COMPACTLY SUPPORTED ORTHONORMAL COMPLEX WAVELETS WITH DILATION 4 AND SYMMETRY BIN HAN AND HUI JI Abstract. In this paper, we provide a family of compactly supported orthonormal complex wavelets with dilation 4 such that the generating wavelet functions are symmetric/antisymmetric and have linearly increasing orders of vanishing moments and smoothness.. Introduction and Main Result It is well known in Daubechies [5] that except the Haar wavelet which is discontinuous, compactly supported dyadic orthonormal real-valued wavelets cannot have symmetry. In order to achieve symmetry, wavelets with other dilations have been considered in the literature. Indeed, a few examples of compactly supported orthonormal real-valued wavelets with symmetry and dilations greater than two have been reported in [, 3, 4, 6, 9]. All such examples available in the literature are obtained by solving systems of nonlinear quadratic algebraic equations. For example, by complicated calculation, only some examples of compactly supported C symmetric orthonormal real-valued wavelets with dilation 4 have been obtained in [6]. To the best of our knowledge, no compactly supported C 2 symmetric orthonormal real-valued wavelets with dilation 4 have been known so far in the literature. On the other hand, it has been observed in the interesting work Lawton [] that symmetry can also be achieved by considering orthonormal complex wavelets. Motivated by [8, ], in this paper we consider symmetric orthonormal complex wavelets with dilation 4 and we provide a family of compactly supported arbitrarily smooth orthonormal complex wavelets with dilation 4 and symmetry. In order to introduce our main result, let us recall some necessary notation. Throughout the paper, i denotes the imaginary unit such that i 2 =. The Fourier transform in this paper is defined to be ˆfξ) := R fx)e iξx dx for f L R) and is naturally extended to L 2 functions and distributions. For a compactly supported complex-valued function φ : R C, we say that φ is refinable with dilation 4 if ˆφ4ξ) = âξ)ˆφξ) for a 2π-periodic trigonometric polynomial â with complex coefficients. Such a trigonometric polynomial â is called the mask for the refinable function φ. We say that φ is an orthonormal refinable function with dilation 4 and mask â, if ˆφ4ξ) = âξ)ˆφξ) and the integer shifts of φ are orthonormal, that is, φ k), φ := φx k)φx)dx = δ k, k Z,.) R where φx) denotes the complex conjugate of φx) and δ denotes the Dirac sequence such that δ = and δ k = for all k. If φ is an orthonormal refinable function with dilation 4 and mask â, then it is well known that â must be an orthogonal mask, that is, âξ) 2 + âξ + π/2) 2 + âξ + π) 2 + âξ + 3π/2) 2 =, ξ R..2) 2 Mathematics Subject Classification. 42C4, 42C5. Key words and phrases. Orthonormal complex wavelets, symmetry, smoothness, vanishing moments. Research supported in part by the Natural Sciences and Engineering Research Council of Canada NSERC Canada) under Grant RGP 2285. February 25, 28.
2 BIN HAN AND HUI JI For a function f L 2 R), its smoothness is measured by the quantity { ν 2 f) := sup ν R : ˆfξ) } 2 + ξ 2 ) ν dξ <..3) R For a smooth function f, we denote f j) the jth derivative of f. For a compactly supported function f : R C, we say that f has m vanishing moments if ˆf j) ) = for all j =,,..., m. The notion of vanishing moments plays an important role in wavelet analysis. For each positive integer m, we define P m to be a polynomial of degree m such that P m x) := [ x) 2x) 2 ] m + Ox m ), x..4) That is, P m is the m )-th Taylor polynomial of the function [ x) 2x) 2 ] m at x =. Now we state the main result of this paper. Theorem. Let m be a positive odd integer and denote P m the polynomial defined in.4). Then there exist two polynomials Pm r and P m i with real coefficients such that P m x) = [P r mx)] 2 + [P i mx)] 2, x R with P r m) =, P i m) =..5) Define a 2π-periodic trigonometric polynomial with complex coefficients by â m ξ) := 4 m e iξ3m )/2 + e iξ + e 2iξ + e 3iξ ) m [P r m sin2 ξ/2)) + ip i m sin2 ξ/2))].6) and define φ m ξ) := â m 4 j ξ), ξ R..7) j= Then φ m is a compactly supported orthonormal refinable function with dilation 4 and mask â m. Moreover, there exist three 2π-periodic trigonometric polynomials b m,, b m,2 and b m,3 such that ) {2 j ψ m, 4 j k), 2 j ψ m,2 4 j k), 2 j ψ m,3 4 j k) : j, k Z} is an orthonormal basis of L 2 R), where ψ m, 4ξ) := b m, ξ) φ m ξ), ψm,2 4ξ) := b m,2 ξ) φ m ξ), ψm,3 4ξ) := b m,3 ξ) φ m ξ)..8) 2) φ ) = φ and ψ m, ) = ψ m,, ψ m,2 ) = ψ m,2, ψ m,3 ) = ψ m,3. 3) All ψ m,, ψ m,2, ψ m,3 have m vanishing moments and lim inf m min l 3 ν 2 ψ m,l ) m >. The polynomials Pm r and P m i can be constructed from P m by [8, Proposition 9]. In Section 2, we shall propose an algorithm to construct the high-pass wavelet masks b m,, bm,2 and b m,3 in Theorem from the low-pass mask â m. In Section 3 we shall present several examples of orthonormal complex wavelets with dilation 4 and symmetry to illustrate the main result in Theorem and Algorithm 3 of this paper. The proof of Theorem will be provided in Section 4. 2. An Algorithm for Constructing High-pass Wavelet Masks In this section, in order to obtain the wavelet masks b m,, b m,2, b m,3 in Theorem, we shall propose an algorithm to construct high-pass wavelet masks with symmetry from a given symmetric low-pass mask with complex coefficients. Throughout the paper, we shall use Rec), Imc), and c to denote the real part, the imaginary part, and the complex conjugate of a complex number c C. For a nonzero 2π-periodic trigonometric polynomial âξ) = k Z a ke ikξ, we denote degâ) := min{n Z : a k = for all k > N}. In order to state the algorithm for deriving high-pass wavelet masks from low-pass masks, we need the following result.
ORTHONORMAL COMPLEX WAVELETS WITH DILATION 4 AND SYMMETRY 3 Lemma 2. Let Aξ) := [A ξ), A 2 ξ), A 3 ξ), A 4 ξ)] T be a column vector of 2π-periodic trigonometric polynomials with complex coefficients such that A satisfies and A ξ) = A ξ), A 2 ξ) = A 2 ξ), A 3 ξ) = A 3 ξ), A 4 ξ) = A 4 ξ) 2.) A ξ) 2 + A 2 ξ) 2 + A 3 ξ) 2 + A 4 ξ) 2 = ξ R. 2.2) Denote dega) := maxdega ), dega 2 ), dega 3 ), dega 4 )). If dega) >, then by 2.) we can write f f 3 f 3 f Aξ) = f 2 g eikξ + f 4 g 3 eik )ξ + + f 4 g 3 e ik )ξ + f 2 g e ikξ, 2.3) g 2 g 4 g 4 g 2 where k = dega) and f, f 2, f 3, f 4, g, g 2, g 3, g 4 are some complex numbers when dega) =, we replace f 3, f 4, g 3, g 4 in 2.3) by f 3 /2, f 4 /2,, respectively). Now we define Bξ) := [B ξ), B 2 ξ), B 3 ξ), B 4 ξ)] T := U A ξ)aξ), 2.4) where c f cos ξ i h ) c c f 2 cosξ i h) ic c g sin ξ ic g 2 sin ξ 2 U A ξ) := f 2 c 2 f c 2 g 2 c 2 g c ic f sin ξ ic f 2 sin ξ c g cosξ + i h) c c g 2 cosξ + i h) c 2.5) with c := f 2 + f 2 2 + g 2 + g 2 2, h := Imf f 3 + f 2 f 4 g g 3 g 2 g 4 ), 2c c := c 2 + h2. 2.6) Then 2.) and 2.2) hold with A being replaced by B, degb) < dega), and U A ξ)u A ξ) T = I 4. Proof. By calculation, it follows from the definition of the vector Bξ) in 2.4) that we have B ξ) := c f A ξ) + f 2 A 2 ξ))cosξ i h ) c ) + ig A 3 ξ) + g 2 A 4 ξ)) sinξ, B 2 ξ) := 2/c f A 2 ξ) f 2 A ξ) ), B 3 ξ) := 2/c g 2 A 3 ξ) g A 4 ξ) ), B 4 ξ) := c if A ξ) + f 2 A 2 ξ)) sin ξ + g A 3 ξ) + g 2 A 4 ξ))cosξ + i h c ) ). 2.7) By 2.), now it is straightforward to see that B ξ) = B ξ), B 2 ξ) = B 2 ξ), B 3 ξ) = B 3 ξ) and B 4 ξ) = B 4 ξ). By 2.3) and 2.7), it is easy to see that degb 2 ) < k, degb 3 ) < k, and the degrees of B and B 4 are no greater than k +. By a direct calculation, it follows from 2.3) and 2.7) that B ξ) = c e ik+)ξ + c 2 e ikξ + + c 2 e ikξ + c e ik+)ξ, B 4 ξ) = c 3 e ik+)ξ c 4 e ikξ + + c 4 e ikξ + c 3 e ik+)ξ,
4 BIN HAN AND HUI JI where c := c f 2 + f 2 2 g 2 g 2 2) /2, c 2 := c f f 3 + f 2 f 4 g g 3 g 2 g 4 ) i2 f 2 + f 2 2 )h/c ) /2, c 3 := c g 2 + g 2 2 f 2 f 2 2) /2, c 4 := c g g 3 + g 2 g 4 f f 3 f 2 f 4 ) + i2 g 2 + g 2 2 )h/c ) /2. To show that degb) < dega) = k, we have to prove that degb ) < k and degb 4 ) < k. In other words, we need to prove that c = c 2 = c 3 = c 4 =. From 2.2), we observe that f 2 + f 2 2 = g 2 + g 2 2 and Ref f 3 + f 2 f 4 ) = Reg g 3 + g 2 g 4 ). 2.8) It follows directly from the first identity in 2.8) that c = c 3 =. By the second identity in 2.8), we deduce that Rec 2 ) = Rec 4 ) =. To show that c 2 = c 4 =, now it suffices to show that Imf f 3 + f 2 f 4 g g 3 g 2 g 4 ) = 2 f 2 + f 2 2 )h/c = 2 g 2 + g 2 2 )h/c. But the above identity is obviously true by the definition of h and c in 2.6). Consequently, we verified that degb) < dega). Now we prove U A ξ)u A ξ) T = I 4. By a direct calculation, we have U, U,4 U A ξ)u A ξ) T = 2 f 2 + f 2 2 )/c 2 g 2 + g 2 2 )/c. U 4, U 4,4 with ) U, := c 2 f 2 + f 2 2 )cos 2 ξ + h 2 /c 2 ) + g 2 + g 2 2 ) sin 2 ξ, U,4 := ic 2 g 2 + g 2 2 f 2 f 2 2 )cosξ ih/c) sin ξ, ) U 4, := ic 2 f 2 + f 2 2 g 2 g 2 2 )cosξ + ih/c) sin ξ, ) U 4,4 := c 2 f 2 + f 2 2 ) sin 2 ξ + g 2 + g 2 2 )cos 2 ξ + h 2 /c 2 ). Note that by 2.8), we have f 2 + f 2 2 = c/2 = g 2 + g 2 2. Now U A ξ)u A ξ) T = I 4 can easily verified by the definition of c, c, h in 2.6). Consequently, we conclude that Bξ) T Bξ) = Aξ) T U A ξ) T U A ξ)aξ) = Aξ) T Aξ) =, which completes the proof. Now we state the algorithm for deriving the high-pass wavelet masks b m,, b m,2 and b m,3 from the low-pass mask â m. Note that â m ξ) = e iξ â m ξ), where â m is defined in.6). Algorithm 3. Let âξ) = k Z a ke ikξ be a 2π-periodic trigonometric polynomial with complex coefficients such that â) =, â satisfies.2) and â ξ) = e iξ âξ). Denote A ξ) := 2[â ξ) + â ξ),â2 ξ) + â2 ξ),â ξ) â ξ),â2 ξ) â2 ξ)] T, 2.9) where â j ξ) := k Z a j+4k e ikξ, ξ R, j Z. 2.) Then A satisfies all the conditions in 2.) and 2.2) with A being replaced by A.
ORTHONORMAL COMPLEX WAVELETS WITH DILATION 4 AND SYMMETRY 5 ) Recursively apply Lemma 2 and define A j ξ) := U A j ξ)u A j 2ξ) U A ξ)u A ξ)a ξ) 2.) until dega j ) = for some nonnegative integer j dega ), where all U A,...,U A j are given in Lemma 2. 2) By dega j ) =, it follows from 2.) and 2.2) that A j = [h, h 2,, ] T for some complex numbers h, h 2 C such that h 2 + h 2 2 =. Define a matrix Uξ) by h h 2 Uξ) := U A ξ) T U A j ξ) T h 2 h. 2.2) 3) Obtain the high-pass wavelet masks b, b 2 and b 3 by 2 b l ξ) := + e iξ )U,l 4ξ) + e iξ + e i2ξ )U 2,l 4ξ) 4 ) + e iξ )U 3,l 4ξ) + e i2ξ e iξ )U 4,l 4ξ), l = 2, 3, 4, 2.3) Then where U j,k ξ) denotes the j, k)-entry of the 4 4 matrix Uξ). b ξ) = e iξ b ξ), b2 ξ) = e iξ b2 ξ), b3 ξ) = e iξ b3 ξ), 2.4) and P [â, b, b 2, b ξ)p 3 ] [â, b, b 2, b 3 ] ξ)t = I 4, that is, P [â, b, b 2, b ξ) is a unitary matrix, where 3 ] âξ) âξ + π/2) âξ + π) âξ + 3π/2) b P [â, b, b 2, b ξ) := ξ) b ξ + π/2) b ξ + π) b ξ + 3π/2) 3 ] b 2 ξ) b2 ξ + π/2) b2 ξ + π) b2 ξ + 3π/2). 2.5) b 3 ξ) b3 ξ + π/2) b3 ξ + π) b3 ξ + 3π/2) Proof. Since â ξ) = e iξ âξ), by the definition of âj in 2.), we have â ξ) = â ξ) and â ξ) = â2 ξ), ξ R. Now by.2), we see that 2.2) holds with A being replaced by A. It is evident that 2.) holds with A being replaced by A. By Lemma 2, we have Uξ)Uξ) T = I 4 and it is not difficult to deduce that the symmetry pattern of Uξ) is the same as U A ξ) T in 2.5). More precisely, we have { U j,k ξ), j, k {, 2} or j, k {3, 4}, U j,k ξ) = U j,k ξ), otherwise. Now the symmetry of b, b 2 and b 3 in 2.4) can be checked easily using 2.3). Note that b in 2.3) is just â. 3. Some Examples of Symmetric Orthonormal Complex Wavelets In this section, using Theorem and Algorithm 3, we present several examples of symmetric orthonormal complex wavelets with dilation 4. Before presenting some examples, let us recall a quantity ν 2 â, 4) from [7]. Let â be a 2πperiodic trigonometric polynomial with â) =. Write âξ) = +e iξ +e i2ξ +e i3ξ ) m ĉξ) for some nonnegative integer m and some 2π-periodic trigonometric polynomial ĉξ) with ĉπ/2) 2 + ĉπ) 2 + ĉ3π/2) 2. Write ĉξ) 2 = K k= K c ke ikξ, where K is some nonnegative integer.
6 BIN HAN AND HUI JI Denote ρâ, 4) the spectral radius of the square matrix c 4j k ) K/3 j,k K/3 and define ν 2 â, 4) := /2 log 4 ρâ, 4) see [6, Theorem2.] or [7, page 6]). The quantity ν2 â, 4) plays an important role in wavelet analysis and subdivision schemes. Define ˆφξ) := j= â4 j ξ). Then φ is an orthonormal refinable function with dilation 4 and mask â, if and only if,.2) holds and ν 2 â, 4) >, which in addition imply that ν 2 φ) = ν 2 â, 4). See [7] for more details on the quantity ν 2 â, 4) and related references. Example. Let m = 3. Then P 3 x) = + 5 x + 26 x 2, P r 3 x) = + 5/2 x, P i 3x) = 3 3/2 x. By Theorem and Algorithm 3, we have and â 3 ξ) = ei4ξ 52 + e iξ + e i2ξ + e i3ξ ) 3[ 38 + i6 3) 5 + i3 3)e iξ + e iξ ) ] b 3, ξ) = e iξ ) 3 52 9 ei2ξ ) [ 89 3 i23) + 3 3 i39)e iξ + e iξ ) + 43 3 i3)e i2ξ + e i2ξ ) + 3 3 i5)e i3ξ + e i3ξ ) ], b 3,2 ξ) = e iξ e iξ ) 3 2, 5 b 3,3 ξ) = e iξ e iξ ) 3 28 [ 366 3 i7) + 65 3 i)e iξ + e iξ ) + 35 3 i7) 95 e i2ξ + e i2ξ ) + 23 3 i3)e i3ξ + e i3ξ ) + 5 3 i5)e i4ξ + e i4ξ ) ]. By calculation, ν 2 â3, 4) = 3 log 6 223.49775. In fact, by [6, Corollary 2.2], we have ν â3, ) = 5/4 log 6 5.66957 see [6, 7]) and φ 3 C.66957 R). See Figure for the graphs of φ 3, ψ 3,, ψ 3,2, and ψ 3,3. Example 2. Let m = 5. Then P 5 x) = + 25 x + 335 x 2 + 395 x 3 + 243 x 4 and P r 5 x) = + 25/2 x + 75 t 2 )/8 x 2, P i 5x) = tx/2 + x85249 + 85t 2 t 4 )/648), where t 9.588 is a real root satisfying t 6 85t 4 66365t 2 + 5959225 =. By Theorem and Algorithm 3, we have â 5 ξ) = b 5,2 ξ) = a 5 k e ikξ + e ik )ξ ), b5, ξ) = k= 9 k= 9 b 5,2 k e ikξ + e ik )ξ ), b5,3 ξ) = k= 9 k= 9 b 5, k e ikξ + e ik )ξ ), b 5,3 k e ikξ + e ik )ξ ),
ORTHONORMAL COMPLEX WAVELETS WITH DILATION 4 AND SYMMETRY 7.5.5.5.5.5.5.5 2 2 3.5 2 2 3 3 2 3 2 2 2 3 2 2 3 3 2 2 3 Figure. The graphs of the orthonormal refinable function φ 3 and its associated three wavelets ψ 3,, ψ 3,2, ψ 3,3 in Example. where a 5 ) := [a 5 9, a5 8,..., a5 ]T and b 5,l ) := [b 5,l 9,...,b5,l ]T, l =, 2, 3 are given by.482893.822.82233.7537.7778.79438.83993.63957.334978.488276.822842.85.85988.56533.66656.4953 a 5 ) =.35849.56536 + i.438589.438589, b 5, ) =.883.6888 + i.99828.5836.25853.394356.922.4485.85426.35892.899899.65285.23534.646825.252284.8594.24376.7374.24848.87882 b 5,2 ) =.27564.362.759572.24355.52.3486 + i.7598.27646.5298.33765.35626.6463, b 5,3 ) =.828275.7658.49392.589.53424.4854.345348.62297.287684.6239 + i.487755.7957.33835.868537.28765.375728.34852.89856.339.3856 By calculation, ν 2 â5, 4).274852. See Figure 2 for the graphs of φ 5, ψ 5,, ψ 5,2, and ψ 5,3.,.
8 BIN HAN AND HUI JI.5.5.5.5.5.5.5 2 2 3.5 2 2 3.5.5.5.5.5.5.5 2 2 3.5 2 2 3 Figure 2. The graphs of the orthonormal refinable function φ 5 and its associated three wavelets ψ 5,, ψ 5,2, ψ 5,3 in Example 2. 4. Proof of Theorem First, we prove that there are two polynomials Pm r and P m i with real coefficients satisfying.5). By [8, Proposition 9],.5) holds if and only if P m ) = and P m x) x R. 4.) By the definition of P m in.4), it is evident that P m ) = and all the coefficients of P m are nonnegative. Therefore, it is straightforward to see that P m x) for all x. To prove 4.), we have to show P m x) for all x <, which is the major part of this proof. Denote and Aξ) := cos 2m ξ/2) cos 2m ξ)p m sin 2 ξ/2)) 4.2) Hξ) := Aξ/4) + Aξ/4 + π/2) + Aξ/4 + π) + Aξ/4 + 3π/2). 4.3) By.4), both A and H are 2π-periodic trigonometric polynomials such that A ξ) = Aξ), H ξ) = Hξ), dega) < 4m, degh) < m. 4.4) By the definition of A in 4.2) and H in 4.3), it follows from.4) that Hξ) = Aξ/4) + O ξ 2m ) = sin 2 ξ/8)) m 2 sin 2 ξ/8)) 2m P m sin 2 ξ/8)) + O ξ 2m ) = + O ξ 2m ), ξ. That is, [H ] j) ) = for all j =,..., 2m. Since H ξ) = Hξ) and degh) < m, from the above relation, we must have Hξ). That is, we have Aξ) + Aξ + π/2) + Aξ + π) + Aξ + 3π/2) =. 4.5)
ORTHONORMAL COMPLEX WAVELETS WITH DILATION 4 AND SYMMETRY 9 Note that there is a unique polynomial Q with real coefficients and degq) < m such that Qsin 2 ξ)) = cos 2m ξ/2)p m sin 2 ξ/2)) + sin 2m ξ/2)p m cos 2 ξ/2)). By sin 2 ξ) = 4 sin 2 ξ/2) sin 2 ξ/2)), the above identity can be rewritten as Q4y y)) = y) m P m y) + y m P m y) with y := sin 2 ξ/2). 4.6) Consequently, by the definition of Q, we have and Aξ) + Aξ + π) = cos 2m ξ)qsin 2 ξ)) = 2y) 2m Q4y y)) Aξ + π/2) + Aξ + 3π/2) = sin 2m ξ)qcos 2 ξ)) = 4y y)) m Q 2y) 2 ). Now by 4.5) and the above two identities, we conclude that 2y) 2m Q4y y)) + 4y y)) m Q 2y) 2 ) = y. 4.7) Taking y = x)/2 with x in 4.7), we get x m Q x) + x) m Qx) = x. 4.8) From the above identity, now it is straightforward to see that Qx) = x) m + Ox m ), x. Since degq) < m, the above relation implies that Q must be the m )th Taylor polynomial of x) m at x =. So, Qx) = m m+j ) j= j x j and all the coefficients of Q are nonnegative. Now we are ready to prove P m x) for all x <. By 4.6) and 4.7), we have 2x) 2m x) m P m x) = 4x x)) m Q 2x) 2 ) 2x) 2m x m P m x) x R. 4.9) Since all the coefficients of P m and Q are nonnegative, noting that m is an odd integer, we have 4x x)) m <, Q 2x) 2 ), x m <, 2x) 2m P m x), x <. Consequently, it follows from 4.9) that 2x) 2m x) m P m x), x <. That is, we must have P m x) > for all x <. Hence, 4.) is verified and by [8, Proposition 9].5) holds. Next we prove that â m is an orthogonal mask. By.6) and.5), we have â m ξ) 2 = cos 2m ξ/2) cos 2m ξ)p m sin 2 ξ/2)) = Aξ). Now by 4.5), we see that â m is an orthogonal mask. Since P m sin 2 ξ/2)) > for all ξ R, it is easy to see that φ m ξ) = if and only if ξ 2πZ\{}. Now it is a standard argument in wavelet analysis to check that φ m L 2 R) is an orthonormal refinable function with dilation 4 and mask â m. Note that â m ξ) = e iξ â m ξ). By Algorithm 3, there exist 2π-periodic trigonometric polynomials b m,, b m,2 and b m,3 such that 2.4) holds with â, b, b 2, b 3 being replaced by â m, b m,, b m,2, b m,3. Moreover, P [â m, b m,, b m,2, b is a unitary matrix, where P m,3 ] [â m, b m,, b m,2, b is defined in 2.5). Now m,3 ] it is easy to verify that Item 2) holds. By the standard argument in wavelet analysis, Item ) is true and all ψ m,, ψ m,2, ψ m,3 have m vanishing moments. Since ν 2 ψ m,l ) ν 2 φ m ) for all l =, 2, 3, to complete the proof, it suffices to prove that ν lim inf 2 φ m ) m >. Let ĉ m m be a 2π-periodic trigonometric polynomial with real coefficients such that ĉ m ) = and ĉ m ξ) 2 = cos 2m ξ/2) cos 2m ξ)p m sin 2 ξ/2)).
BIN HAN AND HUI JI Define φ cm by φ cm ξ) := j= ĉm 4 j ξ). Then φ cm is a refinable function with dilation 4 and mask ĉ m ν. It is known in [2] that lim 2 φ cm ) m = m log m 4 sin 2π ) >. On the other hand, 5 it is easy to see that φ m ξ) 2 = φ cm ξ) 2. Consequently, ν 2 φ m ) = ν 2 φ cm ) and therefore, ν lim 2 φ m ) m = m log m 4sin 2π ) >. This completes the proof. 5 References [] E. Belogay and Y. Wang, Compactly supported orthogonal symmetric scaling functions. Appl. Comput. Harmon. Anal. 7 999), 37 5. [2] N. Bi, X. Dai and Q. Sun, Construction of compactly supported M-band wavelets, Appl. Comput. Harmon. Anal. 6 999), 3 3. [3] N. Bi, B. Han, and Z. Shen, Examples of refinable componentwise polynomials. Appl. Comput. Harmon. Anal. 22 27), 368 373. [4] C. K. Chui and J. A. Lian, Construction of compactly supported symmetric and antisymmetric orthonormal wavelets with scale = 3. Appl. Comput. Harmon. Anal. 2 995), 2 5. [5] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 4 988), 674 996. [6] B. Han, Symmetric orthonormal scaling functions and wavelets with dilation factor 4. Adv. Comput. Math. 8 998), 22 247. [7] B. Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, J. Approx. Theory 24 23), 44 88. [8] B. Han, Symmetric complex Coiflets of arbitary orders, preprint, 27). [9] H. Ji and Z. Shen, Compactly supported bi)orthogonal wavelets generated by interpolatory refinable functions. Adv. Comput. Math. 999), 8 4. [] W. Lawton, Applications of complex valued wavelet transforms to subband decomposition, IEEE Trans. Signal. Proc. 4 993), 3566 3568. [] W. Lawton, S. L. Lee, and Z. Shen, An algorithm for matrix extension and wavelet construction. Math. Comp. 65 996), 723 737. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G. E-mail address: bhan@math.ualberta.ca URL: http://www.ualberta.ca/ bhan Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 7543. E-mail address: matjh@nus.edu.sg URL: http://www.math.nus.edu.sg/~matjh