# Some Bivariate Smooth Compactly Supported Tight Framelets with Three Generators

Authors: A. San Antolín; R. A. Zalik
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

For any2×2 dilation matrix with integer entries and |detA|=2, we construct a family of smooth compactly supported tight wavelet frames with three generators in L2(ℝ2). Our construction involves some compactly supported refinable functions, the oblique extension principle, and a slight generalization of a theorem of Lai and Stöckler. Estimates for the degrees of smoothness are given. With the exception of a polynomial whose coefficients must in general be computed by spectral factorization, the framelets are expressed in closed form in the frequency domain, in terms of elementary transcendental functions. By means of two examples we also show that for low degrees of smoothness the use of spectral factorization may be avoided.


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## Body

## 1. Introduction

Given a2×2 dilation matrix A with integer entries, such that |detA|=2, we construct smooth compactly supported tight framelets with three generators in L2(ℝ2) associated to such a dilation, and with any desired degree of smoothness. Tight wavelet frames have recently become the focus of increased interest because they can be computed and applied just as easily as orthonormal wavelets, but are easier to construct.We begin with notation and definitions. The sets of strictly positive integers, integers, and real numbers will be denoted byℕ, ℤ, and ℝ, respectively. Given a (Lebesgue) measurable set S⊂ℝd, |S| will denote its Lebesgue measure on ℝd and χS will be its characteristic function. Given a matrix M, its transpose will be denoted by MT and the conjugate of its transpose will be denoted by M*. The n×n identity matrix will be denoted by In×n.We say thatA∈ℝd×d is a dilation matrix preserving the lattice ℤd if all eigenvalues of A have modulus greater than 1 and A(ℤd)⊂ℤd. The set of all d×d dilation matrices preserving the lattice ℤd will be denoted by Ed(ℤ). Note that if A∈Ed(ℤ), then dA:=|detA| is an integer greater than 1. The quotient group ℤd/Aℤd is well defined, and by ΔA⊂ℤd we will denote a full collection of representatives of the cosets of ℤd/Aℤd. Recall that there are exactly dA cosets [1].A sequence{ϕn}n=1∞ of elements in a separable Hilbert space ℍ is a frame for ℍ if there exist constants C1,C2>0 such that
(1)C1∥h∥2≤∑n=1∞‍|〈h,ϕn〉|2≤C2∥h∥2,∀h∈ℍ,
where 〈·,·〉 denotes the inner product on ℍ. The constants C1 and C2 are called frame bounds. The definition implies that a frame is a complete sequence of elements of ℍ. A frame {ϕn}n=1∞ is tight if we may choose C1=C2.LetA∈Ed(ℤ). A set of functions Ψ={ψ1,…,ψN}⊂L2(ℝd) is called a wavelet frame or framelet with dilation A if the system
(2){dAj/2ψℓ(Ajx+k);j∈ℤ,k∈ℤd,1≤ℓ≤N}
is a frame for L2(ℝd). If this system is a tight frame for L2(ℝd), then Ψ is called a tight framelet.Letf^ denote the Fourier transform of the function f. Thus, if f∈L1(ℝd), x,t∈ℝd,
(3)f^(t):=∫ℝd‍f(x)e-2πix·tdx,
where x·t denotes the dot product of vectors x and t. The Fourier transform is extended to L2(ℝd) in the usual way.Han [2], and independently Ron and Shen [3], found necessary and sufficient conditions for translates and dilates of a set of functions to be a tight framelet. Ron and Shen also formulated what is known as the Unitary Extension Principle (UEP), which, in addition to its other applications, provides a method for constructing compactly supported framelets. In [4] (see also [5]), Ron and Shen show two bivariate constructions of compactly supported tight framelets in L2(ℝ2) with dilation matrix
(4)B:=(111-1)
and state that constructions of compactly supported tight framelets with dilation matrix B can be made in an analogous way. They also describe an algorithm for constructing compactly supported tight affine frames in L2(ℝd) with any dilation matrix. That algorithm works particularly well with box splines. Using Ron and Shen’s method, Gröchenig and Ron [6] show how to construct, for any dilation matrix, compactly supported framelets with any desired degree of smoothness. Furthermore, based on works by Ron and Shen and by Gröchenig and Ron, Han [7] also constructs compactly supported tight wavelet frames with degree of smoothness and vanishing moments of order as large as desired. Another method for constructing smooth compactly supported tight framelets was described in [8]. Note that while in [4, 6, 8] the number of generators of the tight framelets increases with the degree of smoothness, in [7] the number of generators may be bounded by a constant depending on the dimension and the determinant of the dilation matrix.The remainder of this paper is organized as follows. In Section2 we summarize results we will use for the proof of the main results. In Section 3, for any 2×2 dilation matrix A with integer entries and determinant ±2 we describe an algorithm to construct a family of compactly supported tight framelets with three generators and with any desired degree of smoothness and dilation factor A. In Section 4 we illustrate the results of this paper by means of examples.


## 2. Background

In this section we summarize the results we will use in our construction of tight framelets.The UEP led to the oblique extension principle (OEP), a method based on the UEP; it was developed by Chui et al. [9], and independently by Daubechies et al. [10], who gave the method its name. The OEP may be formulated as follows.Theorem A.
LetA∈Ed(ℤ). Let ϕ∈L2(ℝd) be compactly supported and refinable, that is,
(5)ϕ^(A*t)=P(t)ϕ^(t),
where P(x) is a trigonometric polynomial. Assume moreover that |ϕ^(0)|=1. Let S(t) be another trigonometric polynomial such that S(t)≥0 and S(0)=1. Assume that there are trigonometric polynomials or rational functions Qℓ, ℓ=1,…,N, that satisfy the OEP condition(6)S(A*t)P(t)P(t+j)¯+∑ℓ=1N‍Qℓ(t)Qℓ(t+j)¯6666={S(t)ifj∈ℤd,0ifj∈((A*)-1(ℤd)ℤd)∖ℤd.
If
(7)ψℓ^(A*t)∶=Qℓ(t)ϕ^(t),ℓ=1,…,N,
then Ψ={ψ1,…,ψN} is a tight framelet in L2(ℝd) with dilation factor A and frame constant 1.With an additional decay condition, Theorem A follows from [10, Proposition 1.11], except for the value of the frame constant, which follows from, for example, [3, Theorem 6.5]. However, recent results of Han imply that this decay condition is redundant. Indeed, Theorem A in its present formulation is a consequence of Proposition 4, Corollary 12, and Theorem 17 in [11] (for a simpler version of Han’s results in dimension, see [12]).We also need the following slight generalization of Theorem 3.4 of Lai and Stöckler [13]. The proof is similar and will be omitted. We have also included in the statement a generalization of the algorithm implicit in the proof of Theorem 3.4.Theorem B.
LetA∈Ed(ℤ) and let ΔA={qs}s=0dA-1 and ΔA*={ps}s=0dA-1 be full collections of representatives of the cosets of ℤd/Aℤd and ℤd/A*ℤd, respectively, with q0=p0=0. Let P(t) be a trigonometric polynomial defined on ℝd that satisfies the condition:
(8)∑s=0dA-1‍|P(t+(A*)-1(ps))|2≤1,
let
(9)𝒫(t)∶=(P(t+(A*)-1(ps));s=0,…,dA-1)T,
and let
(10)ℳ(t)∶=dA-1/2(ei2πql·t+((A*)-1(ps));ℓ,s=0,…,dA-1)
be the polyphase matrix, where s denotes the row index and ℓ denotes the column index.
Let thedA×1 matrix function G(t) be defined by
(11)G(t):=ℳ*(t)𝒫(t)=(Lk(A*t);k=0,…,dA-1)T,
and suppose that there exist trigonometric polynomials P~1,…,P~M such that
(12)∑k=0dA-1‍|Lk(t)|2+∑j=1M‍|P~j(t)|2=1.
Let N:=dA+M and let the N×1 matrix function 𝒢(t) be defined by
(13)𝒢(t)∶=(P~j(A*t);1≤j≤MLk(A*t);k=0,…,dA-1,66P~j(A*t);1≤j≤M)T,𝒬~(t)∶=IN×N-𝒢(t)𝒢*(t).
Let K(t) denote the first dA×N block matrix of 𝒬(t)~,
(14)𝒬(t)∶=ℳ(t)K(t),
and let [Q1(t),…,QN(t)] denote the first row of 𝒬(t). Then the trigonometric polynomials P and Qℓ, ℓ=1,…,N, satisfy the identity (6) with S(t)=1.Using the OEP and Theorem B, we will obtain a general method for constructing compactly supported framelets inL2(ℝ2) valid for any dilation matrix.We will use the following theorem of Gröchenig and Madych [1].Theorem C.
LetA∈Ed(ℤ) and let ΔA={qs}s=0dA-1 be a full collection of representatives of the cosets of ℤd/Aℤd with q0=0. Then the characteristic function χE, where the set E is defined by
(15)E∶={x∈ℝd:x=∑j=1∞‍A-jk(j),k(j)∈ΔA},
is a nonnull compactly supported measurable function, ∥χE∥L2(ℝd)≥1, and it satisfies the refinement equation:
(16)χE^(A*(t))=H(t)χE^(t),whereH(t)∶=1dA∑s=0dA-1‍e-2πit·qs.The following statement may be found in [14, Appendix A.2]. The proof is straightforward and will be omitted.Lemma D.
LetC0 be the class of continuous functions in L2(ℝd), and let Cr, r=1,2,… be the class of functions f such that all partial derivatives of f of order not greater than r are continuous and in L2(ℝd). If
(17)|f^(t)|≤C(1+|t|)-N-ε
for some integer N≥d and ε>0, then f is in CN-d.Proofs of the following proposition may be found in [15, Lemma 3.1], [14, Proposition 5.23], or [6, Result 2.6].Proposition E.
LetA∈Ed(ℤ) and let ΔA={qs}s=0dA-1 be a full collection of representatives of the cosets of ℤd/Aℤd with q0=0, and let E⊂ℝd be the set defined by (15). Then there exist two positive  constants ε and C such that
(18)|χE^(t)|≤C|t|-ε.


## 3. A Family of Tight Framelets

LetA∈ℤ2×2 be a dilation matrix with integer entries such that |detA|=2. In this section we construct smooth compactly supported tight framelets with three generators in L2(ℝ2) with dilation factor A and any desired degree of smoothness.Two matricesA and B with integer coefficients are integrally similar if there exists a matrix U with integer entries such that |detU|=1 and A=U-1BU. Let
(19)A1∶=(0210),A2∶=(02-10),A3:=(02-11),A4:=(0-21-1),A5:=(11-11),A6:=(-1-11-1).
The following complete classification of all matrices in E2(ℤ) with |detA|=2 was found by Lagarias and Wang [16, Lemma 5.2].Lemma F.
LetA∈E2(ℤ). If detA=-2, then A is integrally similar to A1. If detA=2, then A is integrally similar to one of the matrices Ak, k=2,…,6.We now focus on the dilation matricesAk. For r∈N, let
(20)Λ(r)∶={q0=(0,0)T,q1=(2r-1,0)T},(21)Γ∶={p0=(0,0)T,p1=(0,1)T}.It is easy to see that for eachr∈ℕ the set Λ(r) is a full collection of representatives of the cosets of ℤ2/Akℤ2, k=1,…,6, and that Γ is a full collection of representatives of the cosets of ℤ2/Ak*ℤ2, k=1,…,6. Moreover, we have
(22){(Ak*)-1p0,(Ak*)-1p1}={(0,0)T,(12,0)T},k=1,2,3,{(A4*)-1p0,(A4*)-1p1}={(0,(-12,0))T},{(A5*)-1p0,(A5*)-1p1}={(0,0)T,(12,12)T},{(A6*)-1p0,(A6*)-1p1}={(0,0)T,(-12,-12)T}.
For simplicity we shall consider Λ(1) as in (20) with r=1, and Γ as in (21). Then the corresponding matrix ℳ(t), t=(t1,t2), defined by (10), where A is any of the matrices Ak, k=1,…,6, is
(23)ℳ(t)=12(1ei2πt11-ei2πt1).
We now construct a family of smooth compactly supported refinable functions in L2(ℝ2) with dilation factor Ak.Proposition 1.
Let the matricesAk, k=1,…,6 be defined as in (19). Let m,n∈ℕ, t=(t1,t2) and P(t):=cos2n(2m-1)πt1. Let
(24)ϕ^(t)∶=||E|-1χ^E(t)|2n,
where E is the set defined by (15) with ΔA=Λ(m), and A is one of the matrices Ak. Then ϕ^(0)=1, and the function ϕ whose Fourier transform ϕ^ is defined by (24) has the following properties: ϕ is nonnull, compactly supported, and square integrable on ℝ2, ∥ϕ∥L2(ℝ2)≤1, and satisfies the refinement equation:
(25)ϕ^(Ak*(t))=P(t)ϕ^(t).
Moreover, if 2εn-2>r>1, where ε is defined in Proposition E, then ϕ is in continuity class Cr.Proof.
From Theorem C we deduce thatχE is a nonnull compactly supported function.
We now prove that∥ϕ∥L2(ℝ2)≤1. Let φ1:=|E|-1χE and for j∈ℕ let φj denote the j-fold convolution of φ1 with itself. Since ∥ϕ∥L2(ℝ2)=∥ϕ∥L2(ℝ2), by Young’s inequality for convolutions and bearing in mind that |E|≥1, we conclude that
(26)∥ϕ∥L2(ℝd)≤∥φn-1∥L2(ℝd)∥|E|-1χE∥L1(ℝd)=∥φn-1∥L2(ℝd)≤⋯≤∥φ1∥L2(ℝd)≤1.
We now show that ϕ^ satisfies the refinement equation (25). From (16) we have
(27)ϕ^(Ak*(t))=|E|-2n|χE^(Ak*(t))|2n=|H(t)|2n|E|-2n|χE^(t)|2n=P(t)ϕ^(t),
where H(t)=(1/2)(1+e2πi(2m-1)t1).
Sinceχ^E(0)=|E|, the definition of ϕ^ implies that ϕ^(0)=1.
We now prove the estimates for the degree of smoothness ofϕ. By Proposition E, we have
(28)|ϕ^(t)|=|E|-2n|χE^(t)|2n≤C|E|-2n|t|-2εn≤C|t|-2εn.
Moreover, since ϕ^ is continuous,
(29)|ϕ^(t)|≤D(1+|t|)-2εn.
Hence, if 2εn-2>r>1, Lemma D implies that ϕ is in continuity class Cr.
Finally, replicating an argument of Wojtaszczyk [14, page 79] it is easy to see that ϕ is a compactly supported function on ℝ2.We now construct tight frameletsΨ={ψ0,ψ1,ψ2} in L2(ℝ2) with dilation matrices A∈E2(ℤ) such that the functions ψℓ are smooth and compactly supported. For this purpose we use the refinable functions that we obtained in Proposition 1, Theorem B, the Oblique Extension Principle, and an appropriate change of variables.LetAk, k=1,…,6 be the matrices defined in (19), let Γ be defined by (21), assume that n,m∈ℕ, t=(t1,t2)∈ℝ2, and that P(t) defined as in Proposition 1. Using (22) we readily see that
(30)|P(t)|2+|P(t+(Ak*)-1p1)|266=cos4n((2m-1)πt1)6666+cos4n((2m-1)π(t1+12)),
and by an elementary computation we conclude that
(31)|P(t)|2+|P(t+(Ak*)-1p1)|2≤1,
and that equality holds only if 2(2m-1)t1∈ℤ. Since the values of the trigonometric polynomial P in Proposition 1 only depend on one variable, from a lemma of Riesz (cf., e.g., [17, Lemma 6.1.3], [14, Lemma 4.6]) we know there is a nonnull trigonometric polynomial L(Ak*t) on ℝ2 such that
(32)|L(Ak*t)|2=1-(|P(t)|2+|P(t+(Ak*)-1p1)|2)=1-cos4n((2m-1)πt1)-sin4n((2m-1)πt1).
The coefficients of L(t) may be obtained by spectral factorization ([18]).The following theorem describes the construction of smooth tight framelets of compact support and arbitrary degree of smoothness with dilation matrixAk.Theorem 2.
LetAk, k=1,…,6. Let m,n∈ℕ, t=(t1,t2), let P(t):=cos2n((2m-1)πt1), and let
(33)Q1(t)∶=12[1-cos4n((2m-1)πt1)6666-cos2n((2m-1)πt1)6666×sin2n((2m-1)πt1)],Q2(t)∶=ei2πt12[1-cos4n((2m-1)πt1)666666+cos2n((2m-1)πt1)666666×sin2n((2m-1)πt1)],Q3(t)∶=-12cos2n((2m-1)πt1)L(Ak*t)¯,
where L(Ak*t) is a trigonometric polynomial that satisfies (32). If ϕ^ is defined by (24),
(34)ψℓ^(Ak*t)∶=Qℓ(t)ϕ^(t),ℓ=1,2,3,
and Ψ={ψ1,ψ2,ψ3} is the set of inverse Fourier transforms of the functions ψℓ^ defined in the preceding displayed identity, then Ψ is a tight framelet with dilation factor Ak and frame constant 1, and the functions ψℓ have compact support. Moreover, if 2εn-2>r>1, where ε is defined in Proposition E, then the functions ψℓ(t) are in continuity class Cr.Proof.
Recall that the setΛ(m) defined by (20) is a full collections of representatives of the cosets of ℤ2/Akℤ2, and that the set Γ defined by (21) is a full collection of representatives of the cosets of ℤ2/Ak*ℤ2. Thus, the inequality (31) allows us to apply Theorem B.
From (9) we see that
(35)𝒫(t)=(cos2n((2m-1)πt1),sin2n((2m-1)πt1))T,
whence (23) implies that
(36)L0(Ak*t)=12(cos2n((2m-1)πt1)+sin2n((2m-1)πt1)),L1(Ak*t)=12e-i2πt1(cos2n((2m-1)πt1)66666666666-sin2n((2m-1)πt1)).
Thus, we have
(37)|L0(Ak*t)|2+|L1(Ak*t)|2=cos4n((2m-1)πt1)+sin4n((2m-1)πt1).
LetL be defined as in (32). Setting P~1:=L we therefore conclude that
(38)|L0(t)|2+|L1(t)|2+|P~1(t)|2=1.
Applying the algorithm described in Theorem B with N=3 and P~1=L, and bearing in mind that L0¯=L0, we obtain
(39)Q1(t)=12[1-|L0(Ak*t)|26666-ei2π(2b-1)t1L1(Ak*t)L0(Ak*t)]=12[1-cos4n((2m-1)πt1)6666-cos2n((2m-1)πt1)sin2n((2m-1)πt1)],Q2(t)=12[ei2πt1-L0(Ak*t)L1(Ak*t)¯6666-ei2πt1|L1(Ak*t)|2]=ei2πt12[1-cos4n((2m-1)πt1)666666+cos2n((2m-1)πt1)666666×sin2n((2m-1)πt1)],Q3(t)=-12[L0(Ak*t)+ei2πt1L1(Ak*t)]L(Ak*t)¯=-12cos2n((2m-1)πt1)L(Ak*t)¯.
Theorem B guarantees that the trigonometric polynomialsP, Q0, Q1 and Q2, satisfy the identity (6) with S(x)=1 and N=3. This may also be verified by direct computation. Applying Theorem A we conclude that Ψ is a tight framelet in L2(ℝ2).
Since the functionsQℓ are trigonometric polynomials and therefore bounded on ℝ2, the smoothness of the functions ψℓ follows from (29) and Lemma D.
Finally, note that the functionsψℓ are compactly supported because ϕ is compactly supported and the Qℓ are trigonometric polynomials.Using Theorem2 and Lemma F we now obtain an algorithm for constructing tight framelets in L2(ℝ2) with dilation factor A, for any matrix A∈ℝ2×2 preserving the lattice ℤ2 and with |detA|=2. These framelets have three compactly supported generators of arbitrary degree of smoothness.Corollary 3.
LetA∈E2(ℤ) with |detA|=2 and let k∈{1,…,6} be such that there exists a matrix U∈ℤ2×2 with det(U)=1, such that A=U-1AkU. Let m,n∈ℕ, t=(t1,t2), P(t):=cos2n(2m-1)πt1, let ϕ^ be defined by (24), and let Ψ={ψ1,ψ2,ψ3} be the set of inverse Fourier transforms of the functions ψℓ^ defined by (34). If
(40)θℓ(t)=ψℓ(Ut),ℓ=1,2,3,
then Θ={θ1,θ2,θ3}⊂L2(ℝ2) is a tight framelet with A, and the functions θℓ have compact support. Moreover, if 2εn-2>r>1, where ε is defined in Proposition E, then the functions θℓ(t) are in continuity class Cr.Proof.
SinceU-1ℤ2=ℤ2, the assertion that Θ is a tight framelet in L2(ℝ2) with frame constant 1 readily follows by a change of variable of the form t→U-1t.
Letr be an integer such that 2εn-2>r>1, where ε is defined in Proposition E. Since ψℓ is in continuity class Cr, applying the chain rule we conclude that also θℓ is in continuity class Cr.


## 4.  Examples

In this section, we illustrate the results of this paper showing examples of tight framelets where the use of spectral factorization may be avoided.Example 4.
LetAk, k∈{1,…,6}, m∈ℕ, n=1, t=(t1,t2), P(t):=cos2((2m-1)πt1) and Γ be the set defined by (21). We can obtain a tight framelet by elementary computations. Since
(41)1-|P(t)|2-|P(t+(Ak*)-1p1)|266666=1-cos4((2m-1)πt1)6666666-sin4((2m-1)πt1)66666=2-1sin2(2(2m-1)πt1),
a choice for L(t) is L(t):=(1/2)sin(2(2m-1)πt1). Thus, if we set
(42)Q1(t)=12[1-cos4((2m-1)πt1)66666-cos2((2m-1)πt1)sin2((2m-1)πt1)],Q2(t)=ei2πt12[1-cos4((2m-1)πt1)666666+cos2((2m-1)πt1)sin2((2m-1)πt1)],Q3(t)=-122cos2((2m-1)πt1)sin(2(2m-1)πt1),
and if ϕ^ is defined by (24) with n=1,
(43)ψℓ^(Ak*t)∶=Qℓ(t)ϕ^(t),ℓ=1,2,3,
and Ψ={ψ1,ψ2,ψ3} is the set of inverse Fourier transforms of the functions ψℓ^ defined in the preceding displayed identity, then Ψ is a tight framelet in L2(ℝ2) with dilation factor Ak, and the functions ψℓ are compactly supported.Example 5.
LetAk, k∈{1,…,6}, m∈ℕ, n=2, t=(t1,t2), P(t):=cos4((2m-1)πt1) and Γ be the set defined by (21). The case when n=2 is also sufficiently simple that it does not require the use of spectral factorization algorithms to compute L(t). We will just apply the arguments used to prove the lemma of Riesz and simple trigonometric identities.
From (37) we readily see that
(44)1-|L0(Ak*t)|2-|L1(Ak*t)|2=1-cos8((2m-1)πt1)-sin8((2m-1)πt1).
We define(45)q(t)∶=1-cos8((2m-1)πt)-sin8((2m-1)πt),t∈ℝ.
If s:=cos((2m-1)πt), we see that q(t)=D(cos((2m-1)πt)), where D(s)=-2s8+4s6-6s4+4s2. Since 4-6y+4y2-2y3=-2(y-1)(y-c)(y-c¯), where c=(1+i7)/2, we conclude that
(46)D(s)=-2s2(s2-1)(s2-c)(s2-c¯).
Setting z=ei(2m-1)πt and bearing in mind that |z|=1, this yields
(47)q(t)=-2[(z+z-12)2][(z+z-12)2-1]×[(z+z-12)2-c][(z+z-12)2-c¯]=-2(2z)-8(z2+1)2[(z2+1)2-4z2]B(z)=-2(2z)-8(z4-1)2B(z),
where
(48)B(z)=[(z2+1)2-4cz2][(z+1)2-4c¯z2].
Let(49)z0=12(27+42)1/2(1+i).
Then z0 is a zero of (z2+1)2-4cz2 and z0¯ is a zero of (z+1)2-4c¯z2.From the discussion in [17] or [14] we see that if z=eiπt, then
(50)B(z)=1|z0|2|(z-z0)(z-z0¯)(z-1z0)(z-1z0¯)|2.
Since q(t)≥0, we have
(51)q(t)=|q(t)|=2-7|z4-1|2B(z).
Thus, a choice for L(Ak*t) is
(52)L(Ak*t)∶=182|z0|(ei4(2m-1)πt1-1)×(ei(2m-1)πt1-z0)(ei(2m-1)πt1-z0¯)×(ei(2m-1)πt1-1z0)(ei(2m-1)πt1-1z0¯).
We now set
(53)Q1(t)=12[1-cos8((2m-1)πt1)6666-cos4((2m-1)πt1)sin4((2m-1)πt1)],(54)Q2(t)=ei2πt12[1-cos8((2m-1)πt1)6666666+cos4((2m-1)πt1)sin4((2m-1)πt1)],(55)Q3(t)=-12cos4((2m-1)πt1)L(Ak*t)¯,
where L(Ak*t) is defined by (52) with z0 given by (49). If ϕ^ is defined by (24) with n=2,
(56)ψℓ^(Ak*t):=Qℓ(t)ϕ^(t),ℓ=1,2,3
and Ψ={ψ1,ψ2,ψ3} is the set of inverse Fourier transforms of the functions ψℓ^ defined in the preceding displayed identity, then Ψ is a tight framelet in L2(ℝ2) with dilation factor Ak, and the functions ψℓ have compact support.



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