j ∈ Z, are pairwise disjoint for sufficiently large |k|, so we may conclude that R(f1,f1) → 0, as k →−∞ by the Lebesgue Dominated Convergence Theorem. Therefore, we have 1 = lim k→−∞ 1 |detB| aj+1−1∑ n=aj L∑ `=1 1 |Hk| ∫ ξ0+Hk ∑ j∈Z ˆ̃ωn` ( A∗jξ ) ω̂n` ( A∗jξ ) dξ = 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∑ j∈Z ˆ̃ωn` ( A∗jξ0 ) ω̂n` (A ∗jξ0), which proves our claim for α = 0. This also shows that M(f,g) = 〈f,g〉, and thus R(f,g) = 0, for f,g ∈D. Now, we choose α0 ∈ Λ(A,B) \{0}, and write R(f,g) = R1(f,g) + R2(f,g), CHARACTERIZATION OF BIORTHOGONAL MULTIWAVELET 79 where R1(f,g) = 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∫ Rd ĝ(ξ)f̂(ξ+α0) ∑ (j,m)∈IA,B(α0) ˆ̃ωn` (A ∗jξ) ω̂n` ( A∗j(ξ + α0) ) dξ, and R2(f,g) = 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∑ α∈Λ(A,B) α 6=0,α0 ∫ Rd ĝ(ξ)f̂(ξ+α) ∑ (j,m)∈IA,B(α) ˆ̃ωn` ( A∗jξ ) ω̂n ` ( A∗j(ξ + α) ) dξ. Let ξ0 ∈ Rd \{0} be a Lebesgue point of differentiability for the functions aj+1−1∑ n=aj L∑ `=1 ∞∑ j=J ∣∣ω̂n` (A∗jξ)∣∣2 and a j+1−1∑ n=aj L∑ `=1 ∞∑ j=J ∣∣∣ˆ̃ωn` (A∗jξ)∣∣∣2 , J ∈ Z. Then, for given k ∈ Z, we define f2 and g2 as follows: f̂2(ξ + α0) = 1 |Hk|1/2 χξ0+Hk(ξ), ĝ2(ξ + α0) = 1 |Hk|1/2 χξ0+Hk(ξ). Using equation (2.3), we obtain lim k→−∞ R1(f2,g2) = 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∑ (j,m)∈IA,B(α0) ˆ̃ωn` ( A∗jξ0 ) ω̂n` ( A∗j(ξ0 + α0) ) . To estimate R2(f2,g2), we observe that ĝ2(ξ)f̂2(ξ + α) 6≡ 0 is only possible when α ∈ α0 + Hk+η. Since α = (A∗)−jm ∈ Λ(A,B) \{0,α0} , there exists J0 ∈ Z such that (A∗)−jm 6∈ α0 + Hη for any m ∈ B∗−1(Zd)\{0} and j ≤ J0. Thus, R2(f2,g2) can be re-written as R2(f2,g2) = 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∞∑ j=J1 ∑ m 6=0, A∗−jm−α0∈Hk+η ∫ Rd ĝ2(ξ)f̂2(ξ + α) ˆ̃ω n ` ( A∗jξ ) ω̂n ` ( A∗j(ξ + α) ) dξ + 1 |detB| aj+1−1∑ n=aj L∑ `=1 J1∑ j=J0 ∑ m 6=0, A∗−jm−α0∈Hk+η ∫ Rd ĝ2(ξ)f̂2(ξ + α) ˆ̃ω n ` ( A∗jξ ) ω̂n ` ( A∗j(ξ + α) ) dξ = R2,1(f2,g2) + R2,2(f2,g2), where J1 ∈ Z. Since R2,2(f2,g2) is now a finite sum, and the number of m’s satisfy- ing the condition A∗−jm−α0 ∈ Hk+η ⊂ Hη may now be estimated independently of k ≤ 0, we have limk→−∞R2,2(f2,g2) = 0 by Lebesgue Dominated Convergence Theorem. To estimate R2,1(f2,g2), we will show that for every ε > 0, there exists 80 SHAH AND ABASS J1 ∈ Z such that |R2,1(f2,g2)| ≤ ε for sufficiently large |k|. In fact, as in the case of R(f1,g1), we have R2,1(f2,g2) ≤ 1 |detB||Hk|   aj+1−1∑ n=aj L∑ `=1 ∑ j≥J1 ∑ m 6=0, A∗−jm−α0∈Hk+η ∫ ξ0+Hk ∣∣∣ˆ̃ωn` (A∗jξ)∣∣∣2 dξ   1/2 ×   aj+1−1∑ n=aj L∑ `=1 ∑ j≥J1 ∑ m 6=0, A∗−jm−α0∈Hk+η ∫ ξ0+Hk ∣∣ω̂n` (A∗jξ)∣∣2 dξ   1/2 . Therefore, it is enough to estimate just one of these factors, namely: 1 |Hk| aj+1−1∑ n=aj L∑ `=1 ∑ j≥J1 ∑ m 6=0, A∗−jm−α0∈Hk+η ∫ ξ0+Hk ∣∣∣ˆ̃ωn` (A∗jξ)∣∣∣2 dξ ≤ 1 |Hk| aj+1−1∑ n=aj L∑ `=1 ∑ j≥J1 ( 1 + C|detA|k+j+η )∫ ξ0+Hk ∣∣∣ˆ̃ωn` (A∗jξ)∣∣∣2 dξ = aj+1−1∑ n=aj L∑ `=1 ∑ j≥J1 1 |Hk| ∫ ξ0+Hk ∣∣∣ˆ̃ωn` (A∗jξ)∣∣∣2 dξ + a j+1−1∑ n=aj L∑ `=1 ∑ j≥J1 ∫ (A∗)j(ξ0+Hk) ∣∣∣ˆ̃ωn` (ξ)∣∣∣2 dξ. Here, we have used the fact that the number of points of the lattice B∗−1(Zd) that are contained in the set A∗j(α0 + Hk+η) = (A ∗)jα0 + Hj+k+η, is smaller than one plus a constant multiple of the volume of this set. Let J1 ∈ Z be such that aj+1−1∑ n=aj L∑ `=1 ∑ j≥J1 ∣∣∣ˆ̃ωn` (A∗jξ0)∣∣∣2 < ε/2. Then, by our choice of ξ0 and equation (2.3), we have lim k→−∞ sup aj+1−1∑ n=aj L∑ `=1 ∑ j≥J1 1 |Hk| ∫ ξ0+Hk ∣∣∣ˆ̃ωn` (A∗jξ)∣∣∣2 dξ < ε/2. Therefore, by virtue of Lebesgue Dominated Convergence Theorem, we get lim k→−∞ sup aj+1−1∑ n=aj L∑ `=1 ∑ j≥J1 ∫ A∗j(ξ0+Hk) ∣∣∣ˆ̃ωn` (ξ)∣∣∣2 dξ = 0. Since the sets A∗j(α0 + Hk+η), j ∈ Z, are pairwise disjoint for sufficiently large |k|, therefore, for every ε > 0, there exist J1 such that lim k→−∞ sup ∣∣R2,1(f2,g2)∣∣ ≤ ε. CHARACTERIZATION OF BIORTHOGONAL MULTIWAVELET 81 Combining these observations with the fact that ε is arbitrary, we obtain 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∑ (j,m)∈IA,B(α0) ˆ̃ωn` ( A∗jξ ) ω̂n` ( A∗j(ξ + α0) ) = 0, for all α0 ∈ Λ(A,B) \{0}. An immediate consequence of the above theorem is the following: Corollary 3.2. Let {ωn` : n ∈ Z +,` = 1, . . . ,L} be the basic multiwavelet packets associated with the scaling vector Φ. Then 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∑ (j,m)∈IA,B(α) ω̂n` ( A∗jξ ) ω̂n` ( A∗j(ξ + A∗−jm) ) = δα,0, (3.4) for a.e. ξ ∈ Rd and for all α ∈ Λ(A,B), if and only if the system F(A,B) given by (2.15) is a normalized tight frame for L2 ( Rd ) . Theorem 3.3. If {ωn` : n ∈ Z +,` = 1, . . . ,L} and {ω̃n` : n ∈ Z +,` = 1, . . . ,L} are Bessel families and have the property that the functions in (3.1) are locally inte- grable. Then, they are biorthogonal if and only if 1 |detB| aj+1−1∑ n=aj L∑ `=1 ∑ (j,m)∈IA,B(α) ˆ̃ωn` ( A∗jξ ) ω̂n` ( A∗j(ξ + A∗−jm) ) = δα,0, (3.5) for a.e. ξ ∈ Rd and for all α ∈ Λ(A,B). Moreover, if ωn` = ω̃ n ` and ‖ω n ` ‖2 = ‖ω̃n` ‖2 = 1 for n ∈ Z +,` = 1, . . . ,L. Then, the system F(A,B) forms an orthonor- mal basis for L2 ( Rd ) . Proof. The proof of this theorem follows from (2.2) and Theorem 3.1. REFERENCES [1] B. Behera, Multiwavelet packets and frame packets of L2 ( Rd ) , Proc. Indian Acad. Sci. 111(4) (2001), 439-463. [2] M. 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