Pairs of Dual Wavelet Frames From Any Two Refinable Functions

Transcription

1 Pairs of Dual Wavelet Frames From Any Two Refinable Functions Ingrid Daubechies Bin Han Abstract Starting from any two compactly supported refinable functions in L 2 (R) with dilation factor d, we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L 2 (R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function φ in L 2 (R), it is possible to construct explicitly easily wavelets that are finite linear combinations of translates φ(d k), that generate a wavelet frame with arbitrarily preassigned number of vanishing moments. We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions. Key words: Dual wavelet frames, wavelet frames, refinable functions, B-spline functions. 2 AMS subject classification: 42C4, 42C15, 41A15. Research was partially supported by grants NSF (DMS , DMS ) AFOSR (F ). Address: Program in Applied Computational Mathematics, Princeton University, Princeton, NJ , USA. ingrid@math.princeton.edu, WWW: icd Research was partially supported by the Natural Sciences Engineering Research Council of Canada (NSERC Canada) under Grant G , by Alberta Innovation Science REE under Grant G Address: Department of Mathematical Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1. bhan@math.ualberta.ca, WWW: bhan 1

2 1 Introduction As a generalization of biorthogonal wavelets, pairs of dual wavelet frames have proved particularly useful in signal denoising many other applications where translation invariance or redundancy is important. By allowing redundancy in a wavelet system, one has much more freedom in the choice of wavelets. It may also be easier to recognize patterns in a redundant transform. From the computational point of view, it is often easier to work with dual wavelet frames that are generated by MRAs. If one works with biorthogonal bases, then the two MRAs have to be linked in special ways [5]. In this paper, we are particularly interested in pairs of dual wavelet frames derived from refinable functions; we shall see that there are then no restrictions on the MRAs. Before proceeding further, let us introduce some notation. Throughout this paper, by d we denote the dilation factor which is an integer with absolute value greater than one. For simplicity, throughout this paper, we further assume that d is a positive dilation factor since all the corresponding results in this paper for a negative dilation factor can be proved almost identically. The inner product, in L 2 (R) is defined to be f, g := f(t)g(t) dt, f, g L 2 (R). R Let {ψ 1,..., ψ r } be a finite set of functions in L 2 (R). We say that {ψ 1,..., ψ r } generates a d-wavelet frame in L 2 (R) if there exist positive constants C 1 C 2 such that C 1 f 2 r f, ψj,k l 2 C 2 f 2 f L 2 (R), (1.1) l=1 j Z k Z where f 2 := f, f ψ l j,k := d j/2 ψ l (d j k), j Z, k Z. In particular, when C 1 = C 2 = 1 in (1.1), we say that {ψ 1,..., ψ r } generates a (normalized) tight d-wavelet frame in L 2 (R). If both {ψ 1,..., ψ r } { ψ 1,..., ψ r } generate d-wavelet frames in L 2 (R) satisfy f, g = r f, ψj,k l ψ j,k, l g l=1 j Z k Z f, g L 2 (R), then we say that {ψ 1,..., ψ r } { ψ 1,..., ψ r } generate a pair of dual d-wavelet frames in L 2 (R). A pair of dual d-wavelet frames is also called a bi-frame in the literature [11]. Consequently, any function f in L 2 (R) has the following wavelet expansions: f = r f, ψj,k l ψ j,k l = l=1 j Z k Z r f, ψ j,k ψ l j,k l l=1 j Z k Z with the series converging absolutely in the L 2 norm. Using the Fourier transform, one can give an explicit characterization for {ψ 1,..., ψ r } { ψ 1,..., ψ r } to generate a pair of dual d-wavelet frames in L 2 (R); see [7, 11]. 2

3 An important property of a wavelet system is its order of vanishing moments. We say that {ψ 1,..., ψ r } has vanishing moments of order n if t k ψ l (t) dt = l = 1,..., r k =,..., n 1. R In this paper, we are particularly interested in obtaining pairs of dual wavelet frames that are derived from pairs of refinable functions with a general dilation factor. Let d be a dilation factor. A function φ is said to be d-refinable if φ = d k Z a k φ(d k), (1.2) where a is a sequence on Z, called the mask for φ. The Fourier series of a sequence a on Z is defined to be â(ξ) := k Z a k e ikξ, ξ R. (1.3) Any mask a for a refinable function in this paper is assumed to be finitely supported with â() = k Z a k = 1. We shall only consider L 2 -solutions φ to (1.2) with a finitely supported mask a; because a is finitely supported, this solution φ is compactly supported, (if it exists) uniquely defined up to normalization by ˆφ(ξ) := j=1 â(d j ξ), ξ R (see [5]), where the Fourier transform is defined to be ˆf(ξ) := f(t)e iξt dt, f L 1 (R). Since â() = 1 ˆφ(ξ) := j=1 â(d j ξ), we always have ˆφ() = 1. R We say that a satisfies the sum rules of order n with respect to the lattice dz if a k k l l =,..., n 1 j Z. (1.4) k dz a k+j (k + j) l = k dz Equivalently, a finitely supported sequence a satisfies the sum rules of order n with respect to the lattice dz if only if (1 + e iξ + + e i(d 1)ξ ) n â(ξ). That is, â(ξ) = (1 + e iξ + + e i(d 1)ξ ) n p(ξ) for some 2π-periodic trigonometric polynomial p. Throughout this paper, we shall use the notation q(ξ) â(ξ) to mean that â(ξ) = q(ξ)p(ξ) for some 2π-periodic trigonometric polynomial p. Let φ φ be two d-refinable functions in L 2 (R) with finitely supported masks a b, respectively. Suppose that a b satisfy the sum rules of orders m n with respect to the lattice dz for some positive integers m n, respectively. For any nonnegative integer N, we show in Section 3 that there exist finitely supported sequences a 1,..., a d, b 1,..., b d such that by defining ψ l = d k Z a l kφ(d k) ψl = d k Z b l k φ(d k), l = 1,..., d, (1.5) 3

4 (or equivalently, in the frequency domain, ψ l (dξ) = âl (ξ) ˆφ(ξ) ψl (dξ) = b l (ξ) ˆ φ(ξ) for l = 1,..., d.) {ψ 1,..., ψ d } { ψ 1,..., ψ d } generate a pair of dual d-wavelet frames in L 2 (R). Moreover, ψ 1, {ψ 2,..., ψ d }, ψ 1 { ψ 2,..., ψ d } have vanishing moments of orders n, n+2n, m+ 2N m, respectively. In addition, if both φ φ are real-valued symmetric d-refinable functions such that the symmetry centers of φ φ differ by a half integer, then the wavelet functions ψ 1,..., ψ d, ψ 1,..., ψ d can be chosen to be real-valued be either symmetric or antisymmetric with a same symmetry center. See Sections 3 4 for more detail. The structure of this paper is as follows. In Section 2, we shall recall a general method for constructing pairs of dual wavelet frames derived from any two refinable functions; we also prove two auxiliary results that will be useful for our main theorems. In Section 3, we shall discuss how to obtain pairs of dual wavelet frames in a concrete constructive way from any two refinable functions. An algorithm for constructing pairs of dual wavelet frames will be presented. Wavelet frames derived from refinable functions will be discussed in Section 3. In Section 4, we shall investigate how to derive pairs of real-valued symmetric dual wavelet frames from any two real-valued symmetric refinable functions. Finally, in Section 5 we give several examples of pairs of dual wavelet frames derived from B-spline functions. A program consisting of a collection of MAPLE routines based on the algorithms constructions of dual wavelet frames in this paper, which comes without warranty, can be downloaded at bhan. The examples in Section 5 are produced by this program. 2 Dual Wavelet Frames of High Vanishing Moments In this section, we shall discuss how to construct dual wavelet frames with high vanishing moments from refinable functions. The following lemma is a direct consequence of results from Cohen Daubechies [3] Villemoes [12]. Lemma 2.1 Let φ L 2 (R) be a d-refinable function with a dilation factor d a finitely supported mask a. Let b be a finitely supported sequence on Z such that ˆb() =. Define a function ψ by ˆψ(dξ) = ˆb(ξ) ˆφ(ξ). Then there exists a positive constant C such that f, ψ j,k 2 C f 2 f L 2 (R), (2.1) j Z k Z where ψ j,k := d j/2 ψ(d j k). Proof: Since φ L 2 (R) is compactly supported, it is well known that there exists a compactly supported function η L 2 (R) in the closure of span{φ( k) : k Z} such that the shifts of η are linearly independent φ = k Z c kη( k) for some finitely supported sequence c (see [1, 1]). By [3, Theorem 5.1], for some α >, η W2 α (R) := {f L 2 (R) : (1 + R ξ 2 ) α ˆf(ξ) 2 dξ < }. Therefore, as a finite linear combination of η( k), φ W2 α (R). So the compactly supported function ψ lies in W2 α (R) ψ(t) dt =. By [3, Theorem 5.1] or [12, Theorem 3.3], there R exists a positive constant C such that (2.1) holds. 4

5 We point out to the reader that Lemma 2.1 has been generalized to the case of the multivariate multiwavelets in [8]. Let M be an s s integer matrix such that all its eigenvalues are greater than one in modulus. Suppose that φ = (φ 1,..., φ r ) T ( L 2 (R s ) ) r is compactly supported ˆφ(M T ξ) = â(ξ) ˆφ(ξ) for some r r matrix â(ξ) of 2π-periodic trigonometric polynomials. Then it was proved in [8] that there exists α > such that R s (1 + ξ 2 ) α φ l (ξ) 2 dξ < (1 + ) α φ l L for all l = 1,..., r. Moreover, for any ψ = (ψ 1,..., ψ r ) T which is defined by ˆψ(M T ξ) = ˆb(ξ) ˆφ(ξ) for some r r matrix ˆb(ξ) of 2π-periodic trigonometric polynomials, if ψ l (t) dt = for all l = 1,..., r, then there exists a positive constant C such that R s r l=1 j Z k Z s f, ψ l j,k 2 C f 2 for all f L 2 (R s ), where ψ l j,k := det M j/2 ψ l (M j k). Pairs of dual wavelet frames can be obtained from refinable functions by the following result. Theorem 2.2 Let φ φ be two d-refinable functions in L 2 (R) with the dilation factor d finitely supported masks a b, respectively. Suppose that there are finitely supported sequences a 1,..., a r, b 1,..., b r a 2π-periodic trigonometric polynomial Θ such that Θ() = 1, â l () = b l () = l = 1,..., r (2.2) â(ξ) â 1 (ξ) â r (ξ) Θ(dξ)ˆb(ξ) Θ(ξ) â(ξ + 2π) d â1 (ξ + 2π) â d r (ξ + 2π) d b 1 (ξ) =.. (2.3) â(ξ + 2π(d 1) ) â d 1 (ξ + 2π(d 1) ) â d r (ξ + 2π(d 1) ) b r (ξ) d Define wavelet functions ψ 1,..., ψ r, ψ 1,..., ψ r as follows: ψ l = d a l kφ(d k) ψl = d b l φ(d k k), l = 1,..., r. (2.4) k Z k Z Then {ψ 1,..., ψ r } { ψ 1,..., ψ r } generate a pair of dual d-wavelet frames in L 2 (R). Proof: By Lemma 2.1, there exists a positive constant C such that r [ f, ψ lj,k 2 + f, ψ lj,k ] 2 C f 2 f L 2 (R), l=1 j Z k Z where ψ l j,k = d j/2 ψ l (d j k) ψ l j,k = d j/2 ψl (d j k). Define η by ˆη(ξ) := Θ(ξ) ˆ φ(ξ). By (2.3) a simple calculation, for every j Z, one has r l=1 k Z f, ψ l j,k ψ l j,k, g = k Z f, φ j+1,k η j+1,k, g k Z f, φ j,k η j,k, g, f, g L 2 (R). The rest of the proof follows directly from Daubechies, Han, Ron Shen [6, Corollary 5.3]. It is easy to see that (2.3) can be rewritten as follows: â(ξ + 2πj/d)ˆb(ξ)Θ(dξ) + r â l (ξ + 2πj/d) b l (ξ) = δ j Θ(ξ), j =,..., d 1, (2.5) l=1 5

6 where δ denotes the Dirac sequence such that δ = 1 δ j = for all j Z\{}. Tight wavelet frames dual wavelet frames have been investigated in [2, 6] for the case d = 2. In this paper, we shall give a systematic study of dual wavelet frames with a general dilation factor. We mention that Theorem 2.2 can also be verified using the characterization of dual wavelet frames in [7, 11]. In order to prove the main results in this paper, the following result is crucial in our construction of dual wavelet frames from refinable functions. Lemma 2.3 Let d be a dilation factor. Let A B be two finitely supported sequences on Z such that Â() = ˆB(). Let Θ(ξ) := k Z Θ ke ikξ be a 2π-periodic trigonometric polynomial we denote by Θ (j) the jth derivative of the trigonometric polynomial Θ. Then for any positive integer n, Θ() = 1 (1 e iξ ) n [Θ(ξ)Â(ξ) Θ(dξ) ˆB(ξ)] (2.6) if only if i l Θ (l) () = k Z Θ k k l = λ l, l =,..., n 1, (2.7) where λ = 1 λ l (l N) are uniquely determined by the following recursive formula: [ 1 l 1 l! λ l = (d l 1)Â() A k k l j d ] j B k k l j λ j, l N. (2.8) j!(l j)! j= k Z k Z Consequently, for any positive integer n there exists a 2π-periodic trigonometric polynomial Θ such that (2.6) holds. In particular, if A B are finitely supported real-valued sequences on Z such that A s k = A k B s k = B k for all k Z for some integers s s such that c = s s is an integer, then Θ can be chosen to be a 2π-periodic trigonometric polynomial with d 1 real coefficients such that Θ(ξ) = e icξ Θ(ξ); that is, Θ c k = Θ k for all k Z. Proof: By the Leibniz differentiation formula Â() = ˆB(), (2.6) is equivalent to Θ() = 1 (d l 1)Â()Θ(l) () = l 1 j= l! j!(l j)! Θ(j) ()[Â(l j) () d j ˆB(l j) ()], l = 1,..., n 1. It follows directly from the above equations that (2.6) is equivalent to (2.7). When A B are real-valued sequences, by (2.8), it is clear that we can choose Θ to be a 2πperiodic trigonometric polynomial with real coefficients. Note that A s k = A k for all k Z if only if Â(ξ) = e isξ Â(ξ). Let θ be a 2π-periodic trigonometric polynomial with real coefficients such that θ() = 1 (1 e iξ ) n [θ(ξ)â(ξ) θ(dξ) ˆB(ξ)]. Set Θ(ξ) := [θ(ξ) + e icξ θ(ξ)]/2. Since e icξ θ(ξ)â(ξ) e idcξ θ(dξ) ˆB(ξ) = e i(s+c)ξ [θ(ξ)e isξ Â(ξ) θ(dξ)e i sξ ˆB(ξ)] it is easy to check that (2.6) holds Θ(ξ) = e icξ Θ(ξ). i(s+c)ξ = e [θ(ξ)â(ξ) θ(dξ) ˆB(ξ)], Lemma 2.3 was also given in [9, Lemma 3.2] which generalized the special case A(ξ) 1 in an earlier version of this paper. The following result is important for us to construct pairs of symmetric dual wavelet frames from symmetric refinable functions. 6

7 Proposition 2.4 Let d be a positive integer such that d 2. For any positive integer N any integer s, there exist 2π-periodic trigonometric polynomials c 1,..., c d with real coefficients such that (a) det C(), where C(ξ) is the matrix defined by c 1 (ξ) c 2 (ξ) c d (ξ) c 1 (ξ + 2π d C(ξ) := ) c2 (ξ + 2π) d cd (ξ + 2π) d ; (2.9) c 1 (ξ + 2π(d 1) ) c 2 (ξ + 2π(d 1) ) c d (ξ + 2π(d 1) ) d d d (b) (1 e iξ ) 2N c j (ξ) for all j = 2,..., d; (c) The 2π-periodic trigonometric polynomials c 1,..., c d have real coefficients c j (ξ) = e is ξ c j (ξ), j = 1,..., N d,s c j (ξ) = e is ξ c j (ξ), j = N d,s + 1,..., d, where the integer N d,s is defined to be { d+2 2 N d,s :=, when s is an even integer, d+1, when s 2 is an odd integer, (2.1) with x denoting the greatest integer which is no greater than x. In particular, when d = 2 s is even, one can choose c 1 (ξ) = e is ξ/2 c 2 (ξ) = e is ξ/2 (1 cos ξ) N. When d = 2 s is odd, one can choose c 1 (ξ) = e i(s 1)ξ/2 (1 + e iξ ) c 2 (ξ) = e i(s 1 2N)ξ/2 (1 e iξ ) 2N+1. Proof: Observe that a 2π-periodic trigonometric polynomial c with real coefficients satisfies c(ξ) = c(ξ) if only if c(ξ) = p(cos ξ) for some polynomial p with real coefficients. Let m := d+2. The main idea in the following proof is that we divide the set {2πj/d : j = 2,..., d 1} into three subsets I 1, I 2 I 3, where I 1 := {jπ : j =,..., 2m d 1}, I 2 := {2πj/d : j = 1,..., d m} I 3 := {2π 2πj/d : j = 1,..., d m}. It is not difficult to see that there exist polynomials p,..., p m 1 with real coefficients such that p j ( cos(2πk/d) ) = δj k, j, k =,..., m 1 When s is even, we define p (l) j (1) = l =,..., N 1 j = 1,..., m 1. c j (ξ) = e is ξ/2 p j 1 (cos ξ), j = 1,..., m c j+m (ξ) = e is ξ/2 (e iξ e iξ )p j (cos ξ), j = 1,..., d m. 7

8 In order to prove that det C(), it suffices to prove it for the case s =. When s =, by the choice of the polynomials p,..., p m 1, it is easy to see that after performing suitable permutations on rows columns of the matrix C(), the matrix C() becomes I 2m d I d m E E with E := diag ( e i2π/d, e i4π/d,..., e i2(d m)π/d). I d m E E Evidently, det C() = 4 d m d m 2πj j=1 sin. d When s is odd d is odd, we have 2m 1 = d we define c j (ξ) = e i(s 1)ξ/2 (1 + e iξ )p j 1 (cos ξ), j = 1,..., m c j+m (ξ) = e i(s 1)ξ/2 (1 e iξ )p j (cos ξ), j = 1,..., m 1. When s is odd d is even, we have 2m 2 = d we define c j (ξ) = e i(s 1)ξ/2 (1 + e iξ )p j 1 (cos ξ), j = 1,..., m 1 c j+m 1 (ξ) = e i(s 1)ξ/2 (1 e iξ )p j (cos ξ), j = 1,..., m 1. In order to prove that det C(), it is easy to see that it suffices to prove it for the case s = 1. When s = 1, by the choice of the polynomials p,..., p m 1, it is easy to see that after performing suitable permutations on rows columns of the matrix C(), the matrix C() becomes I 2m d I d m + E I d m E with E := diag ( e i2π/d, e i4π/d,..., e i2(d m)π/d). I d m + E I d m E Evidently, det C() = 4 d m d m 2πj j=1 sin d. All other claims can be easily verified. Proposition 2.4 still holds if we take p = 1 in the above proof. We observe that the degrees of the 2π-periodic trigonometric polynomials c 1,..., c d constructed in the proof of Proposition 2.4 can be made even smaller. 3 Construction of Dual Wavelet Frames In this section, we shall discuss how to construct pairs of dual d-wavelet frames from any two d-refinable functions. Let φ φ be two d-refinable functions in L 2 (R) with finitely supported masks a b, respectively. Suppose that a b satisfy the sum rules of orders m n with respect to the lattice dz for some positive integers m n, respectively; in other words, (1 + e iξ + + e i(d 1)ξ ) m â(ξ) (1 + e iξ + + e i(d 1)ξ ) n ˆb(ξ). In order to construct a pair of dual d-wavelet frames by Theorem 2.2, we need to construct finitely supported sequences a 1,..., a r, b 1,..., b r on Z a 2π-periodic trigonometric polynomial 8

9 Θ such that (2.2) (2.3) are satisfied. In this section, let us consider the special case r = d. When r = d, the relation in (2.3) can be rewritten as follows: â 1 (ξ) â 2 (ξ) â d (ξ) b â 1 (ξ + 2π) d â2 (ξ + 2π) â d d (ξ + 2π) 1 (ξ) Θ(ξ) Θ(dξ)â(ξ)ˆb(ξ) d b 2 (ξ) = Θ(dξ)â(ξ + 2π)ˆb(ξ) d.. â 1 (ξ + 2π(d 1) ) â d 2 (ξ + 2π(d 1) ) â d d (ξ + 2π(d 1) ) b d d (ξ) Θ(dξ)â(ξ + 2(d 1)π )ˆb(ξ) d (3.1) Define the wavelet functions ψ 1,..., ψ d, ψ 1,..., ψ d as in (1.5). Since ˆφ() = 1, it is easy to see that {ψ 1,..., ψ d } has vanishing moments of order n if only if (1 e iξ ) n â l (ξ) for all l = 1,..., d. So, in order to achieve high vanishing moments, it is necessary natural to require that â l (ξ) = (1 e iξ ) n g(ξ)c l (ξ), l = 1,..., d, (3.2) where g, c l, l = 1,..., d, are 2π-periodic trigonometric polynomials with g being a certain common divisor of all the 2π-periodic trigonometric polynomials âl, l = 1,..., d. Consequently, we have â 1 (ξ) â 2 (ξ) â d (ξ) â 1 (ξ + 2π) d â2 (ξ + 2π) â d d (ξ + 2π) d = D(ξ)C(ξ) â 1 (ξ + 2π(d 1) ) â d 2 (ξ + 2π(d 1) ) â d d (ξ + 2π(d 1) ) d with the matrix C(ξ) being defined in (2.9) (1 e iξ ) n g(ξ) D(ξ) := Denote (1 e i(ξ+ 2π d ) ) n g(ξ + 2π d )... f 1 (ξ) f 2 (ξ). := f d (ξ) (1 e 2π(d 1) i(ξ+ d ) ) n g(ξ + 2π(d 1) d ). h(ξ) := det C(ξ) (3.3) Θ(ξ) Θ(dξ)â(ξ)ˆb(ξ) (1 e iξ ) n g(ξ) h(ξ) Θ(dξ)â(ξ+2π/d)ˆb(ξ) (1 e i(ξ+2π/d) ) n g(ξ+2π/d) h(ξ). Θ(dξ)â(ξ+2π(d 1)/d)ˆb(ξ) (1 e i(ξ+2π(d 1)/d) ) n g(ξ+2π(d 1)/d) h(ξ). (3.4) Now, the equation in (3.1) can be rewritten as follows: b 1 (ξ) Θ(ξ) Θ(dξ)â(ξ)ˆb(ξ) f b 2 (ξ). = Θ(dξ)â(ξ + [C(ξ)] 1 [D(ξ)] 1 2π)ˆb(ξ) 1 (ξ) d. = adjc(ξ) f 2 (ξ)., (3.5) b d (ξ) Θ(dξ)â(ξ + 2(d 1)π )ˆb(ξ) f d (ξ) d 9

10 where adjc(ξ) denotes the adjacent matrix of C(ξ), that is, C(ξ) 1 = [det C(ξ)] 1 adjc(ξ). Clearly, all the entries in adjc(ξ) are 2π-periodic trigonometric polynomials since all the entries of C(ξ) are 2π-periodic trigonometric polynomials. Now, according to Theorem 2.2, the challenging question that remains is to choose an appropriate 2π-periodic trigonometric polynomial Θ with Θ() = 1 such that f 1,..., f d are 2π-periodic trigonometric polynomials b l () = for all l = 1,..., d. We have the following result on pairs of dual wavelet frames. Theorem 3.1 Let φ L 2 (R) φ L 2 (R) be two arbitrary d-refinable functions with dilation factor d finitely supported masks a b, respectively. Suppose that a b satisfy the sum rules of orders m n with respect to the lattice dz for some positive integers m n, respectively. Construct a 2π-periodic trigonometric polynomial Θ by Lemma 2.3 such that Θ() = 1, (1 e iξ ) n+m [Θ(ξ) Θ(dξ)â(ξ)ˆb(ξ)]. (3.6) Define the finitely supported sequences a 1,..., a d, b 1,..., b d on Z as in (3.2) (3.5) by taking g(ξ) = 1 c l (ξ) = e i(l 1)ξ, l = 1,..., d. Then {ψ 1,..., ψ d } { ψ 1,..., ψ d }, which are defined in (1.5), generate a pair of dual d-wavelet frames in L 2 (R). Moreover, {ψ 1,..., ψ d } { ψ 1,..., ψ d } have vanishing moments of orders n m, respectively. Note that ψ l = ψ 1 ( (l 1)/d) for l = 1,..., d. Proof: Since the mask b satisfies the sum rules of order n with respect to the lattice dz, we have (1+e iξ + +e i(d 1)ξ ) n ˆb(ξ). Consequently, (1 e i(ξ+2πj/d) ) n ˆb(ξ) for all j = 1,..., d 1. By c l (ξ) = e i(l 1)ξ, l = 1,..., d, we observe that h(ξ) := det C(ξ) is a monomial since C(ξ)C(ξ) T = di d, where the matrix C(ξ) is defined in (2.9). Now by the fact that g(ξ) = 1 h(ξ) is a monomial, it is straightforward to see that f 2,..., f d are 2π-periodic trigonometric polynomials. Since (1 + e iξ + + e i(d 1)ξ ) m â(ξ), we have (1 e iξ ) m â(ξ + 2πj/d) for all j = 1,..., d 1 therefore, (1 e iξ ) m f j (ξ) for all j = 2,..., d. On the other h, since g(ξ) = 1 h(ξ) is a monomial, it follows directly from (3.6) that f 1 is a 2π-periodic trigonometric polynomial (1 e iξ ) m f 1 (ξ). The proof is completed by our discussion before this theorem. We point out that Theorem 3.1 holds for general 2π-periodic trigonometric polynomials c 1,..., c d provided that det C(ξ) is a monomial, where C(ξ) is defined in (2.9). For a 2π-periodic trigonometric polynomial p, we define Z(p, ξ ) := sup{l N {} : (e iξ e iξ ) l p(ξ)} = inf { l N {} : p (l) (ξ ) }. (3.7) That is, Z(p, ξ ) denotes the multiplicity of the zeros of p(ξ) at the point ξ = ξ. Now we can generalize Theorem 3.1 the following is the main result in this section. Theorem 3.2 Let φ φ be two d-refinable functions in L 2 (R) with the dilation factor d finitely supported masks a b, respectively. Suppose that a b satisfy the sum rules of orders m n with respect to the lattice dz for some positive integers m n, respectively; that is, (1+e iξ + +e i(d 1)ξ ) m â(ξ) (1+e iξ + +e i(d 1)ξ ) n ˆb(ξ). Let g, c 1,..., c d be 2π-periodic trigonometric polynomials. Define h(ξ) := det C(ξ), where the matrix C(ξ) is defined in (2.9). Then there exists a 2π-periodic trigonometric polynomial Θ such that 1

11 (a) Θ() = 1, (b) All f l, l = 1,..., d, which are defined in (3.4), are 2π-periodic trigonometric polynomials, (c) (1 e iξ ) m f l (ξ) for all l = 1,..., d, if only if (1 + e iξ + + e i(d 1)ξ ) n+z(g,)+z(h,) ˆb(ξ) (3.8) Z(â, 2πj/d) Z(g, 2πj/d) Z(h, ) m j = 1,..., d 1. (3.9) (For example, when g(ξ) 1 h(), then (3.8) (3.9) are automatically satisfied.) For any 2π-periodic trigonometric polynomial Θ such that (a), (b) (c) are satisfied (such Θ can be easily obtained by solving a system of linear equations which are induced by the three conditions in (a), (b) (c) by long division), let a 1,..., a d, b 1,..., b d be defined in (3.2) (3.5). Define the wavelet functions ψ 1,..., ψ d, ψ 1,..., ψ d as in (1.5). Then {ψ 1,..., ψ d } { ψ 1,..., ψ d } generate a pair of dual d-wavelet frames with compact support in L 2 (R). Moreover, {ψ 1,..., ψ d } { ψ 1,..., ψ d } have vanishing moments of orders n m, respectively. Proof: Sufficiency: For simplicity of the presentation, let us assume here that g(ξ) 1 h() ; the complete proof for the general case can be found in the Appendix. Since h(), we define θ 1 (ξ) := e iξ(d 1)/2 h(ξ/d)/[h()] 2. By the fact h(ξ + 2π/d) = ( 1) d 1 h(ξ), we have θ 1 (ξ + 2π) = e i(ξ+2π)(d 1)/2 h(ξ/d + 2π/d)/[h()] 2 = θ 1 (ξ) therefore, θ 1 is a 2π-periodic trigonometric polynomial. Since h()θ 1 () = h()θ 1 ()â()ˆb() = 1, by Lemma 2.3, there exists a 2π-periodic trigonometric polynomial θ 2 such that θ 2 () = 1 (1 e iξ ) n+m θ 2 (ξ)[h(ξ)θ 1 (ξ)] θ 2 (dξ)[h(dξ)θ 1 (dξ)â(ξ)ˆb(ξ)]. (3.1) Now we take Θ(ξ) := θ 2 (ξ)h(ξ)θ 1 (ξ). Obviously, Θ() = 1. Noting that (1 + e iξ + + e i(d 1)ξ ) m â(ξ) (1+e iξ + +e i(d 1)ξ ) n ˆb(ξ), by a simple calculation (see the Appendix for more detail), we can verify that indeed all f l are 2π-periodic trigonometric polynomials (1 e iξ ) m f l (ξ) for all l = 1,..., d. Necessity: Since all f l are 2π-periodic trigonometric polynomials (1 e iξ ) m f l (ξ) for all l = 2,..., d, we have Z(f j+1, ) m Z(f j+1, 2πj/d) for all j = 1,..., d 1. By the definition of f l in (3.4) Θ() = ˆb() = 1, we deduce Z(â, 2πj/d) Z(g, 2πj/d) Z(h, ) = Z(f j+1, ) m j = 1,..., d 1. So, (3.9) must hold. Similarly, by the definition of f l in (3.4) Θ() = â() = 1, we have Z(ˆb, 2πj/d) n Z(g, ) Z(h, ) = Z(f j+1, 2πj/d) j = 1,..., d 1 which is equivalent to (3.8). B-spline functions are of great interest in many applications. The B-spline function of order m (m N), denoted by B m throughout this paper, can be obtained via the following recursive formula: B 1 = χ [,1], the characteristic function of the interval [, 1], B m (x) := 1 B m 1 (x t) dt, x R, m = 2, 3,.... (3.11) 11

12 The B-spline function B m C m 2 (R) is a function of piecewise polynomials of degree less than m, vanishes outside the interval [, m] is symmetric about the point x = m/2 (that is, B m (m x) = B m (x) for all x R). It is well known that the B-spline function B m is a d-refinable function satisfying ( ) 1 + e B iξ + + e i(d 1)ξ m m (dξ) = Bm (ξ). d Now we have the following result on wavelet frames which is a direct consequence of Theorem 3.2. Corollary 3.3 Let φ L 2 (R) be a d-refinable function with the dilation factor d a finitely supported mask a. Choose 2π-periodic trigonometric polynomials g, c 1,..., c d such that Z(â, 2πj/d) > Z(g, 2πj/d) + Z(h, ) j = 1,..., d 1, (3.12) where h(ξ) := det C(ξ) the matrix C(ξ) is defined in (2.9). For any positive integer n, define the wavelet functions ψ 1,..., ψ d by ψ l (dξ) = (1 e iξ ) n g(ξ)c l (ξ) ˆφ(ξ), l = 1,..., d. (3.13) Then {ψ 1,..., ψ d } generates a d-wavelet frame in L 2 (R) has vanishing moments of order n. Moreover, there exist compactly supported functions ψ 1,..., ψ d, which can be derived explicitly from any d-refinable function in L 2 (R) whose mask is finitely supported satisfies the sum rules of order n + Z(g, ) + Z(h, ) with respect to the lattice dz, such that {ψ 1,..., ψ d } { ψ 1,..., ψ d } generate a pair of dual d-wavelet frames in L 2 (R). Proof: Take φ to be any d-refinable function in L 2 (R) whose mask is finitely supported satisfies the sum rules of order n + Z(g, ) + Z(h, ) with respect to the lattice dz. For example, we can take φ = B n+z(g,)+z(h,) to be the B-spline function of order n + Z(g, ) + Z(h, ) which is defined in (3.11). Observe that (3.12) is equivalent to the condition in (3.9) with m = 1. Now Corollary 3.3 follows directly from Theorem 3.2. If in Corollary 3.3 we choose c l (ξ) = e i(l 1)ξ, l = 1,..., d, then we have the following result. Corollary 3.4 Let φ be a d-refinable function in L 2 (R) with the dilation factor d a finitely supported mask a. For any 2π-periodic trigonometric polynomial g such that g() = Z(â, 2πj/d) > Z(g, 2πj/d) j = 1,..., d 1, (3.14) define a wavelet function ψ by ˆψ(ξ) = g(ξ) ˆφ(ξ). Then ψ generates a d-wavelet frame in L 2 (R); that is, {ψ j,k : j, k Z} is a frame in L 2 (R), where ψ j,k := d j/2 ψ(d j k). Moreover, there exist compactly supported functions ψ 1,..., ψ d with arbitrary smoothness such that {ψ(d ), ψ(d 1),..., ψ(d d + 1)} { ψ 1, ψ 2,..., ψ d } generate a pair of dual d-wavelet frames in L 2 (R). Proof: Let c l (ξ) = e i(l 1)ξ, l = 1,..., d. Then it is easy to see that h() = det C() since C(ξ)C(ξ) T = di d, where the matrix C(ξ) is defined in (2.9). Clearly, by (3.14), (3.12) holds g()c l () = for all l = 1,..., d. Therefore, by Corollary 3.3, {ψ 1,..., ψ d }, which is defined in 12

13 (3.13) with n =, generates a d-wavelet frame. It is easy to check that ψ l = d ψ(d l + 1) for l = 1,..., d. Now the claim follows from the fact that {ψ 1,..., ψ d } { d 1/2 ψ} generate the same d-wavelet frame in L 2 (R). Let us make some remarks here for the above results. Through the proof of Theorem 3.2 in the Appendix the following argument, we shall see that the quantities Z(âl, 2πj/d), j =,..., d 1, which denote the multiplicity of zeros of the trigonometric polynomials âl (ξ) at ξ = 2πj/d, play a critical role in the construction of pairs of dual wavelet frames via Theorem 2.2. In the following, we shall see that (3.14) in Corollary 3.4 is not only a sufficient condition for having an MRA wavelet frame, but also a necessary condition. More precisely, (3.14) must hold if via Theorem 2.2 there exist compactly supported functions ψ 1,..., ψ d such that {ψ(d ), ψ(d 1),..., ψ(d d+1)}, which is given in Corollary 3.4, { ψ 1, ψ 2,..., ψ d } generate a pair of dual d-wavelet frames in L 2 (R). To see this point, by Theorem 2.2, (2.5) must hold with Θ, â, ˆb, âl, b l, l = 1,..., r being some 2π-periodic trigonometric polynomials such that (2.2) holds. In particular, it follows from (2.5) that r â(ξ + 2πj/d)ˆb(ξ)Θ(dξ) = â l (ξ + 2πj/d) b l (ξ) j = 1,..., d 1. (3.15) Therefore, by Θ() = ˆb() = 1, it follows from (3.15) that we must have l=1 Z(â, 2πj/d) min{z(âl, 2πj/d) + Z( b l, ) : l = 1,..., r} j = 1,..., d 1. (3.16) Similarly, by Θ() = â() = 1, it follows from (3.15) that Z(ˆb, 2πj/d) min{z(âl, ) + Z( b l, 2πj/d) : l = 1,..., r} j = 1,..., d 1. (3.17) By our choice of â l (ξ) = (1 e iξ ) n g(ξ)c l (ξ) in (3.2) r = d, we have Z(âl, 2πj/d) = Z(g, 2πj/d) + Z(c l, 2πj/d) for all j = 1,..., d 1 Z(âl, ) = n + Z(g, ) + Z(c l, ) for all l = 1,..., d. Consequently, we conclude from (3.16) that Z(â, 2πj/d) Z(g, 2πj/d) min{z(c l, 2πj/d) : l = 1,..., d} min{z( b l, ) : l = 1,..., d} j = 1,..., d 1, (3.18) which is quite similar to (3.9). Similarly, it follows from (3.17) that That is, Z(ˆb, 2πj/d) n + Z(g, ) + min{z(c l, ) : l = 1,..., d} j = 1,..., d 1. which is quite similar to (3.8). (1 + e iξ + + e i(d 1)ξ ) n+z(g,)+min{z(cl,) : l=1,...,d} ˆb(ξ), (3.19) By Theorem 2.2, it is necessary that b l () = therefore, min{z( b l, ) : l = 1,..., d} >. Since in Corollary 3.4 we set c l (ξ) = e i(l 1)ξ, l = 1,..., d, we deduce from (3.18) that Z(â, 2πj/d) > Z(g, 2πj/d) for all j = 1,..., d 1. So, (3.14) must be a necessary condition in Corollary 3.4. Let us consider the following simple example. 13

14 Example 3.5 Let B m be the B-spline function of order m defined in (3.11). Then B m is d- refinable with mask â(ξ) = d m (1 + e iξ + + e i(d 1)ξ ) m for any dilation factor d 2. For any positive integer n, define n ψ = ( 1) k n! k!(n k)! B m( k). k= (That is, ˆψ(ξ) = (1 e iξ ) n Bm (ξ).) Then ψ has vanishing moments of order n. We apply Corollary 3.4 with the special choice g(ξ) = (1 e iξ ) n ; note that g does not depend on d g(ξ) for all ξ (, 2π). We then conclude that ψ generates a d-wavelet frame in L 2 (R) for any dilation factor d 2. Finally, we demonstrate that by appropriately choosing the 2π-periodic trigonometric polynomials c l, l = 1,..., d, the set {ψ 2,..., ψ d } of wavelet functions can have vanishing moments of arbitrary order. Corollary 3.6 Let φ φ be two d-refinable functions in L 2 (R) with the dilation factor d finitely supported masks a b, respectively. Suppose that a b satisfy the sum rules of orders m n with respect to the lattice dz for some positive integers m n, respectively. Let N be an arbitrary nonnegative integer. Then one can construct finitely supported sequences a 1,..., a d, b 1,..., b d such that by defining the functions ψ 1,..., ψ d, ψ 1,..., ψ d as in (1.5), (a) {ψ 1,..., ψ d } { ψ 1,..., ψ d } generate a pair of dual d-wavelet frames in L 2 (R); (b) ψ 1, {ψ 2,..., ψ d }, ψ 1 { ψ 2,..., ψ d } have vanishing moments of orders n, n + 2N, m + 2N m, respectively. Proof: Let c 1,..., c d be the 2π-periodic trigonometric polynomials obtained in Proposition 2.4 with s =. By Proposition 2.4, h() = det C(). Take g(ξ) = 1. The rest of the claim can be verified similarly as in the proof of Theorem 3.1 with the modification: replace the factor (1 e iξ ) n+m in (3.1) by (1 e iξ ) n+m+2n. When φ φ are real-valued symmetric d-refinable functions, in this section, we didn t discuss whether one can obtain a pair of real-valued symmetric dual d-wavelet frames from φ φ. In the next section, we shall address such an issue in detail. 4 Real-valued Symmetric Dual Wavelet Frames Given two real-valued symmetric d-refinable functions in L 2 (R), it is of interest to construct from such two d-refinable functions pairs of dual d-wavelet frames which are also real-valued symmetric. In this section, we shall discuss in detail how to obtain pairs of real-valued symmetric dual wavelet frames from two real-valued symmetric refinable functions. Proposition 4.1 Let a, a 1,..., a r, b, b 1,..., b r be finitely supported sequences on Z let Θ be a 2π-periodic trigonometric polynomial such that (2.2) (2.5) are satisfied. Suppose that 14

15 a, b a 1,..., a r are sequences of real numbers. Define new sequences b 1,..., b r a new 2π-periodic trigonometric polynomial Θ as follows: Θ(ξ) := [Θ(ξ) + Θ( ξ)]/2 b l (ξ) := [ b l (ξ) + b l ( ξ)]/2, l = 1,..., r. Then Θ is a 2π-periodic trigonometric polynomial with real coefficients all b 1,..., b r sequences of real numbers satisfying are Θ() = 1, â l () = bl () =, l = 1,..., r (4.1) â(ξ + 2πj/d)ˆb(ξ) Θ(dξ) + r â l (ξ + 2πj/d) bl (ξ) = δ j Θ(ξ), j =,..., d 1. (4.2) l=1 Proof: Note that a is a sequence of real numbers if only if â( ξ) = â(ξ). Since a, a 1,..., a r are sequences of real numbers, taking the complex conjugate on both sides of (2.5) replacing ξ by ξ, we deduce from (2.5) that â(ξ 2πj/d) ˆb( ξ) Θ( dξ) + r â l (ξ 2πj/d) b l ( ξ) = δ j Θ( ξ), j =,..., d 1. l=1 Since b is a sequence of real numbers, we have ˆb( ξ) = ˆb(ξ). Therefore, â(ξ + 2πj/d)ˆb(ξ)Θ( dξ) + r â l (ξ + 2πj/d) b l ( ξ) = δ j Θ( ξ), j =,..., d 1. l=1 Equation (4.2) can be easily verified by adding the above identity to (2.5). When φ φ are real-valued refinable functions in L 2 (R), by Theorem 3.2 Proposition 4.1, we can always obtain pairs of real-valued dual wavelet frames. Proposition 4.2 Let a, a 1,..., a r, b, b 1,..., b r be finitely supported sequences on Z such that for some integers s s such that c = s s d 1 a s k = a k b s k = b k k Z (4.3) is an integer, a l s l k = ε l a l k k Z l = 1,..., r (4.4) for some ε l { 1, 1} some integers s l, l = 1,..., r, such that s s l d is an integer for all l = 1,..., r. Suppose that (2.2) (2.5) are satisfied with a 2π-periodic trigonometric polynomial Θ. Define new sequences b 1,..., b r a new 2π-periodic trigonometric polynomial Θ as follows: Θ(ξ) := [Θ(ξ) + Θ(ξ)e icξ ]/2 b l (ξ) := [ b l (ξ) + ε l bl (ξ)e i(s l+c)ξ ]/2, l = 1,..., r. Then both (4.1) (4.2) are satisfied. Moreover, Θ(ξ) = e icξ Θ(ξ) bl sl +c k = ε l bl k k Z l = 1,..., r. (4.5) 15

16 Proof: Note that a l s l k = ε la l k for all k Z if only if âl (ξ) = ε l e is lξ â l (ξ). Take the complex conjugate on (2.5), we have â(ξ + 2πj/d)ˆb(ξ) Θ(dξ) + r â l (ξ + 2πj/d) b l (ξ) = δ j Θ(ξ), j =,..., d 1. l=1 Note that â(ξ) = e isξ â(ξ) ˆb(ξ) = e i sξˆb(ξ). Since â l (ξ) = ε l e is lξ â l (ξ) for l = 1,..., r, the above equation becomes e is(ξ+2πj/d) â(ξ + 2πj/d)e i sξˆb(ξ)θ(dξ) + r ε l e is l(ξ+2πj/d)â l (ξ + 2πj/d) b l (ξ) = δ j Θ(ξ), (4.6) l=1 for j =,..., d 1. By assumption, s s = (d 1)c s s l dz for all l = 1,..., r. Multiplying the factor e i(s2πj/d cξ) with both sides of the equation (4.6), we have that for j =,..., d 1, δ j e icξ Θ(ξ) = δ j e i(s2πj/d cξ) Θ(ξ) = â(ξ + 2πj/d)ˆb(ξ)e i( s s c)ξ Θ(dξ) + = â(ξ + 2πj/d)ˆb(ξ)e idcξ Θ(dξ) + r ε l e i(s sl)2πj/d â l (ξ + 2πj/d)e i(s l+c)ξ bl (ξ) l=1 r â l (ξ + 2πj/d)ε l e i(s l+c)ξˆbl (ξ). l=1 Equation (4.2) can be verified by adding the above identity to (2.5). All other claims can be easily checked by computation. Let φ φ be two d-refinable functions with finitely supported masks a b, respectively. Then (4.3) implies that φ = φ( s ) φ s = φ( ). Define ψ d 1 d 1 l (dξ) := âl (ξ) ˆφ(ξ) ψ l (dξ) := bl (ξ) ˆ φ(ξ) for l = 1,..., r. Then (4.4) (4.5) in Proposition 4.2 imply that ψ l = ε l ψ l ( (d 1)sl + s d(d 1) ) ( (d 1)sl + s ) ψl = ε l ψl, l = 1,..., r. d(d 1) Now we have the following result on constructing pairs of real-valued symmetric dual wavelet frames from two real-valued symmetric refinable functions. Theorem 4.3 Let φ φ be two real-valued d-refinable functions in L 2 (R) with the dilation factor d finitely supported masks a b, respectively. Let N be a nonnegative integer J be an integer. Suppose that a b satisfy the sum rules of orders m n with respect to the lattice dz for some positive integers m n, respectively. Further assume that a s k = a k b s k = b k k Z (4.7) for some integers s s such that s s is an integer. Then one can construct finitely supported d 1 sequences a 1,..., a d, b 1,..., b d of real numbers a 2π-periodic trigonometric polynomial Θ satisfying (2.2) (2.3) such that the wavelet functions ψ 1,..., ψ d, ψ 1,..., ψ d, which are defined in (1.5), satisfy 16

17 (a) {ψ 1,..., ψ d } { ψ 1,..., ψ d } generate a pair of dual d-wavelet frames in L 2 (R); (b) ψ 1, {ψ 2,..., ψ d }, ψ 1 { ψ 2,..., ψ d } have vanishing moments of orders n, n + 2N, m + 2N m, respectively; (c) All the functions ψ 1,..., ψ d, ψ 1,..., ψ d are real-valued are either symmetric or antisymmetric about the point J +. More precisely, for all l = 1,..., d, 2 ψ l (x) = ε l ψ l (J + s 2(d 1) s d 1 x) ψl (x) = ε ψl l (J + s x) x R, d 1 where ε l = ( 1) n, l = 1,..., N d,s n ε l = ( 1) n+1, l = N d,s n + 1,..., d with the integer N d,s n being defined in (2.1). Proof: Let c 1,..., c d be 2π-periodic trigonometric polynomials satisfying all the conditions in Proposition 2.4 with s = s + dj n. Let g(ξ) 1 define the sequences a 1,..., a d as in (3.2). It is evident that a l s+dj k = ε la l k for all k Z l = 1,..., d. Let h(ξ) := det C(ξ), where the matrix C(ξ) is defined in (2.9). Observe that h(ξ + 2π/d) = ( 1) d 1 h(ξ). In the proof of Theorem 3.2, we can take θ 1 (ξ) = h(ξ/d) when d is odd θ 1 (ξ) = e iξ/2 h(ξ/d) when d is even. The claim follows directly from Theorem 3.2, Corollary 3.6, Propositions When a b are symmetric sequences satisfying (4.7), in order to have a symmetric 2πperiodic trigonometric polynomial Θ such that Θ(ξ) Θ(dξ)â(ξ)ˆb(ξ) is also either symmetric or antisymmetric, it is not difficult to see that one naturally require s s to be an integer. The d 1 restriction that s s should be an integer automatically disappears when d = 2. d 1 Finally, we mention that the earlier version of this paper inspired the authors of [9] to generalize the results in this paper to the case of d-refinable function vectors. For detail on constructing pairs of dual d-wavelet frames from d-refinable function vectors, see [9]. 5 Examples of Dual Wavelet Frames In this section, we shall give several examples to illustrate the main results in this paper on construction of pairs of dual wavelet frames from pairs of refinable functions. The following examples follow easily from the results in Sections 3 4 are produced by the program which consists of a collection of MAPLE routines is available from bhan. Example 5.1 Let the dilation factor d = 2. Let φ = φ = B 2 be the B-spline function of order 17

18 (a) (b) (c) (d) Figure 1: (a) (b) are the graphs of the wavelet functions ψ 1 ψ 2 in Example 5.1. (c) (d) are the graphs of their dual wavelet functions ψ 1 ψ 2. The functions ψ 1, ψ 2, ψ 1, ψ 2 have vanishing moments of orders 2, 4, 4, 2, respectively they are either symmetric or antisymmetric about the point 1. {ψ 1, ψ 2 } { ψ 1, ψ 2 } generate a pair of dual 2-wavelet frames. 2 given in (3.11). Taking N = 1 J =, by Theorem 4.3, we have [ ] 71(z 4 + z 4 ) 252(z 3 + z 3 ) + 12(z 2 + z 2 ) + 492(z 1 + z) + 98, Θ = 1 96 â 1 = (1 z) 2, â 2 = z 1 (1 z) 4, b 1 = (1 z) [ 71(z 8 + z 6 ) + 426(z 7 + z 5 ) (z 6 + z 4 ) (z 5 + z 3 ) (z 4 + z 2 ) (z 3 + z) (z 2 + 1) z ], 1 (1 [ b 2 z)2 = 71(z 6 + z 6 ) + 284(z 5 + z 5 ) + 458(z 4 + z 4 ) + 412(z 3 + z 3 ) 384 ] + 85(z 2 + z 2 ) 726(z 1 + z) 1228, where z = e iξ. Then {ψ 1, ψ 2 } { ψ 1, ψ 2 } generate a pair of dual 2-wavelet frames. The functions ψ 1, ψ 2, ψ 1, ψ 2 have vanishing moments of orders 2, 4, 4, 2, respectively they are either symmetric or antisymmetric about the point 1. See Figure 1 for their graphs. 18

19 (a) (b) (c) (d) Figure 2: (a) (b) are the graphs of the wavelet functions ψ 1 ψ 2 in Example 5.2. (c) (d) are the graphs of their dual wavelet functions ψ 1 ψ 2. Both {ψ 1, ψ 2 } { ψ 1, ψ 2 } have vanishing moments of order 4 generate a pair of dual 2-wavelet frames. Example 5.2 Let the dilation factor d = 2 φ = φ = B 4. By Theorem 3.1, we have Θ = 1 [ ] 311(z 3 + z 3 ) (z 2 + z 2 ) 14913(z 1 + z) , 1512 â 1 = (1 z) 4, â 2 := z(1 z) 4, (1 [ b 1 z)4 = 311(z 6 + z 6 ) (z 5 + z 5 ) (z 4 + z 4 ) (z 3 + z 3 ) ] (z 2 + z 2 ) (z 1 + z) , (1 [ b 2 z)4 = 311(z 4 + z 6 ) (z 3 + z 5 ) + 125(z 2 + z 4 ) (z 1 + z 3 ) ] (1 + z 2 ) z, where z = e iξ. Then {ψ 1, ψ 2 } { ψ 1, ψ 2 } generate a pair of dual 2-wavelet frames. Both {ψ 1, ψ 2 } { ψ 1, ψ 2 } have vanishing moments of order 4. See Figure 2 for their graphs. When the dilation factor d = 2 φ = φ = B m, pairs of dual 2-wavelet frames derived from φ φ have been also constructed in [6]. It turns out that up to some integer shifts the construction in [6] for B-spline functions coincides with the construction in Theorem 3.1 for B-spline functions with the particular choice Θ as given by Θ(ξ) = P m (sin 2 ξ/2) with ( 1 + j=1 So Example 5.2 was also obtained in [6]. (2j 1)!! (2j)!!(2j + 1) xj ) 2m = Pm (x) + O( x 2m ), x. 19

20 (a) (b) (c) (d) Figure 3: (a) (b) are the graphs of the wavelet functions ψ 1 ψ 2 in Example 5.3. (c) (d) are the graphs of their dual wavelet functions ψ 1 ψ 2. {ψ 1, ψ 2 } { ψ 1, ψ 2 } have vanishing moments of orders 2 4, respectively they generate a pair of dual 2-wavelet frames. Example 5.3 Let the dilation factor d = 2. Let φ = B 4 φ = B 2. By Theorem 3.1, we have Θ = 1 [ ] 13(z 1 + z 3 ) 112(1 + z 2 ) + 438z, 24 â 1 = (1 z) 2, â 2 = z(1 z) 2, (1 [ ] b 1 z)4 = 13(z 4 + z 4 ) + 78(z 3 + z 3 ) + 356(z 2 + z 2 ) (z 1 + z) , 1536 (1 [ ] b 2 z)4 = 39(z 2 + z 4 ) + 234(z 1 + z 3 ) + 613(1 + z 2 ) + 948z, 768 where z = e iξ. Then {ψ 1, ψ 2 } { ψ 1, ψ 2 } generate a pair of dual 2-wavelet frames have vanishing moments of orders 2 4, respectively. See Figure 3 for their graphs. Example 5.4 Let the dilation factor d = 3 φ = φ = B 3. By Theorem 3.1 Proposition 4.1, we have Θ = 1 [ ] 13(z 2 + z 2 ) 112(z 1 + z) + 438, 24 â 1 = (1 z) 3, â 2 = z(1 z) 3, â 3 = z 2 (1 z) 3, (1 [ b 1 z)3 = 13(z 9 + z 9 ) + 78(z 8 + z 8 ) + 273(z 7 + z 7 ) (z 6 + z 6 ) (z 5 + z 5 ) ] (z 4 + z 4 ) (z 3 + z 3 ) (z 2 + z 2 ) (z 1 + z) , (1 [ b 2 z)3 = 91z z z z z z b 3 = z 6 b2 (1/z), z z z z z z z z z 9 ], 2

21 (a) (b) (c) (d) (e) (f) Figure 4: (a), (b) (c) are the graphs of the wavelet functions ψ 1, ψ 2, ψ 3 in Example 5.4. (d), (e) (f) are the graphs of their dual wavelet functions ψ 1, ψ 2, ψ 3. {ψ 1, ψ 2, ψ 3 } { ψ 1, ψ 2, ψ 3 } have vanishing moments of order 3 generate a pair of dual 3-wavelet frames. where z = e iξ. Then {ψ 1, ψ 2, ψ 3 } { ψ 1, ψ 2, ψ 3 } generate a pair of dual 3-wavelet frames. Both {ψ 1, ψ 2, ψ 3 } { ψ 1, ψ 2, ψ 3 } have vanishing moments of order 3. See Figure 4 for their graphs. Finally, let us present an example using Theorem 3.2. Example 5.5 Let the dilation factor d = 2 φ = φ = B 3. Take n = 2, g = 1 e iξ, c 1 = 1 c 2 = e iξ e iξ in (3.2). Therefore, we have h = det C = 2(e iξ e iξ ) with the matrix C being defined in (2.9). Clearly, g(π) = h() =. By Theorem 3.2, we set Θ = z 2 (1 + z) 2 ( z z 2 )/8, â 1 = (z 1) 2 (1 + z), â 2 := z 1 (1 z) 3 (1 + z) 2, where z := e iξ. By the definition of f 1 f 2 in (3.4), we have f 1 = (1 z)[(z 3 + z 5 ) + 8(z 2 + z 4 ) + 3(z 1 + z 3 ) + 72(1 + z 2 ) z], f 2 = z 3 (1 z)(1 + z 2 ) 2 (1 4 z 2 + z 4 ). Consequently, it follows from (3.5) that b 1 = (1 z)2 (1 + z)[(z 4 + z 4 ) + 4(z 3 + z 3 ) + 14(z 2 + z 2 ) + 36(z 1 + z) + 58], b 2 = 1 [ ] 128 (z 1) (z 2 + z 4 ) + 4(z 1 + z 3 ) + 9(1 + z 2 ) + 16 z. Then {ψ 1, ψ 2 } { ψ 1, ψ 2 } generate a pair of dual 2-wavelet frames. See Figure 5 for their graphs. 21

22 (a) (b) (c) (d) Figure 5: (a) (b) are the graphs of the wavelet functions ψ 1 ψ 2 in Example 5.5. (c) (d) are the graphs of their dual wavelet functions ψ 1 ψ 2. {ψ 1, ψ 2 } { ψ 1, ψ 2 } have vanishing moments of orders 2 1, they generate a pair of dual 2-wavelet frames. 6 Appendix Proof of Theorem 3.2: We only need to prove the sufficiency part for the general case. Note that h(ξ+2π/d) = ( 1) d 1 h(ξ). By the definition of Z(p, ξ ) in (3.7), rewrite g(ξ) = (1 e iξ ) Z(g,) g 1 (ξ) h(ξ) = (1 e iξ ) Z(h,) h 1 (ξ) for some 2π-periodic trigonometric polynomials g 1 h 1 such that g 1 () h 1 (). Define 2π-periodic trigonometric polynomials F k G k by F (ξ) G (ξ) := â(ξ)ˆb(ξ) g 1 (ξ)h 1 (ξ) F k (ξ) G k (ξ) := â(ξ)ˆb(ξ 2πk/d), k = 1,..., d 1, (6.1) (1 e iξ ) n g(ξ) h(ξ) where F k G k have no common zeros on the set 2π Z for k =,..., d 1. Now we claim that d G k (2πj/d) j =,..., d 1 k =,..., d 1. (6.2) Since F k G k have no common zeros on 2π Z, in order to show (6.2), it suffices to show that d Z(F k, 2πj/d) Z(G k, 2πj/d) j =,..., d 1 k =,..., d 1 (6.3) which implies that Z(G k, 2πj/d) = therefore, (6.2) holds. Note that (3.8) implies that Z(ˆb, 2πk/d) n + Z(g, ) + Z(h, ) for all k = 1,..., d 1. By our assumptions in (3.8), it follows from (6.1) that for every k = 1,..., d 1, Z(F, ) Z(G, ) = Z(F k, ) Z(G k, ) = Z(ˆb, 2πk/d) n Z(g, ) Z(h, ). Similarly, for every j = 1,..., d 1, by (3.9), we have Z(F, 2πj/d) Z(G, 2πj/d) = Z(ˆb, 2πj/d) + Z(â, 2πj/d) Z(g 1, 2πj/d) Z(h 1, 2πj/d) for every k = 1,..., d 1, = Z(ˆb, 2πj/d) + Z(â, 2πj/d) Z(g, 2πj/d) Z(h, ) Z(F k, 2πj/d) Z(G k, 2πj/d) = Z(ˆb, 2π(j k)/d) + Z(â, 2πj/d) Z(g, 2πj/d) Z(h, ). 22

23 So, (6.3) holds therefore, (6.2) must be true. Define d 1 d 1 θ 1 (ξ) := G k (ξ/d + 2πj/d), ξ R. (6.4) k= j= By (6.2), we have θ 1 () θ 1 is a 2π-periodic trigonometric polynomial since θ 1 (ξ + 2π) = θ 1 (ξ). By Lemma 2.3, there exists a 2π-periodic trigonometric polynomial θ 2 such that θ 2 () = 1 (1 e iξ ) n+m+z(g,)+z(h,) θ 2 (ξ)[θ 1 (ξ)g 1 (ξ)h 1 (ξ)] θ 2 (dξ)[θ 1 (dξ)g 1 (dξ)h 1 (dξ)â(ξ)ˆb(ξ)]. (6.5) Since θ 1 (), g 1 () h 1 () are nonzero numbers, now we define Clearly, Θ() = 1 by (6.5), it is easy to see that Θ(ξ) := θ 2(ξ)θ 1 (ξ)g 1 (ξ)h 1 (ξ). (6.6) θ 1 ()g 1 ()h 1 () (1 e iξ ) n+m+z(g,)+z(h,) [Θ(ξ) Θ(dξ)a(ξ)b(ξ)]. (6.7) In the following, we show that with the choice of Θ in (6.6), all f l must be 2π-periodic trigonometric polynomials satisfying (1 e iξ ) m f l (ξ) for all l = 1,..., d. By computation, we have f 1 (ξ) = = = = Θ(ξ) Θ(dξ)â(ξ)ˆb(ξ) (1 e iξ ) n+z(g,)+z(h,) g 1 (ξ)h 1 (ξ) (1 e iξ ) m [ Θ(ξ) (1 e iξ ) n+m+z(g,)+z(h,) g 1 (ξ)h 1 (ξ) Θ(dξ)â(ξ)ˆb(ξ) ] g 1 (ξ)h 1 (ξ) (1 e iξ ) m [ θ 1 (ξ)θ 2 (ξ) (1 e iξ ) n+m+z(g,)+z(h,) θ 1 ()g 1 ()h 1 () Θ(dξ) ] G (ξ) F (ξ) (1 e iξ ) m [ (1 e iξ ) n+m+z(g,)+z(h,) θ 1 (ξ)θ 2 (ξ) θ 1 ()g 1 ()h 1 () θ 1(dξ) G (ξ) g 1 (dξ)h 1 (dξ)θ 2 (dξ) F (ξ) θ 1 ()g 1 ()h 1 () By the definition of θ 1 in (6.4), we see that θ 1 (dξ)/g (ξ) is a 2π-periodic trigonometric polynomial. Consequently, it follows from (6.7) the above identity that f 1 is a 2π-periodic trigonometric polynomial satisfying (1 e iξ ) m f 1 (ξ). For j = 1,..., d 1, by computation the fact h(ξ + 2π/d) = ( 1) d 1 h(ξ), we have f j+1 Θ(dξ)â(ξ + 2πj/d)ˆb(ξ) (ξ) = (1 e i(ξ+2πj/d) ) n g(ξ + 2πj/d) h(ξ) = Θ(dξ) ( 1)(d 1)j+1 G j (ξ + 2πj/d) F j(ξ + 2πj/d) = ( 1) (d 1)j+1 θ 1 (dξ) G j (ξ + 2πj/d) g 1 (dξ)h 1 (dξ)θ 2 (dξ) F j (ξ + 2πj/d). θ 1 ()g 1 ()h 1 () By the definition of θ 1 in (6.4), we see that θ 1 (dξ)/g j (ξ + 2πj/d) is a 2π-periodic trigonometric polynomial. By (3.4) (3.9), we have Z(f j+1, ) = Z(â, 2πj/d) Z(g, 2πj/d) Z(h, ) m, j = 1,..., d 1. Consequently, f j, j = 2,..., d, are 2π-periodic trigonometric polynomials satisfying (1 e iξ ) m f j (ξ) for all j = 2,..., d. So, by Theorem 2.2, {ψ 1,..., ψ d } { ψ 1,..., ψ d } generate a pair of dual d-wavelet frames in L 2 (R). Moreover, {ψ 1,..., ψ d } has vanishing moments of order n { ψ 1,..., ψ d } has vanishing moments of order m. 23 ].

24 Let â, ˆb, g h be given. Using long division, we observe that the conditions in (a), (b), (c) of Theorem 3.2 are equivalent to a set of linear equations on the coefficients of the 2π-periodic trigonometric polynomial Θ. Therefore, one can obtain a desirable 2π-periodic trigonometric polynomial Θ with smallest degree by solving a set of linear equations; the existence of such desirable Θ is guaranteed by the above proof of Theorem 3.2. Acknowledgment: We would like to thank the referees for their helpful comments suggestions which improved the presentation of this paper. In particular, we thank one of the referees for suggesting the first part of the proof of Lemma 2.3 which shortens our original proof. The authors also thank Dr. Qun Mo at the University of Alberta for several discussions with the second author which led to the remarks after Corollary 3.4. The second author also thanks PACM at Princeton University for their hospitality during his visit at PACM in the year References [1] A. Ben-Artzi A. Ron, On the integer translates of a compactly supported function: dual bases linear projectors, SIAM J. Math. Anal. 21 (199), [2] C. K. Chui, W. He, J. Stöckler, Compactly supported tight sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal. 13 (22), [3] A. Cohen I. Daubechies, A stability criterion for biorthogonal wavelet bases their related subb coding scheme, Duke Math. J. 68 (1992), [4] I. Daubechies, The wavelet transform, time-frequency localization signal analysis, IEEE Trans. Inform. Theory 36 (199), [5] I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61, SIAM, Philadelphia, PA, [6] I. Daubechies, B. Han, A. Ron Z. W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (23), [7] B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. 4 (1997), [8] B. Han, Compactly supported tight wavelet frames orthonormal wavelets of exponential decay with a general dilation matrix, J. Comput. Appl. Math. 155 (23), [9] B. Han Q. Mo, Multiwavelet frames from refinable function vectors, Adv. Comput. Math. 18 (23), [1] A. Ron, Factorization theorems for univariate splines on regular grids, Israel J. Math. 7 (199), [11] A. Ron Z. W. Shen, Affine systems in L 2 (R d ) II: dual systems, J. Fourier Anal. Appl. 3 (1997), [12] L. Villemoes, Sobolev regularity of wavelets stability of iterated filter banks, in Progress in wavelet analysis applications, Y. Meyer S. Roques eds., (1993),