CONSTRUCTIVE APPROXIMATION

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1 Constr Approx 24) 2: DOI: 117/s CONSTRUCTIVE APPROXIMATION 24 Springer-Verlag New York, LLC Pairs of Dual Wavelet Frames from Any Two Refinable Functions Ingri Daubechies an Bin Han Abstract Starting from any two compactly supporte refinable functions in L 2 R) with ilation factor, we show that it is always possible to construct 2 wavelet functions with compact support such that they generate a pair of ual -wavelet frames in L 2 R) Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation orer of the ual MRA; this is the highest possible In particular, when we consier symmetric refinable functions, the constructe ual wavelets are also symmetric or antisymmetric As a consequence, for any compactly supporte refinable function ϕ in L 2 R), it is possible to construct, explicitly an easily, wavelets that are finite linear combinations of translates ϕ k), an that generate a wavelet frame with an arbitrarily preassigne number of vanishing moments We illustrate the general theory by examples of such pairs of ual wavelet frames erive from B-spline functions 1 Introuction As a generalization of biorthogonal wavelets, pairs of ual wavelet frames have prove particularly useful in signal enoising an many other applications where translation invariance or reunancy is important By allowing reunancy in a wavelet system, one has much more freeom in the choice of wavelets It may also be easier to recognize patterns in a reunant transform From the computational point of view, it is often easier to work with ual wavelet frames that are generate by MRAs If one works with biorthogonal bases, then the two MRAs have to be linke in special ways [5] In this paper, we are particularly intereste in pairs of ual wavelet frames erive from refinable functions; we shall see that there are then no restrictions on the MRAs Before proceeing further, let us introuce some notation Throughout this paper, by we enote the ilation factor which is an integer with absolute value greater than one For simplicity, throughout this paper, we further assume that is a positive ilation factor since all the corresponing results in this paper for a negative ilation factor can be prove almost ientically The inner prouct, in L 2 R) is efine to be f, g := f t)gt) t, f, g L 2 R) R Date receive: July 31, 2 Date revise: April 1, 22 Date accepte: January 14, 24 Communicate by Amos Ron AMS classification: 42C4, 42C15, 41A15 Key wors an phrases: Dual wavelet frames, Wavelet frames, Refinable functions, B-Spline functions 325

2 326 I Daubechies an B Han Let {ψ 1,,ψ r } be a finite set of functions in L 2 R) We say that {ψ 1,,ψ r } generates a -wavelet frame in L 2 R) if there exist positive constants C 1 an C 2 such that 11) r C 1 f 2 f,ψj,k l 2 C 2 f 2 f L 2 R), l=1 j Z k Z where f 2 := f, f an ψ l j,k := j/2 ψ l j k), j Z, k Z In particular, when C 1 = C 2 = 1 in 11), we say that {ψ 1,,ψ r } generates a normalize) tight -wavelet frame in L 2 R) If both {ψ 1,,ψ r } an { ψ 1,, ψ r } generate -wavelet frames in L 2 R) an satisfy f, g = r f,ψj,k l ψ j,k l, g f, g L 2R), l=1 j Z k Z then we say that {ψ 1,,ψ r } an { ψ 1,, ψ r } generate a pair of ual -wavelet frames in L 2 R) A pair of ual -wavelet frames is also calle a bi-frame in the literature [11] Consequently, any function f in L 2 R) has the following wavelet expansions: f = r r f,ψj,k l ψ j,k l = f, ψ j,k l ψ j,k l l=1 j Z k Z 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 } an { ψ 1,, ψ r } to generate a pair of ual -wavelet frames in L 2 R); see [7], [11] An important property of a wavelet system is its orer of vanishing moments We say that {ψ 1,,ψ r } has vanishing moments of orer n if t k ψ l t) t = l = 1,,r an k =,,n 1 R In this paper, we are particularly intereste in obtaining pairs of ual wavelet frames that are erive from pairs of refinable functions with a general ilation factor Let be a ilation factor A function ϕ is sai to be -refinable if 12) ϕ = k Z a k ϕ k), where a is a sequence on Z, calle the mask for ϕ The Fourier series of a sequence a on Z is efine to be 13) âξ) := k Z a k e ikξ, ξ R Any mask a for a refinable function in this paper is assume to be finitely supporte with â) = k Z a k = 1 We shall only consier L 2 -solutions ϕ to 12) with a finitely supporte mask a; because a is finitely supporte, this solution ϕ is compactly supporte,

3 Pairs of Dual Wavelet Frames from Any Two Refinable Functions 327 an if it exists) uniquely efine up to normalization by ˆϕξ) := j=1 â j ξ),ξ R see [5]), where the Fourier transform is efine to be ˆ f ξ) := R f t)e iξt t, f L 1 R) Since â) = 1 an ˆϕξ) := j=1 â j ξ), we always have ˆϕ) = 1 We say that a satisfies the sum rules of orer n with respect to the lattice Z if 14) a k+ j k + j) l = a k k l l =,,n 1 an j Z k Z k Z Equivalently, a finitely supporte sequence a satisfies the sum rules of orer n with respect to the lattice Z if an only if 1 + e iξ + +e i 1)ξ ) n âξ) That is, âξ) = 1 + e iξ + +e i 1)ξ ) n pξ) for some 2π-perioic trigonometric polynomial p Throughout this paper, we shall use the notation qξ) âξ) to mean that âξ) = qξ) pξ) for some 2π-perioic trigonometric polynomial p Let ϕ an ϕ be two -refinable functions in L 2 R) with finitely supporte masks a an b, respectively Suppose that a an b satisfy the sum rules of orers m an n with respect to the lattice Z for some positive integers m an n, respectively For any nonnegative integer N, we show in Section 3 that there exist finitely supporte sequences a 1,,a, b 1,,b such that by efining 15) ψ l = ak l ϕ k) an k Z ψ l = bk l ϕ k), l = 1,,, k Z or equivalently, in the frequency omain, ψ l ξ) = âl ξ) ˆϕξ) an ψ l ξ) = b l ξ) ˆ ϕξ)for l = 1,,) {ψ 1,,ψ } an { ψ 1,, ψ } generate a pair of ual -wavelet frames in L 2 R) Moreover, ψ 1, {ψ 2,,ψ }, ψ 1 an { ψ 2,, ψ } have vanishing moments of orers n, n + 2N, m + 2N, an m, respectively In aition, if both ϕ an ϕ are real-value an symmetric -refinable functions, such that the symmetry centers of ϕ an ϕ iffer by a half-integer, then the wavelet functions ψ 1,,ψ, ψ 1,, ψ can be chosen to be real-value an be either symmetric or antisymmetric with the same symmetry center See Sections 3 an 4 for more etails The structure of this paper is as follows In Section 2, we shall recall a general metho for constructing pairs of ual wavelet frames erive from any two refinable functions; we also prove two auxiliary results that will be useful for our main theorems In Section 3, we shall iscuss how to obtain pairs of ual wavelet frames in a concrete an constructive way from any two refinable functions An algorithm for constructing pairs of ual wavelet frames will be presente Wavelet frames erive from refinable functions will be iscusse in Section 3 In Section 4, we shall investigate how to erive pairs of real-value an symmetric ual wavelet frames from any two real-value an symmetric refinable functions Finally, in Section 5 we give several examples of pairs of ual wavelet frames erive from B-spline functions

4 328 I Daubechies an B Han A program consisting of a collection of MAPLE routines base on the algorithms an constructions of ual wavelet frames in this paper, which comes without warranty, can be ownloae at bhan The examples in Section 5 are prouce by this program 2 Dual Wavelet Frames of High Vanishing Moments In this section, we shall iscuss how to construct ual wavelet frames with high vanishing moments from refinable functions The following lemma is a irect consequence of results from Cohen an Daubechies [3] an Villemoes [12]: Lemma 21 Let ϕ L 2 R) be a -refinable function with a ilation factor an a finitely supporte mask a Let b be a finitely supporte sequence on Z such that ˆb) = Define a function ψ by ˆψξ) = ˆbξ) ˆϕξ) Then there exists a positive constant C such that 21) f,ψ j,k 2 C f 2 f L 2 R), j Z k Z where ψ j,k := j/2 ψ j k) Proof Since ϕ L 2 R) is compactly supporte, it is well-known that there exists a compactly supporte function η L 2 R) in the closure of span{ϕ k) : k Z} such that the shifts of η are linearly inepenent an ϕ = k Z c kη k) for some finitely supporte sequence c see [1], [1]) By [3, Theorem 51], for some α>, η W2 αr) := {f L 2R) : R 1 + ξ 2 ) α f ˆξ) 2 ξ< } Therefore, as a finite linear combination of η k), ϕ W2 α R) So the compactly supporte function ψ lies in W2 αr) an R ψt) t = By [3, Theorem 51] or [12, Theorem 33], there exists a positive constant C such that 21) hols We point out to the reaer that Lemma 21 has been generalize 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 moulus Suppose that ϕ = ϕ 1,,ϕ r ) T L2 R s ) ) r is compactly supporte an ˆϕM T ξ) = âξ) ˆϕξ) for some r r matrix âξ) of 2π-perioic trigonometric polynomials Then it was prove in [8] that there exists α > such that R 1 + ξ 2 ) α ϕ l ξ) 2 ξ < an 1 + ) α ϕ l L s for all l = 1,,r Moreover, for any ψ = ψ 1,,ψ r ) T which is efine by ˆψM T ξ) = ˆbξ) ˆϕξ) for some r r matrix ˆbξ) of 2π-perioic trigonometric polynomials, if R ψ l t) t = for all l = 1,,r, then there exists a positive constant C such that r s l=1 j Z k Z s f,ψl j,k 2 C f 2 for all f L 2 R s ), where ψj,k l := et M j/2 ψ l M j k) Pairs of ual wavelet frames can be obtaine from refinable functions by the following result: Theorem 22 Let ϕ an ϕ be two -refinable functions in L 2 R) with the ilation factor an finitely supporte masks a an b, respectively Suppose that there are

5 Pairs of Dual Wavelet Frames from Any Two Refinable Functions 329 finitely supporte sequences a 1,,a r, b 1,,b r polynomial such that an a 2π-perioic trigonometric 22) ) = 1, â l ) = b l ) = l = 1,,r, an 23) â âξ) ) â 1 ξ) â r ξ) â 1 ξ + 2π ) â ξ + 2π ξ + ) ) 2π 1) 2π 1) â 1 ξ + ξ)ˆbξ) ξ) b 1 ξ) = b r ξ) â r ξ + 2π ) ) 2π 1) â r ξ + Define wavelet functions ψ 1,,ψ r, ψ 1,, ψ r as follows: 24) ψ l = ak l ϕ k) an k Z ψ l = bk l ϕ k), l = 1,,r k Z Then {ψ 1,,ψ r } an { ψ 1,, ψ r } generate a pair of ual -wavelet frames in L 2 R) Proof By Lemma 21, there exists a positive constant C such that r [ f,ψj,k l 2 + f, ψ j,k l 2 ] C f 2 f L 2 R), l=1 j Z k Z where ψj,k l = j/2 ψ l j k) an ψ j,k l = j/2 ψ l j k) Define η by ˆηξ) := ξ)ˆ ϕξ) By 23) an a simple calculation, for every j Z, one has r l=1 f,ψj,k k Z l ψ j,k l, g = f,ϕ j+1,k η j+1,k, k Z g k Z f, g L 2 R) f,ϕ j,k η j,k, g, The rest of the proof follows irectly from Daubechies, Han, Ron, an Shen [6, Corollary 53]

6 33 I Daubechies an B Han It is easy to see that 23) can be rewritten as follows: 25) r âξ + 2π j/) ˆbξ) ξ)+ â l ξ + 2π j/) b l ξ) = δ j ξ), l=1 j =,, 1, where δ enotes the Dirac sequence such that δ = 1 an δ j = for all j Z\{} Tight wavelet frames an ual wavelet frames have been investigate in [2], [6] for the case = 2 In this paper, we shall give a systematic stuy of ual wavelet frames with a general ilation factor We mention that Theorem 22 can also be verifie using the characterization of ual wavelet frames in [7], [11] In orer to prove the main results in this paper, the following result is crucial in our construction of ual wavelet frames from refinable functions: Lemma 23 Let be a ilation factor Let A an B be two finitely supporte sequences on Z such that Â) = ˆB) Let ξ) := k Z ke ikξ be a 2π-perioic trigonometric polynomial an we enote by j) the jth erivative of the trigonometric polynomial Then, for any positive integer n, 26) ) = 1 an 1 e iξ ) n [ ξ)âξ) ξ) ˆBξ)] if an only if 27) i l l) ) = k Z k k l = λ l, l =,,n 1, where λ = 1 an λ l l N) are uniquely etermine by the following recursive formula: 28) 1 λ l = l 1)Â) l 1 j= [ l! A k k l j ] j B k k l j λ j, l N j! l j)! k Z k Z Consequently, for any positive integer n there exists a 2π-perioic trigonometric polynomial such that 26) hols In particular, if A an B are finitely supporte real-value sequences on Z such that A s k = A k an B s k = B k for all k Z for some integers s an s such that c = s s)/ 1) is an integer, then can be chosen to be a 2πperioic trigonometric polynomial with real coefficients such that ξ) = e icξ ξ); that is, c k = k for all k Z Proof By the Leibniz ifferentiation formula an Â) = ˆB), 26) is equivalent to ) = 1 an l 1)Â) l) ) = l 1 j= l = 1,,n 1 l! j! l j)! j) )[Â l j) ) j ˆB l j) )], It follows irectly from the above equations that 26) is equivalent to 27)

7 Pairs of Dual Wavelet Frames from Any Two Refinable Functions 331 When A an B are real-value sequences, by 28), it is clear that we can choose to be a 2π-perioic trigonometric polynomial with real coefficients Note that A s k = A k for all k Z if an only if Âξ) = e isξ Âξ) Let θ be a 2π-perioic trigonometric polynomial with real coefficients such that θ) = 1 an 1 e iξ ) n [θξ)âξ) θξ) ˆBξ)] Set ξ) := [θξ)+ e icξ θξ)]/2 Since e icξ θξ)âξ) e icξ θξ) ˆBξ) = e is+c)ξ [θξ)e isξ Âξ) θξ)e i sξ ˆBξ)] = e is+c)ξ [θξ)âξ) θξ) ˆBξ)], it is easy to check that 26) hols an ξ) = e icξ ξ) Lemma 23 was also given in [9, Lemma 32] which generalize the special case Aξ) 1 in an earlier version of this paper The following result is important for us to construct pairs of symmetric ual wavelet frames from symmetric refinable functions: Proposition 24 Let be a positive integer such that 2 For any positive integer N an any integer s, there exist 2π-perioic trigonometric polynomials c 1,,c with real coefficients such that: a) et C), where Cξ) is the matrix efine by 29) Cξ) c 1 ξ) c 2 ξ) c ξ) c 1 ξ + 2π ) c 2 ξ + 2π ) c ξ + 2π := ) ) c 1 ξ + c 2 ξ + c ξ + 2π 1) 2π 1) ) 2π 1) ) b) 1 e iξ ) 2N c j ξ) for all j = 2,, c) The 2π-perioic trigonometric polynomials c 1,,c have real coefficients an an c j ξ) = e is ξ c j ξ), j = 1,,N,s, c j ξ) = e is ξ c j ξ), j = N,s + 1,,, 21) where the integer N,s is efine to be + 2, when s is an even integer, 2 N,s := + 1, when s is an o integer, 2 with x enoting the greatest integer which is no greater than x

8 332 I Daubechies an B Han In particular, when = 2 an s is even, one can choose c 1 ξ) = e is ξ/2 an c 2 ξ) = e is ξ/2 1 cos ξ) N When = 2 an s is o, one can choose c 1 ξ) = e is 1)ξ/2 1+ e iξ ) an c 2 ξ) = e is 1 2N)ξ/2 1 e iξ ) 2N+1 Proof Observe that a 2π-perioic trigonometric polynomial c with real coefficients satisfies cξ) = cξ) if an only if cξ) = pcos ξ) for some polynomial p with real coefficients Let m := + 2)/2 The main iea in the following proof is that we ivie the set {2π j/ : j =,, 1} into three subsets I 1, I 2, an I 3, where I 1 :={jπ : j =,,2m 1}, I 2 :={2π j/ : j = 1,, m}, an I 3 :={2π 2π j/ : j = 1,, m} It is not ifficult to see that there exist polynomials p,,p m 1 with real coefficients such that p j cos2πk/)) = δ j k, j, k =,,m 1, an p l) j 1) = l =,,N 1 an j = 1,,m 1 When s is even, we efine c j ξ) = e is ξ/2 p j 1 cos ξ), j = 1,,m, an c j+m ξ) = e isξ/2 e iξ e iξ )p j cos ξ), j = 1,, m In orer to prove that et 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 an columns of the matrix C), the matrix C) becomes I 2m I m E E with E := iage i2π/, e i4π/,,e i2 m)π/ ) I m E E Eviently, et C) =4 m m j=1 sin2π j/) When s is o an is o, we have 2m 1 = an we efine c j ξ) = e is 1)ξ/2 1 + e iξ )p j 1 cos ξ), j = 1,,m, an c j+m ξ) = e is 1)ξ/2 1 e iξ )p j cos ξ), j = 1,,m 1 When s is o an is even, we have 2m 2 = an we efine c j ξ) = e is 1)ξ/2 1 + e iξ )p j 1 cos ξ), j = 1,,m 1, an c j+m 1 ξ) = e is 1)ξ/2 1 e iξ )p j cos ξ), j = 1,,m 1

9 Pairs of Dual Wavelet Frames from Any Two Refinable Functions 333 In orer to prove that et 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 an columns of the matrix C), the matrix C) becomes I 2m I m + E I m E I m + E I m E with E := iage i2π/, e i4π/,,e i2 m)π/ ) Eviently, et C) =4 m m j=1 sin2π j/) All other claims can be easily verifie Proposition 24 still hols if we take p = 1 in the above proof We observe that the egrees of the 2π-perioic trigonometric polynomials c 1,,c constructe in the proof of Proposition 24 can be mae even smaller 3 Construction of Dual Wavelet Frames In this section, we shall iscuss how to construct pairs of ual -wavelet frames from any two -refinable functions Let ϕ an ϕ be two -refinable functions in L 2 R) with finitely supporte masks a an b, respectively Suppose that a an b satisfy the sum rules of orers m an n with respect to the lattice Z for some positive integers m an n, respectively; in other wors, 1 + e iξ + +e i 1)ξ ) m âξ) an 1 + e iξ + +e i 1)ξ ) n ˆbξ) In orer to construct a pair of ual -wavelet frames by Theorem 22, we nee to construct finitely supporte sequences a 1,,a r, b 1,,b r on Z ana2π-perioic trigonometric polynomial such that 22) an 23) are satisfie In this section, let us consier the special case r = When r =, the relation in 23) can be rewritten as follows: â 1 ξ) â 2 ξ) â ξ) â 1 ξ + 2π ) â 2 ξ + 2π ) â ξ + 2π ) 31) ) ) ) 2π 1) 2π 1) 2π 1) â 1 ξ + â 2 ξ + â ξ + ξ) ξ)âξ) ˆbξ) b 1 ξ) b 2 ξ) ξ)â ξ + 2π ) ˆbξ) = ) b ξ) 2 1)π ξ)â ξ + ˆbξ)

10 334 I Daubechies an B Han Define the wavelet functions ψ 1,,ψ, ψ 1,, ψ as in 15) Since ˆϕ) = 1, it is easy to see that {ψ 1,,ψ } has vanishing moments of orer n if an only if 1 e iξ ) n âl ξ) for all l = 1,, So, in orer to achieve high vanishing moments, it is necessary an natural to require that 32) â l ξ) = 1 e iξ ) n gξ)c l ξ), l = 1,,, where g, c l,l = 1,,, are 2π-perioic trigonometric polynomials with g being a certain common ivisor of all the 2π-perioic trigonometric polynomials âl,l = 1,, Consequently, we have â 1 ξ) â 2 ξ) â ξ) â 1 ξ + 2π ) â 2 ξ + 2π ) â ξ + 2π ) ) ) ) 2π 1) 2π 1) 2π 1) â 1 ξ + â 2 ξ + â ξ + = Dξ)Cξ) with the matrix Cξ) being efine in 29) an Dξ) being the following iagonal matrix: 1 e iξ ) n gξ) Dξ) := Denote 1 e iξ+2π/) ) n g ξ + 2π ) 1 e iξ+2π 1)/) ) n g ξ + 2π 1) ) 33) an 34) hξ) := et Cξ) ξ) ξ)âξ) ˆbξ) f 1 1 e ξ) iξ ) n gξ) hξ) f 2 ξ) ξ)âξ + 2π/) ˆbξ) := 1 e iξ+2π/) ) n gξ + 2π/) hξ) f ξ) ξ)âξ + 2π 1)/) ˆbξ) 1 e iξ+2π 1)/) ) n gξ + 2π 1)/) hξ)

11 Pairs of Dual Wavelet Frames from Any Two Refinable Functions ) Now, the equation in 31) can be rewritten as follows: ξ) ξ)âξ) ˆbξ) b 1 ξ) b 2 ξ) ξ)â ξ + 2π ) ˆbξ) = [Cξ)] 1 [Dξ)] 1 ) b ξ) 2 1)π ξ)â ξ + ˆbξ) f 1 ξ) f 2 ξ) = aj Cξ), f ξ) where aj Cξ) enotes the ajacent matrix of Cξ), that is, Cξ) 1 = [et Cξ)] 1 aj Cξ) Clearly, all the entries in aj Cξ) are 2π-perioic trigonometric polynomials since all the entries of Cξ) are 2π-perioic trigonometric polynomials Now, accoring to Theorem 22, the challenging question that remains is to choose an appropriate 2π-perioic trigonometric polynomial with ) = 1 such that f 1,, f are 2π-perioic trigonometric polynomials an b l ) = for all l = 1,, We have the following result on pairs of ual wavelet frames: Theorem 31 Let ϕ L 2 R) an ϕ L 2 R) be two arbitrary -refinable functions with ilation factor an finitely supporte masks a an b, respectively Suppose that a an b satisfy the sum rules of orers m an n with respect to the lattice Z for some positive integers m an n, respectively Construct a 2π-perioic trigonometric polynomial by Lemma 23 such that 36) ) = 1, 1 e iξ ) n+m [ ξ) ξ)âξ) ˆbξ)] Define the finitely supporte sequences a 1,,a, b 1,,b on Z as in 32) an 35) by taking gξ) = 1 an c l ξ) = e il 1)ξ,l = 1,, Then {ψ 1,,ψ } an { ψ 1,, ψ }, which are efine in 15), generate a pair of ual -wavelet frames in L 2 R) Moreover, {ψ 1,,ψ } an { ψ 1,, ψ } have vanishing moments of orers n an m, respectively Note that ψ l = ψ 1 l 1)/) for l = 1,, Proof Since the mask b satisfies the sum rules of orer n with respect to the lattice Z, we have 1+e iξ + +e i 1)ξ ) n ˆbξ) Consequently, 1 e iξ+2π j/) ) n ˆbξ) for all j = 1,, 1 By c l ξ) = e il 1)ξ,l= 1,,, we observe that hξ) := et Cξ) is a monomial since Cξ)Cξ) T = I, where the matrix Cξ) is efine in 29) Now, by the fact that gξ) = 1 an hξ) is a monomial, it is straightforwar to see that f 2,, f are 2π-perioic trigonometric polynomials Since 1+e iξ + +e i 1)ξ ) m âξ),we have 1 e iξ ) m âξ +2π j/) for all j = 1,, 1an, therefore, 1 e iξ ) m f j ξ) for all j = 2,,

12 336 I Daubechies an B Han On the other han, since gξ) = 1 an hξ) is a monomial, it follows irectly from 36) that f 1 isa2π-perioic trigonometric polynomial an 1 e iξ ) m f 1 ξ) The proof is complete by our iscussion before this theorem We point out that Theorem 31 hols for general 2π-perioic trigonometric polynomials c 1,,c provie that et Cξ) is a monomial, where Cξ) is efine in 29) Fora2π-perioic trigonometric polynomial p, we efine 37) Zp,ξ ) := sup{l N {} : e iξ e iξ ) l pξ)} = inf{l N {} : p l) ξ ) } That is, Zp,ξ ) enotes the multiplicity of the zeros of pξ) at the point ξ = ξ Now we can generalize Theorem 31 an the following is the main result in this section: Theorem 32 Let ϕ an ϕ be two -refinable functions in L 2 R) with the ilation factor an finitely supporte masks a an b, respectively Suppose that a an b satisfy the sum rules of orers m an n with respect to the lattice Z for some positive integers m an n, respectively; that is, 1 + e iξ + +e i 1)ξ ) m âξ) an 1 + e iξ + +e i 1)ξ ) n ˆbξ) Let g, c 1,,c be 2π-perioic trigonometric polynomials Define hξ) := et Cξ), where the matrix Cξ) is efine in 29) Then there exists a 2π-perioic trigonometric polynomial such that: a) ) = 1 b) All f l,l = 1,,, which are efine in 34), are 2π-perioic trigonometric polynomials c) 1 e iξ ) m f l ξ) for all l = 1,,, if an only if the following two conitions hol: 38) 1 + e iξ + +e i 1)ξ ) n+zg,)+zh,) ˆbξ) an 39) Zâ, 2π j/) Zg, 2π j/) Zh, ) m j = 1,, 1 For example, when gξ) 1 an h), then 38) an 39) are automatically satisfie) For any 2π-perioic trigonometric polynomial such that a), b), an c) are satisfie such can be easily obtaine by solving a system of linear equations which are inuce by the three conitions in a), b), an c) by long ivision), let a 1,,a, b 1,,b be efine in 32) an 35) Define the wavelet functions ψ 1,,ψ, ψ 1,, ψ as in 15) Then {ψ 1,,ψ } an { ψ 1,, ψ } generate a pair of ual -wavelet frames with compact support in L 2 R) Moreover, {ψ 1,,ψ } an { ψ 1,, ψ } have vanishing moments of orers n an m, respectively Proof Sufficiency For simplicity of the presentation, let us assume here that gξ) 1 an h) ; the complete proof for the general case can be foun in the Appenix Since h), we efine θ 1 ξ) := e iξ 1)/2 hξ/)/[h)] 2 By the fact hξ + 2π/) = 1) 1 hξ),wehaveθ 1 ξ +2π) = e iξ+2π) 1)/2 hξ/ + 2π/)/[h)] 2 =

13 Pairs of Dual Wavelet Frames from Any Two Refinable Functions 337 θ 1 ξ) an, therefore, θ 1 isa2π-perioic trigonometric polynomial Since h)θ 1 ) = h)θ 1 )â) ˆb) = 1, by Lemma 23, there exists a 2π-perioic trigonometric polynomial θ 2 such that θ 2 ) = 1 an 31) 1 e iξ ) n+m θ 2 ξ)[hξ)θ 1 ξ)] θ 2 ξ)[hξ)θ 1 ξ)âξ) ˆbξ)] Now we take ξ) := θ 2 ξ)hξ)θ 1 ξ) Obviously, ) = 1 Noting that 1 + e iξ + +e i 1)ξ ) m âξ) an 1 + e iξ + +e i 1)ξ ) n ˆbξ), by a simple calculation see the Appenix for more etail), we can verify that inee all f l are 2π-perioic trigonometric polynomials an 1 e iξ ) m f l ξ) for all l = 1,, Necessity Since all f l are 2π-perioic trigonometric polynomials an 1 e iξ ) m f l ξ) for all l = 2,,,wehaveZ f j+1, ) m an Z f j+1, 2π j/) for all j = 1,, 1 By the efinition of f l in 34) an ) = ˆb) = 1, we euce Zâ, 2π j/) Zg, 2π j/) Zh, ) = Z f j+1, ) m j = 1,, 1 So, 39) must hol Similarly, by the efinition of f l in 34) an ) =â) = 1, we have Z ˆb, 2π j/) n Zg, ) Zh, ) = Z f j+1, 2π j/) j = 1,, 1, which is equivalent to 38) B-Spline functions are of great interest in many applications The B-spline function of orer m m N), enote by B m throughout this paper, can be obtaine via the following recursive formula: B 1 = χ [,1], the characteristic function of the interval [, 1], an 311) B m x) := 1 B m 1 x t) t, x R, m = 2, 3, The B-spline function B m C m 2 R) is a function of piecewise polynomials of egree less than m, vanishes outsie the interval [, m], an is symmetric about the point x = m/2 ie, B m m x) = B m x) for all x R) It is well-known that the B-spline function B m is a -refinable function satisfying 1 + e B iξ + +e i 1)ξ ) m m ξ) = B m ξ) Now we have the following result on wavelet frames which is a irect consequence of Theorem 32: Corollary 33 Let ϕ L 2 R) be a -refinable function with the ilation factor an a finitely supporte mask a Choose 2π-perioic trigonometric polynomials g, c 1,,c such that 312) Zâ, 2π j/) >Zg, 2π j/) + Zh, ) j = 1,, 1,

14 338 I Daubechies an B Han where hξ) := et Cξ) an the matrix Cξ) is efine in 29) For any positive integer n, efine the wavelet functions ψ 1,,ψ by 313) ψ l ξ) = 1 e iξ ) n gξ)c l ξ) ˆϕξ), l = 1,, Then {ψ 1,,ψ } generates a -wavelet frame in L 2 R) an has vanishing moments of orer n Moreover, there exist compactly supporte functions ψ 1,, ψ, which can be erive explicitly from any -refinable function in L 2 R) whose mask is finitely supporte an satisfies the sum rules of orer n + Zg, ) + Zh, ) with respect to the lattice Z, such that {ψ 1,,ψ } an { ψ 1,, ψ } generate a pair of ual -wavelet frames in L 2 R) Proof Take ϕ to be any -refinable function in L 2 R) whose mask is finitely supporte an satisfies the sum rules of orer n + Zg, ) + Zh, ) with respect to the lattice Z For example, we can take ϕ = B n+zg,)+zh,) to be the B-spline function of orer n + Zg, ) + Zh, ) which is efine in 311) Observe that 312) is equivalent to the conition in 39) with m = 1 Now Corollary 33 follows irectly from Theorem 32 If in Corollary 33 we choose c l ξ) = e il 1)ξ,l = 1,,, then we have the following result: Corollary 34 Let ϕ be a -refinable function in L 2 R) with the ilation factor an a finitely supporte mask a For any 2π-perioic trigonometric polynomial g such that 314) g) = an Zâ, 2π j/) >Zg, 2π j/) j = 1,, 1, efine a wavelet function ψ by ˆψξ) = gξ) ˆϕξ) Then ψ generates a -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 := j/2 ψ j k) Moreover, there exist compactly supporte functions ψ 1,, ψ with arbitrary smoothness such that {ψ ), ψ 1),, ψ + 1)} an { ψ 1, ψ 2,, ψ } generate a pair of ual -wavelet frames in L 2 R) Proof Let c l ξ) = e il 1)ξ,l = 1,, Then it is easy to see that h) = et C) since Cξ)Cξ) T = I, where the matrix Cξ) is efine in 29) Clearly, by 314), 312) hols an g)c l ) = for all l = 1,, Therefore, by Corollary 33, {ψ 1,,ψ }, which is efine in 313) with n =, generates a -wavelet frame It is easy to check that ψ l = ψ l + 1) for l = 1,, Now the claim follows from the fact that {ψ 1,,ψ } an { 1/2 ψ} generate the same -wavelet frame in L 2 R) Let us make some remarks here for the above results Through the proof of Theorem 32 in the Appenix an the following argument, we shall see that the quantities Zâl, 2π j/), j =,, 1, which enote the multiplicity of zeros of the trigonometric polynomials âl ξ) at ξ = 2π j/, play a critical role in the construction of pairs of ual wavelet frames via Theorem 22 In the following, we shall see that 314) in Corollary 34 is not only a sufficient conition for having an MRA wavelet frame, but also a

15 Pairs of Dual Wavelet Frames from Any Two Refinable Functions 339 necessary conition More precisely, 314) must hol if via Theorem 22 there exist compactly supporte functions ψ 1,, ψ such that {ψ ), ψ 1),,ψ +1)}, which is given in Corollary 34, an { ψ 1, ψ 2,, ψ } generate a pair of ual -wavelet frames in L 2 R) To see this point, by Theorem 22, 25) must hol with,â, ˆb, âl, b l,l= 1,,r, being some 2π-perioic trigonometric polynomials such that 22) hols In particular, it follows from 25) that r 315) âξ + 2π j/) ˆbξ) ξ) = â l ξ + 2π j/) b l ξ) l=1 j = 1,, 1 Therefore, by ) = ˆb) = 1, it follows from 315) that we must have 316) Zâ, 2π j/) min{zâl, 2π j/) + Z b l, ) : l = 1,,r} j = 1,, 1 Similarly, by ) =â) = 1, it follows from 315) that 317) Z ˆb, 2π j/) min{zâl, ) + Z b l, 2π j/) : l = 1,,r} j = 1,, 1 By our choice of âl ξ) = 1 e iξ ) n gξ)c l ξ) in 32) an r =, wehave Zâl, 2π j/) = Zg, 2π j/) + Zc l, 2π j/) for all j = 1,, 1, an Zâl, ) = n + Zg, ) + Zc l, ) for all l = 1,, Consequently, we conclue from 316) that 318) Zâ, 2π j/) Zg, 2π j/) min{zc l, 2π j/) : l = 1,,} min{z b l, ) : l = 1,,} j = 1,, 1, which is quite similar to 39) Similarly, it follows from 317) that Z ˆb, 2π j/) n + Zg, ) + min{zc l, ) : l = 1,,} j = 1,, 1 That is, 319) 1 + e iξ + +e i 1)ξ ) n+zg,)+min{zcl,) : l=1,,} ˆbξ), which is quite similar to 38) By Theorem 22, it is necessary that b l ) = an, therefore, min{z b l, ) : l = 1,,} > Since in Corollary 34 we set c l ξ) = e il 1)ξ,l= 1,,, we euce from 318) that Zâ, 2π j/) >Zg, 2π j/) for all j = 1,, 1 So, 314) must be a necessary conition in Corollary 34 Let us consier the following simple example: Example 35 Let B m be the B-spline function of orer m efine in 311) Then B m is -refinable with mask âξ) = m 1 + e iξ + +e i 1)ξ ) m for any ilation factor 2 For any positive integer n, efine ψ = n k= 1) k n! k! n k)! B m k)

16 34 I Daubechies an B Han That is, ˆψξ) = 1 e iξ ) n B m ξ)) Then ψ has vanishing moments of orer n We apply Corollary 34 with the special choice gξ) = 1 e iξ ) n ; note that g oes not epen on an gξ) for all ξ, 2π) We then conclue that ψ generates a -wavelet frame in L 2 R) for any ilation factor 2 Finally, we emonstrate that by appropriately choosing the 2π-perioic trigonometric polynomials c l,l = 1,,, the set {ψ 2,,ψ } of wavelet functions can have vanishing moments of arbitrary orer Corollary 36 Let ϕ an ϕ be two -refinable functions in L 2 R) with the ilation factor an finitely supporte masks a an b, respectively Suppose that a an b satisfy the sum rules of orers m an n with respect to the lattice Z for some positive integers m an n, respectively Let N be an arbitrary nonnegative integer Then one can construct finitely supporte sequences a 1,,a, b 1,,b such that by efining the functions ψ 1,,ψ, ψ 1,,ψ as in 15): a) {ψ 1,,ψ } an { ψ 1,, ψ } generate a pair of ual -wavelet frames in L 2 R) b) ψ 1, {ψ 2,,ψ }, ψ 1 an { ψ 2,, ψ } have vanishing moments of orers n, n+ 2N, m + 2N, an m, respectively Proof Let c 1,,c be the 2π-perioic trigonometric polynomials obtaine in Proposition 24 with s = By Proposition 24, h) = et C) Take gξ) = 1 The rest of the claim can be verifie similarly as in the proof of Theorem 31 with the moification: replace the factor 1 e iξ ) n+m in 31) by 1 e iξ ) n+m+2n When ϕ an ϕ are real-value an symmetric -refinable functions, in this section, we in t iscuss whether one can obtain a pair of real-value an symmetric ual -wavelet frames from ϕ an ϕ In the next section, we shall aress such an issue in etail 4 Real-Value an Symmetric Dual Wavelet Frames Given two real-value an symmetric -refinable functions in L 2 R), it is of interest to construct from two such -refinable functions pairs of ual -wavelet frames which are also real-value an symmetric In this section, we shall iscuss in etail how to obtain pairs of real-value an symmetric ual wavelet frames from two real-value an symmetric refinable functions Proposition 41 Let a, a 1,,a r, b, b 1,,b r be finitely supporte sequences on Z an let be a 2π-perioic trigonometric polynomial such that 22) an 25) are satisfie Suppose that a, b an a 1,,a r are sequences of real numbers Define new sequences b 1,, b r ananew2π-perioic trigonometric polynomial as follows: ξ) := [ ξ) + ξ)]/2 an b l ξ) := [ b l ξ) + b l ξ)]/2, l = 1,,r

17 Pairs of Dual Wavelet Frames from Any Two Refinable Functions 341 Then is a 2π-perioic trigonometric polynomial with real coefficients an all b 1,, b r are sequences of real numbers satisfying 41) an 42) ) = 1, â l ) = b l ) =, l = 1,,r, r âξ + 2π j/) ˆbξ) ξ)+ â l ξ + 2π j/) b l ξ) = δ j ξ), l=1 j =,, 1 Proof Note that a is a sequence of real numbers if an only if â ξ) =âξ) Since a, a 1,,a r are sequences of real numbers, taking the complex conjugate on both sies of 25) an replacing ξ by ξ, we euce from 25) that âξ 2π j/) ˆb ξ) ξ)+ r â l ξ 2π j/) b l ξ) = δ j ξ), l=1 j =,, 1 Since b is a sequence of real numbers, we have ˆb ξ) = ˆbξ) Therefore, âξ + 2π j/) ˆbξ) ξ)+ r â l ξ + 2π j/) b l ξ) = δ j ξ), l=1 j =,, 1 Equation 42) can be easily verifie by aing the above ientity to 25) When ϕ an ϕ are real-value refinable functions in L 2 R), by Theorem 32 an Proposition 41, we can always obtain pairs of real-value ual wavelet frames Proposition 42 Let a, a 1,,a r, b, b 1,,b r be finitely supporte sequences on Z such that 43) a s k = a k an b s k = b k k Z, for some integers s an s such that c = s s)/ 1) is an integer, an 44) a l s l k = ε la l k k Z an l = 1,,r, for some ε l { 1, 1} an some integers s l,l = 1,,r, such that s s l )/ isan integer for all l = 1,,r Suppose that 22) an 25) are satisfie with a 2π-perioic trigonometric polynomial Define new sequences b 1,, b r an a new 2π-perioic trigonometric polynomial as follows: an ξ) := [ ξ) + ξ)e icξ ]/2 b l ξ) := [ b l ξ) + ε l bl ξ)e is l+c)ξ ]/2, l = 1,,r

18 342 I Daubechies an B Han Then both 41) an 42) are satisfie Moreover, ξ) = e icξ ξ) an 45) b l s l +c k = ε l b l k k Z an l = 1,,r Proof Note that as l l k = ε lak l for all k Z if an only if âl ξ) = ε l e islξ â l ξ) Take the complex conjugate on 25), we have âξ + 2π j/) ˆbξ) ξ)+ r â l ξ + 2π j/) b l ξ) = δ j ξ), j =,, 1 l=1 Note that âξ) = e isξ âξ) an ˆbξ) = e i sξ ˆbξ) Since âl ξ) = ε l e is lξ â l ξ) for l = 1,,r, the above equation becomes 46) e isξ+2π j/) âξ + 2π j/)e i sξ ˆbξ) ξ) r + ε l e is lξ+2π j/)â l ξ + 2π j/) b l ξ) = δ j ξ), l=1 for j =,, 1 By assumption, s s = 1)c an s s l Z for all l = 1,,r Multiplying the factor e is2π j/ cξ) with both sies of equation 46), we have that for j =,, 1, δ j e icξ ξ) = δ j e is2π j/ cξ) ξ) = âξ + 2π j/) ˆbξ)e i s s c)ξ ξ) r + ε l e is s l)2π j/ â l ξ + 2π j/)e is l+c)ξ bl ξ) l=1 = âξ + 2π j/) ˆbξ)e icξ ξ)+ r â l ξ + 2π j/)ε l e is l+c)ξ ˆb l ξ) Equation 42) can be verifie by aing the above ientity to 25) All other claims can be easily checke by computation Let ϕ an ϕ be two -refinable functions with finitely supporte masks a an b, respectively Then 43) implies that ϕ = ϕs/ 1) ) an ϕ = ϕ s/ 1) ) Define ψ l ξ) := âl ξ) ˆϕξ) an ψ l ξ) := b l ξ) ˆ ϕξ) for l = 1,,r Then 44) an 45) in Proposition 42 imply that an ψ l ψ l = ε l ψ l 1)sl + s 1) = ε l ψ l 1)sl + s 1) ) l=1 ), l = 1,,r Now we have the following result on constructing pairs of real-value an symmetric ual wavelet frames from two real-value an symmetric refinable functions:

19 Pairs of Dual Wavelet Frames from Any Two Refinable Functions 343 Theorem 43 Let ϕ an ϕ be two real-value -refinable functions in L 2 R) with the ilation factor an finitely supporte masks a an b, respectively Let N be a nonnegative integer an let J be an integer Suppose that a an b satisfy the sum rules of orers m an n with respect to the lattice Z for some positive integers m an n, respectively Further, assume that 47) a s k = a k an b s k = b k k Z, for some integers s an s such that s s)/ 1) is an integer Then one can construct finitely supporte sequences a 1,,a, b 1,,b of real numbers an a 2π-perioic trigonometric polynomial satisfying 22) an 23) such that the wavelet functions ψ 1,,ψ, ψ 1,, ψ, which are efine in 15), satisfy: a) {ψ 1,,ψ } an { ψ 1,, ψ } generate a pair of ual -wavelet frames in L 2 R) b) ψ 1, {ψ 2,,ψ }, ψ 1 an { ψ 2,, ψ } have vanishing moments of orers n, n+ 2N, m + 2N, an m, respectively c) All the functions ψ 1,,ψ, ψ 1,, ψ are real-value an are either symmetric or antisymmetric about the point J/2 + s/2 2) More precisely, for all l = 1,,, ψ l x) = ε l ψ l J + s ) 1 x an ψ l x) = ε l ψ l J + s ) 1 x x R, where ε l = 1) n,l= 1,,N,s n, an ε l = 1) n+1,l= N,s n +1,,, with the integer N,s n being efine in 21) Proof Let c 1,,c be 2π-perioic trigonometric polynomials satisfying all the conitions in Proposition 24 with s = s + J n Let gξ) 1 an efine the sequences a 1,,a as in 32) It is evient that as+j k l = ε lak l for all k Z an l = 1,, Let hξ) := et Cξ), where the matrix Cξ) is efine in 29) Observe that hξ + 2π/) = 1) 1 hξ) In the proof of Theorem 32, we can take θ 1 ξ) = hξ/) when is o an θ 1 ξ) = e iξ/2 hξ/) when is even The claim follows irectly from Theorem 32, Corollary 36, an Propositions 41 an 42 When a an b are symmetric sequences satisfying 47), in orer to have a symmetric 2π-perioic trigonometric polynomial, such that ξ) ξ)âξ) ˆbξ) is also either symmetric or antisymmetric, it is not ifficult to see that one naturally requires s s)/ 1) to be an integer The restriction that s s)/ 1) shoul be an integer automatically isappears when = 2 Finally, we mention that the earlier version of this paper inspire the authors of [9] to generalize the results in this paper to the case of -refinable function vectors For etails on constructing pairs of ual -wavelet frames from -refinable function vectors, see [9]

20 344 I Daubechies an B Han 5 Examples of Dual Wavelet Frames In this section, we shall give several examples to illustrate the main results in this paper on the construction of pairs of ual wavelet frames from pairs of refinable functions The following examples follow easily from the results in Sections 3 an 4 an are prouce by the program which consists of a collection of MAPLE routines an is available from bhan Example 51 Let the ilation factor = 2 Let ϕ = ϕ = B 2 be the B-spline function of orer 2 given in 311) Taking N = 1 an J =, by Theorem 43, we have = 1 96 [71z 4 + z 4 ) 252z 3 + z 3 ) + 12z 2 + z 2 ) + 492z 1 + z) + 98], â 1 = 1 z) 2, â 2 = z 1 1 z) 4, 1 b 1 z)4 = 1536 [71z 8 + z 6 ) + 426z 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 = 384 [71z 6 + z 6 ) + 284z 5 + z 5 ) + 458z 4 + z 4 ) + 412z 3 + z 3 ) + 85z 2 + z 2 ) 726z 1 + z) 1228], where z = e iξ Then {ψ 1,ψ 2 } an { ψ 1, ψ 2 } generate a pair of ual 2-wavelet frames The functions ψ 1,ψ 2, ψ 1, ψ 2 have vanishing moments of orers 2, 4, 4, 2, respectively, an they are either symmetric or antisymmetric about the point 1 See Figure 1 for their graphs Example 52 have Let the ilation factor = 2 an ϕ = ϕ = B 4 By Theorem 31, we 1 = 1512 [ 311z 3 + z 3 ) z 2 + z 2 ) 14913z 1 + z) ], â 1 = 1 z) 4, â 2 := z1 z) 4, 1 b 1 z)4 = [311z 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 = [311z 4 + z 6 ) z 3 + z 5 ) + 125z 2 + z 4 ) z 1 + z 3 ) z 2 ) z], where z = e iξ Then {ψ 1,ψ 2 } an { ψ 1, ψ 2 } generate a pair of ual 2-wavelet frames Both {ψ 1,ψ 2 } an { ψ 1, ψ 2 } have vanishing moments of orer 4 See Figure 2 for their graphs

21 Pairs of Dual Wavelet Frames from Any Two Refinable Functions a) b) c) ) Fig 1 Parts a) an b) are the graphs of the wavelet functions ψ 1 an ψ 2 in Example 51 Parts c) an ) are the graphs of their ual wavelet functions ψ 1 an ψ 2 The functions ψ 1,ψ 2, ψ 1, ψ 2 have vanishing moments of orers 2, 4, 4, 2, respectively, an they are either symmetric or antisymmetric about the point 1 {ψ 1,ψ 2 } an { ψ 1, ψ 2 } generate a pair of ual 2-wavelet frames When the ilation factor = 2 an ϕ = ϕ = B m, pairs of ual 2-wavelet frames erive from ϕ an ϕ have also been constructe in [6] It turns out that up to some integer shifts the construction in [6] for B-spline functions coincies with the construction in Theorem 31 for B-spline functions with the particular choice as given by 1 + ξ) = P m sin 2 ξ/2) with ) 2m 2 j 1)!! 2 j)!! 2 j + 1) x j = P m x) + O x 2m ), x j=1 So Example 52 was also obtaine in [6] Example 53 Let the ilation factor = 2 Let ϕ = B 4 an ϕ = B 2 By Theorem 31, we have = 1 24 [13z 1 + z 3 ) z 2 ) + 438z], â 1 = 1 z) 2, â 2 = z1 z) 2, 1 b 1 z)4 = 1536 [13z 4 + z 4 ) + 78z 3 + z 3 ) + 356z 2 + z 2 ) z 1 + z) ], 1 b 2 z)4 = 768 [39z 2 + z 4 ) + 234z 1 + z 3 ) z 2 ) + 948z],

22 346 I Daubechies an B Han a) b) c) ) Fig 2 Parts a) an b) are the graphs of the wavelet functions ψ 1 an ψ 2 in Example 52 Parts c) an ) are the graphs of their ual wavelet functions ψ 1 an ψ 2 Both {ψ 1,ψ 2 } an { ψ 1, ψ 2 } have vanishing moments of orer 4 an generate a pair of ual 2-wavelet frames where z = e iξ Then {ψ 1,ψ 2 } an { ψ 1, ψ 2 } generate a pair of ual 2-wavelet frames an have vanishing moments of orers 2 an 4, respectively See Figure 3 for their graphs Example 54 Let the ilation factor = 3 an ϕ = ϕ = B 3 By Theorem 31 an Proposition 41, we have = 1 24 [13z 2 + z 2 ) 112z 1 + z) + 438], â 1 = 1 z) 3, â 2 = z1 z) 3, â 3 = z 2 1 z) 3, 1 b 1 z)3 = [13z 9 + z 9 ) + 78z 8 + z 8 ) + 273z 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 = 5832 [91z z z z z z z z z z z 5 b 3 = z 6 b 2 1/z), z z z z 9 ], where z = e iξ Then {ψ 1,ψ 2,ψ 3 } an { ψ 1, ψ 2, ψ 3 } generate a pair of ual 3-wavelet frames Both {ψ 1,ψ 2,ψ 3 } an { ψ 1, ψ 2, ψ 3 } have vanishing moments of orer 3 See Figure 4 for their graphs

23 Pairs of Dual Wavelet Frames from Any Two Refinable Functions a) b) c) ) Fig 3 Parts a) an b) are the graphs of the wavelet functions ψ 1 an ψ 2 in Example 53 Parts c) an ) are the graphs of their ual wavelet functions ψ 1 an ψ 2 {ψ 1,ψ 2 } an { ψ 1, ψ 2 } have vanishing moments of orers 2 an 4, respectively, an they generate a pair of ual 2-wavelet frames a) b) c) ) e) f) Fig 4 Parts a), b), an c) are the graphs of the wavelet functions ψ 1,ψ 2,ψ 3 in Example 54 Parts ), e), an f) are the graphs of their ual wavelet functions ψ 1, ψ 2, ψ 3 {ψ 1,ψ 2,ψ 3 } an { ψ 1, ψ 2, ψ 3 } have vanishing moments of orer 3 an generate a pair of ual 3-wavelet frames

24 348 I Daubechies an B Han Finally, let us present an example using Theorem 32 Example 55 Let the ilation factor = 2 an ϕ = ϕ = B 3 Take n = 2, g = 1 e iξ, c 1 = 1, an c 2 = e iξ e iξ in 32) Therefore, we have h = et Cξ) = 2e iξ e iξ ) with the matrix Cξ) being efine in 29) Clearly, gπ) = h) = By Theorem 32, we set = z z) z z 2 )/8, â 2 := z 1 1 z) z) 2, â 1 = z 1) z), where z := e iξ By the efinition of f 1 an f 2 in 34), we have f 1 = z)[z 3 + z 5 ) + 8z 2 + z 4 ) + 3z 1 + z 3 ) z 2 ) + 122z], f 2 = z 3 1 z)1 + z 2 ) 2 1 4z 2 + z 4 ) Consequently, it follows from 35) that b 1 = z)2 1 + z)[z 4 + z 4 ) + 4z 3 + z 3 ) + 14z 2 + z 2 ) + 36z 1 + z) + 58], b 2 = z 1)[z 2 + z 4 ) + 4z 1 + z 3 ) z 2 ) + 16z] Then {ψ 1,ψ 2 } an { ψ 1, ψ 2 } generate a pair of ual 2-wavelet frames See Figure 5 for their graphs 6 Appenix Proof of Theorem 32 We only nee to prove the sufficiency part for the general case Note that hξ + 2π/) = 1) 1 hξ) By the efinition of Zp,ξ ) in 37), rewrite gξ) = 1 e iξ ) Zg,) g 1 ξ) an hξ) = 1 e iξ ) Zh,) h 1 ξ) for some 2π-perioic trigonometric polynomials g 1 an h 1 such that g 1 ) an h 1 ) Define 2πperioic trigonometric polynomials F k an G k by 61) F ξ) G ξ) := F k ξ) G k ξ) := âξ) ˆbξ) g 1 ξ)h 1 ξ) an âξ) ˆbξ 2πk/), k = 1,, 1, 1 e iξ ) n gξ) hξ) where F k an G k have no common zeros on the set 2π/)Z for k =,, 1 Now we claim that 62) G k 2π j/) j =,, 1 an k =,, 1

25 Pairs of Dual Wavelet Frames from Any Two Refinable Functions a) b) c) ) Fig 5 Parts a) an b) are the graphs of the wavelet functions ψ 1 an ψ 2 in Example 55 Parts c) an ) are the graphs of their ual wavelet functions ψ 1 an ψ 2 {ψ 1,ψ 2 } an { ψ 1, ψ 2 } have vanishing moments of orers 2 an 1, an they generate a pair of ual 2-wavelet frames Since F k an G k have no common zeros on 2π/)Z, in orer to show 62), it suffices to show that 63) ZF k, 2π j/) ZG k, 2π j/) j =,, 1 an k =,, 1, which implies that ZG k, 2π j/) = an therefore, 62) hols Note that 38) implies that Z ˆb, 2πk/) n + Zg, ) + Zh, ) for all k = 1,, 1 By our assumptions in 38), it follows from 61) that for every k = 1,, 1, ZF, ) ZG, ) = an ZF k, ) ZG k, ) = Z ˆb, 2πk/) n Zg, ) Zh, ) Similarly, for every j = 1,, 1, by 39), we have ZF, 2π j/) ZG, 2π j/) = Z ˆb, 2π j/) + Zâ, 2π j/) Zg 1, 2π j/) Zh 1, 2π j/) = Z ˆb, 2π j/) + Zâ, 2π j/) Zg, 2π j/) Zh, ) an for every k = 1,, 1, ZF k, 2π j/) ZG k, 2π j/) = Z ˆb, 2πj k)/) + Zâ, 2π j/) Zg, 2π j/) Zh, )

26 35 I Daubechies an B Han So, 63) hols an, therefore, 62) must be true Define ) θ 1 ξ) := G k ξ/ + 2π j/), ξ R k= j= By 62), we have θ 1 ) an θ 1 isa2π-perioic trigonometric polynomial since θ 1 ξ +2π) = θ 1 ξ) By Lemma 23, there exists a 2π-perioic trigonometric polynomial θ 2 such that θ 2 ) = 1 an 65) 1 e iξ ) n+m+zg,)+zh,) θ 2 ξ)[θ 1 ξ)g 1 ξ)h 1 ξ)] θ 2 ξ)[θ 1 ξ)g 1 ξ)h 1 ξ)âξ) ˆbξ)] Since θ 1 ), g 1 ), an h 1 ) are nonzero numbers, we now efine ξ) := θ 2ξ)θ 1 ξ)g 1 ξ)h 1 ξ) 66) θ 1 )g 1 )h 1 ) Clearly, ) = 1 an by 65), it is easy to see that 67) 1 e iξ ) n+m+zg,)+zh,) [ ξ) ξ)aξ)bξ)] In the following, we show that with the choice of in 66), all f l must be 2π-perioic trigonometric polynomials satisfying 1 e iξ ) m f l ξ) for all l = 1,, By computation, we have f 1 ξ) ξ)âξ) ˆbξ) ξ) = 1 e iξ ) n+zg,)+zh,) g 1 ξ)h 1 ξ) [ ] 1 e iξ ) m ξ) = 1 e iξ ) n+m+zg,)+zh,) g 1 ξ)h 1 ξ) ξ)âξ) ˆbξ) g 1 ξ)h 1 ξ) 1 e iξ ) m [ θ 1 ξ)θ 2 ξ) = 1 e iξ ) n+m+zg,)+zh,) θ 1 )g 1 )h 1 ) ξ) ] G ξ) F ξ) 1 e iξ ) m = 1 e iξ ) [ n+m+zg,)+zh,) θ 1 ξ)θ 2 ξ) θ 1 )g 1 )h 1 ) θ ] 1ξ) g 1 ξ)h 1 ξ)θ 2 ξ) F ξ) G ξ) θ 1 )g 1 )h 1 ) By the efinition of θ 1 in 64), we see that θ 1 ξ)/g ξ) is a 2π-perioic trigonometric polynomial Consequently, it follows from 67) an the above ientity that f 1 is a 2π-perioic trigonometric polynomial satisfying 1 e iξ ) m f 1 ξ) For j = 1,, 1, by computation an the fact hξ + 2π/) = 1) 1 hξ),we have f j+1 ξ)âξ + 2π j/) ˆbξ) ξ) = 1 e iξ+2π j/) ) n gξ + 2π j/) hξ) = 1) 1) j+1 ξ) G j ξ + 2π j/) F jξ + 2π j/) = 1) 1) j+1 θ 1 ξ) g 1 ξ)h 1 ξ)θ 2 ξ) F j ξ + 2π j/) G j ξ + 2π j/) θ 1 )g 1 )h 1 )

27 Pairs of Dual Wavelet Frames from Any Two Refinable Functions 351 By the efinition of θ 1 in 64), we see that θ 1 ξ)/g j ξ + 2π j/) is a 2π-perioic trigonometric polynomial By 34) an 39), we have Z f j+1, ) = Zâ, 2π j/) Zg, 2π j/) Zh, ) m, j = 1,, 1 Consequently, f j, j = 2,,, are 2π-perioic trigonometric polynomials satisfying 1 e iξ ) m f j ξ) for all j = 2,, So, by Theorem 22, {ψ 1,,ψ } an { ψ 1,, ψ } generate a pair of ual -wavelet frames in L 2 R) Moreover, {ψ 1,, ψ } has vanishing moments of orer n an { ψ 1,, ψ } has vanishing moments of orer m Let â, ˆb, g, an h be given Using long ivision, we observe that the conitions in a), b), c) of Theorem 32 are equivalent to a set of linear equations on the coefficients of the 2π-perioic trigonometric polynomial Therefore, one can obtain a esirable 2π-perioic trigonometric polynomial with smallest egree by solving a set of linear equations; the existence of such esirable is guarantee by the above proof of Theorem 32 Acknowlegments We woul like to thank the referees for their helpful comments an suggestions which improve the presentation of this paper In particular, we thank one of the referees for suggesting the first part of the proof of Lemma 23 which shortens our original proof The authors also thank Dr Qun Mo at the University of Alberta for several iscussions with the secon author which le to the remarks after Corollary 34 The secon author also thanks the Program in Applie an Computational Mathematics at Princeton University for their hospitality uring his visit at the Program in Applie an Computational Mathematics in the year Research of Ingri Daubechies was partially supporte by grants NSF DMS , DMS ) an AFOSR F ) Research of Bin Han was partially supporte by the Natural Sciences an Engineering Research Council of Canaa NSERC Canaa) uner Grant G , an by Alberta Innovation an Science REE uner Grant G References 1 A BEN-ARTZI, A RON 199): On the integer translates of a compactly supporte function: Dual bases an linear projectors SIAM J Math Anal, 21: C K CHUI, WHE, JSTÖCKLER 22): Compactly supporte tight an sibling frames with maximum vanishing moments Appl Comput Harmon Anal, 13: A COHEN, I DAUBECHIES 1992): A stability criterion for biorthogonal wavelet bases an their relate subban coing scheme Duke Math J, 68: I DAUBECHIES 199): The wavelet transform, time-frequency localization an signal analysis IEEE Trans Inform Theory, 36: I DAUBECHIES 1992): Ten lectures on wavelets In: CBMS-NSF Regional Conference Series in Applie Mathematics, Vol 61 Philaelphia, PA: SIAM 6 I DAUBECHIES, B HAN, A RON, Z W SHEN 23): Framelets: MRA-base constructions of wavelet frames Appl Comput Harmon Anal, 14: B HAN 1997): On ual wavelet tight frames Appl Comput Harmon Anal, 4: B HAN 23): Compactly supporte tight wavelet frames an orthonormal wavelets of exponential ecay with a general ilation matrix J Comput Appl Math, 155:43 67

28 352 I Daubechies an B Han 9 B HAN, Q MO 23): Multiwavelet frames from refinable function vectors Av in Comput Math, 18: A RON 199): Factorization theorems for univariate splines on regular gris Israel J Math, 7: A RON,ZWSHEN 1997): Affine systems in L 2 R ) II: Dual systems J Fourier Anal Appl, 3: L VILLEMOES 1993): Sobolev regularity of wavelets an stability of iterate filter banks In: Progress in Wavelet Analysis an Applications Y Meyer, S Roques, es), pp I Daubechies Program in Applie an Computational Mathematics Princeton University Princeton, NJ 8544 USA ingri@mathprincetoneu www: ic B Han Department of Mathematical an Statistical Sciences University of Alberta Emonton, Alberta Canaa T6G 2G1 bhan@mathualbertaca www: bhan