PSEUDO-SPLINES, WAVELETS AND FRAMELETS

Transcription

1 PSEUDO-SPLINES, WAVELETS AND FRAMELETS BIN DONG AND ZUOWEI SHEN Abstract The first type of pseudo-spines were introduced in [1, ] to construct tight frameets with desired approximation orders via the unitary extension principe of [] In the spirit of the first type of pseudo-spines, we introduce here a new type the second type of pseudo-spines to construct symmetric or antisymmetric tight frameets with desired approximation orders Pseudo-spines provide a rich famiy of refinabe functions B-spines are one of the specia casses of pseudo-spines; orthogona refinabe functions whose shifts form an orthonorma system given in [11] are another cass of pseudo-spines; and so are the interpoatory refinabe functions which are the Lagrange interpoatory functions at Z and were first discussed in [1] The other pseudo-spines with various orders fi in the gaps between the B-spines and orthogona refinabe functions for the first type, and between B-spines and interpoatory refinabe functions for the second type This gives a wide range of choices of refinabe functions that meets various demands for baancing the approximation power, the ength of the support, and the reguarity in appications This paper wi give a reguarity anaysis of pseudo-spines of the both types and provide various constructions of waveets and frameets It is easy to see that the reguarity of the first type of pseudo-spines is between B-spine and orthogona refinabe function of the same order However, there is no precise reguarity estimate for pseudo-spines in genera In this paper, an optima estimate of the decay of the Fourier transform of the pseudo-spines is given The reguarity of pseudo-spines can then be deduced and hence, the reguarity of the corresponding waveets and frameets The asymptotica reguarity anaysis, as the order of the pseudo-spines goes to infinity, is aso provided Furthermore, we show that in a tight frame systems constructed from pseudo-spines by methods provided both in [1] and this paper, there is one tight frameet from the generating set of the tight frame system whose diations and shifts aready form a Riesz basis for L R Key words and phrases B-spines, frameets, pseudo-spines, Riesz waveets, tight frame Corresponding author E-mai address: BDong@nusedusg Bin Dong, matzuows@nusedusg Zuowei Shen Address: Department of Mathematics, Science Drive, Nationa University of Singapore, Singapore, The research is supported by severa grants at Department of Mathematics, Nationa University of Singapore 1

2 BIN DONG AND ZUOWEI SHEN 1 Introductions Pseudo-spines were first introduced in [1, ] in order to construct tight frameets with required approximation order of the truncated frame series Pseudo-spines are refinabe and compacty supported They give a wide variety of choices of refinabe functions and provide arge fexibiities in waveet and frameet constructions and fiter designs Functions such as B-spines, interpoatory, or orthogona refinabe functions are specia cases of them An optima reguarity anaysis of pseudo-spines does not come easiy, as it has aready been iustrated in a reguarity estimate of the orthonorma refinabe functions, which is a specia case of pseudo-spines see [] and [10] This paper gives a systematic reguarity anaysis of both types of pseudo-spines Furthermore, the technique used to estimate the reguarity of pseudo-spines can be appied to discover that the tight frame systems derived from the methods given in both [1] and this paper have one frameet whose diations and shifts aready form a Riesz basis for L R This eads to a new understanding of the structure of the pseudo-spine tight frame systems A function φ L R is refinabe if it satisfies the refinement equation 11 φ = k Z akφ k, for some sequence a Z, caed refinement mask of φ By L p R, for 1 p <, we denote a the functions fx satisfying fx := LpR fx p dx R 1 p < ; and p Z the set of a sequences c defined on Z such that c := pz c p 1 p < Z The Fourier transform of a function f L 1 R is defined by fξ := fte iξt dt, ξ R, R which can be extended to more genera function spaces eg L R naturay Simiary, the Fourier series ĉ of a sequence c Z is defined by ĉξ := Z ce iξ, ξ R With these, the refinement equation 11 can be written in terms of its Fourier Transform as φξ = âξ/ φξ/, ξ R We aso ca â a refinement mask for convenience Pseudo-spines are defined in terms of their refinement masks It starts with the simpe identity, for given nonnegative integers and m with m 1, 1 1 = cos ξ/ + sin ξ/ m+ The refinement masks of pseudo-spines are defined by the summation of the first + 1 terms of the binomia expansion of 1 In particuar, the refinement mask

3 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 3 of a pseudo-spine of Type I with order m, is given by, for ξ [ π, π], 13 1 âξ := 1 â m, ξ := cos m ξ/ m + sin ξ/ cos ξ/ and the refinement mask of a pseudo-spine of Type II with order m, is given by, for ξ [ π, π], 1 âξ := â m, ξ := cos m ξ/ =0 =0 m + sin ξ/ cos ξ/ Except for some specia circumstances, we aways drop the subscript m, in 1âξ m, and âξ m, for simpicity We note that the mask of Type I is obtained by taking the square root of the mask of Type II using the Feér-Riesz emma see eg [10] and [0], ie âξ = 1 âξ Type I was introduced and used in [1] in their constructions of tight frameets The corresponding pseudo-spines can be defined in terms of their Fourier transforms, ie 15 φξ k := kâ ξ, k = 1, =1 The pseudo-spines with order m, 0 for both types are B-spines Reca that a B-spine with order m and its refinement mask are defined by m B m ξ = e i ξ sinξ/ and âξ = e i ξ cos m ξ/, ξ/ where = 0 when m is even, = 1 when m is odd for detaied discussions about B-spines, one may refer to [] The pseudo-spines of Type I with order m, m 1 are the refinabe functions with orthonorma shifts caed orthogona refinabe functions given in [11] The pseudo-spines of Type II with order m, m 1 are the interpoatory refinabe functions which were first introduced in [1] and a systematic construction was given in [11] Reca that a continuous function φ L R is interpoatory if φ = δ, Z, ie φ0 = 1, and φ = 0, for 0 see eg [1] The other pseudo-spines fi in the gap between the B-spines and orthogona or interpoatory refinabe functions For fixed m, since the vaue of the mask k âξ, for k = 1, and ξ R, increases with by 1 of Lemma in section, and the ength of the mask k a aso increases with, we concude that the decay rate of the Fourier transform of a pseudo-spine decreases with and the support of the corresponding pseudo-spine increases with In particuar, for fixed m, the pseudo-spine with order m, 0 has the highest order of smoothness with the shortest support, the pseudo-spine with order m, m 1 has the owest order of smoothness with the argest support in the famiy When we move from B-spines to orthogona or interpoatory refinabe functions, we sacrifice the smoothness and short support of the B-spines to gain some other desirabe properties, such as orthogonaity or interpoatory property What do we get for the pseudo-spines of the other orders? When we move from B-spines to pseudo-spines, we gain the approximation power of the truncated tight frame systems derived from them, as we wi discuss beow

4 BIN DONG AND ZUOWEI SHEN For a given φ L R, a shift integer transation invariant space generated by φ L R is defined by 16 V 0 φ := Span{φ k, k Z} Let 17 V n φ := {f n : f V 0 φ, n Z} The function φ is the generator of V 0, hence the generator of V n φ, n Z It is easy to see that for fixed m and type, pseudo-spines of a orders m,, 0 m 1, satisfy the same order of the Strang-Fix SF condition The Type I pseudo-spines are of order m and Type II are of order m Reca that a function φ satisfies the SF condition of order m if φ0 0, φ πk = 0, = 0, 1,,, m 1, k Z\{0} Assume that φ satisfies the SF condition of order m 0 Then, the order of the best approximation of a sufficienty smooth function f from V n n Z is m 0 Reca that V n φ n Z provides approximation order m 0 or we can say that the refinabe function φ provides approximation order m 0, if for a the f in the Soboev space W m 0 R, distf, V n := min{ f g L R : g V n } = O nm 0 Therefore, even though the V n n Z may be generated by a different pseudo-spine for the fixed type with order m,, 0 m 1, the corresponding spaces V n n Z provide the same approximation order However, in many appications of waveets and frameets, we normay use 18 P n : f k Z f, φ n,k φ n,k to approximate f, where φ n,k := n/ φ n k The operation P n f may not provide the best approximation of f from V n We say that the operator P n provides approximation order m 1, if for a f in the Soboev space W m1 R f P n f L R = O nm1 As shown in [1], the approximation order of P n f depends on the order of the zero of 1 âξ at the origin In fact, if 1 â = O m at the origin, then m 1 = min{m 0, m } For B-spines, m never exceeds This indicates that the approximation order of P n can never exceed even if a high order B-spine is used On the other hand, for the pseudo-spine of either type with order m,, 0 m 1, the corresponding m = + see Theorem 310 Therefore, the approximation order of P n, with a pseudo-spine with order m,, 0 m 1, as the underying refinabe function, is min{m, + } for Type I and + for Type II More importanty, the approximation order of P n determines the approximation order of the truncated series of a tight frame system For given Ψ := {ψ 1, ψ,, ψ r }, the system XΨ := {ψ n,k = n/ ψ n k, ψ Ψ, n, k Z} is a tight frame for L R if f, g = f L R, g XΨ f L R

5 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 5 For XΨ, define the truncated operator as 19 Q n : f f, ψ,k ψ,k ψ Ψ,k Z,<n When the tight frameets Ψ are obtained via the unitary extension principe see eg Section from the mutiresoution anaysis generated by the same φ, then Lemma in [1] shows that P n f = Q n f, for a f L R Reca that for a compacty supported refinabe function φ L R, we define V 0 and V n as in 16 and 17 Then, the sequence of spaces V n n Z forms a mutiresoution anaysis MRA generated by φ, ie i V n V n+1, n Z; ii n Z V n = L R, n Z V n = {0} see eg [3] and [18] The waveet system XΨ is said to be MRA-based if there exists an MRA V n n Z, such that Ψ V 1 If, in addition, the system XΨ is a tight frame system, we refer to the eements of Ψ as tight frameets Therefore, the tight frame system derived from a pseudo-spine normay gives better approximation order when the truncated series is used to approximate the underying functions than that derived from B-spines For fixed m, the choice of depends entirey on appications One needs to baance among the approximation order, the ength of support of the waveet, and reguarity according to the practica probems in hand The rest of the paper is organized as foows: Section gives some technica emmata used in other sections Section 3 focuses on the anaysis of reguarity and approximation order In particuar, the exact decay of the Fourier transforms of the pseudo-spines for a orders of both types are given The asymptotica anaysis is aso provided In Section, we show that in a tight frame systems constructed from pseudo-spines by methods provided both in [1] and this paper, there is one tight frameet from the generating set of the tight frame system whose diations and shifts aready form a Riesz basis for L R Furthermore, antisymmetric tight frameets, which have a Riesz waveet as one of the frameets, are designed Two Lemmata This section gives two key technica emmata that wi be used to prove severa key resuts of this paper We start with the foowing emma on binomia coefficients, where 1 is we known see eg [8] and the proof of 3 is rather technica but needed in Section Lemma 1 For given nonnegative integers m,,, we have: 1 m+1 = m + m 1 for 1 and + 1 m+ +1 = m + m 1+ m+1 1 m+ =0 m+ =0 0, for m 1 and 1 m 1 3 m+ 1 1, for a m 1 and 0 m 1 =0 m+ Proof The identities in 1 are we known and can be proven directy by the definition of the binomia coefficients For, since m >, we ony need to check if 1 m + m + m =0

6 6 BIN DONG AND ZUOWEI SHEN hods, which foows from the identity m + 1 m+ 1 = m+ Finay, we prove 3 by induction with respect to m Since 3 is obviousy true for = 0, we now focus on 1 m 1 When m = 1, the inequaity triviay hods Assume 3 hods when m = m 0, ie m0 + m0 +, for a 1 m 0 1 Consider the case m = m We first show that 3 hods for a, where 1 m 0 1 For 1 m 0 1, we have m m m0 + by induction hypothesis m = < = =0 =0 =0 m0 + + m m m0 + =0 + m 0 + =0 m0 + =0 m0 + m0 + 1 m0 + + from =0 =0 m from 1 =0 This shows that 3 hods for a 1 m 0 1, it remains to show 3 hods for = m 0, ie to show 1 m m0 + 1 m0 m Observe that m 0 m 0 m0 + 1 = 1 =0 m 0+1 =0 =0 m0 + 1 = m0 Then 1 is equivaent to m0 + 1 m 0 m0 + 1, m 0 =0 which is obviousy true This concudes the proof of 3 Define P m, y := and m + y 1 y =0 3 R m, y := 1 y m P m, y,

7 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 7 where y = sin ξ/ and m, are nonnegative integers with m 1 Then, it is obvious that R m, sin ξ/ = âξ Next, we give severa basic properties of the poynomias P m, y and R m, y Part - of the foowing emma are mainy used in Section 33 and Section Lemma For nonnegative integers m and with m 1, et P m, y and R m, y be the poynomias defined in and 3 Then: 1 P m, y = m 1+ =0 y R m, y = m + m+ 1 y 1 y m 1 3 Define Qy := R m, y + R m, 1 y Then, m + min Qy = Q1 y [0,1] = 1 m =0 Define Sy := Rm, y + R m, 1 y Then, min Sy = S1 y [0,1] = 1 m =0 m + Proof For fixed m, we prove 1 by induction with respect to It is obviousy true for = 0 Now suppose 1 hods for 0 Consider = 0 + 1, 0+1 m P m, y = y 1 y 0 +1 =0 = 1 y =1 m Appying the first identity in 1 of Lemma 1, we have, 0 +1 m + P m, y = 1 y y 1 y 0 +1 =1 y 1 y m y +1 1 y 0 =0 m + 0 = 1 yp m,0 y + yp m,0 y + y m 1 + m + = y 0 + y 0+1 by induction hypothesis = =0 0+1 =0 m 1 + y We prove by induction with respect to for given m It is obviousy true when = 0 Suppose hods for 0, ie m + R m, y = m + 0 y 0 1 y m 1, 0

8 8 BIN DONG AND ZUOWEI SHEN and consider the case = m 1 Using 1 and the definition of R m, y in 3, we have R m,0 +1y = 1 y m P m,0 +1y = 1 y m P m,0 y + m + 0 y Since R m,0 y = 1 y m P m,0 y, we have m + 0 R m,0 +1y = y y m + R m,0 y Then, m + R m, 0 0+1y = y 0 m + 1 y m 0 m y y m m m + 0 y 0 1 y m 1 0 Puing the common factor y 0 1 y m 1 out, one obtains R m, 0 +1y = y 0 m + 1 y m 1 0 m y m + 0 m m y m By using the second identity in 1 of Lemma 1, one obtains m + R m, 0 0+1y = m y y m This concudes the proof of For 3, we compute Q y, ie Q y = R m,y + R m, 1 y = R m,y R m,1 y Appying, one obtains m + 1 Q y = m + y m 1 1 y 1 y m 1 y Now we show that Q y 0 on [0, 1 ], Q y 0 on [ 1, 1] Note that Mutipying both sides by y 1 y, Simiary we have y m 1 1 y m 1, for a y [0, 1 ] y m 1 1 y 1 y m 1 y, for a y [0, 1 ] y m 1 1 y 1 y m 1 y, for a y [ 1, 1] 0 We concude that Q y { 0, y [0, 1 ] 0, y [ 1, 1]

9 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 9 This means that Qy reaches its minimum vaue at the point y = 1 Now we compute Q 1 Note that Q 1 = R m, 1 = 1 m P m, 1 Reca that P m,y is defined in, ie P m, y = =0 y 1 y Then min Qy = Q1 y [0,1] = 1 m m+ m + m + = 1 m =0 With 3, the proof of is simper Since using the identities and we obtain S y m + = m+ 1 =0 S y = R m, yr m,y + R m, 1 y R m, 1 y, R m, y = 1 y m P m, y, m + 1 R m,y = m + y 1 y m 1 Rm, 1 y m + 1 = m + y m 1 1 y, m y + y m 1 y + 1 y m 1 =0 For each 0 and y [0, 1 ], we have ym 1 1 y m 1 ; and for y [ 1, 1], we have ym 1 1 y m 1 Then by simiar arguments as in 3 we concude, { S 0, y [0, 1 y ] 0, y [ 1, 1] Thus min y [0,1] Sy = S 1 Since R m, 1 = m =0 =0 m+, we have min Sy = S1 y [0,1] = R m, 1 m + = 1 m Remark 3 From 1 of Lemma we know that the refinement mask of the pseudo-spine of Type I in 13 can be written as 1 âξ = cos m ξ/ m 1 + sin ξ/ =0 Hence, the pseudo-spine of Type I with order m, m 1 is indeed the refinabe function whose shifts form an orthonorma system constructed in [11] and the pseudo-spine of Type II with order m, m 1 is indeed the autocorreation of the orthogona refinabe function, which is interpoatory

10 10 BIN DONG AND ZUOWEI SHEN 3 Basics of pseudo-spines This section is devoted to a systematic anaysis of the reguarity of pseudo-spines and approximation order of quasi-interpoatory operator P n see 18 defined by pseudo-spines These two are basic and essentia properties of pseudo-spines Indeed, the reguarity of pseudo-spines determines the reguarity of the corresponding waveets and frameets; and the approximation order of P n determines that of the truncated waveet and frameet series These two properties, together with the ength of support, are the key criteria in seecting waveets or frameets in various appications 31 Reguarity In this section the reguarity of the pseudo-spines is anayzed For α = n + β, n N, 0 β < 1, the Höder space C α see eg [10] is defined to be the set of functions which are n times continuousy differentiabe and such that the n th derivative f n satisfies the condition, f n x + h f n x C h β, x, h It is we known see [10] that if fξ 1 + ξ α <, R then f C α In particuar, if fξ C1 + ξ 1 α ε, then f C α The main idea here is to estimate the decay of the Fourier transform of pseudospines with order m, in order to get the ower bound of the reguarity of the pseudo-spines It turns out that this ower bound coincides with the upper bound when m goes to infinity, as shown in Section 3 It is we known that the exact Soboev reguarity of a given refinabe function can be obtained via its mask by appying the transfer operator and it is a very we studied area see eg [10, 3] and references in there Athough the exact Soboev exponent of a give refinabe function can be computed exacty by computing the spectrum of the transfer operator derived from the corresponding refinement mask, it is hard to anayze the Soboev exponents of a cass of refinabe functions, such as pseudo-spines discussed here, systematicay from the transfer operator approach This is simpy because a different refinabe function wi ead to a different transfer operator This is the main reason why we give here a systematic estimate of the decay of the Fourier transform of pseudo-spines instead Since for any compacty supported refinabe function φ in L R with φ0 = 1, the refinement mask a must satisfy â0 = 1 and âπ = 0 see eg [10] or [19], then âξ can be factorized as 1 + e iξ nlξ, âξ = where n is the maximum mutipicity of zeros of â at π and Lξ is a trigonometric poynomia with L0 = 1 Hence, we have 1 + e φξ = â i ξ n 1 e ξ = L iξ n ξ = L ξ iξ =1 =1 This shows that any compacty supported refinabe function in L R is the convoution of a B-spine of some order, say n, with a distribution see [1] Indeed, a =1 =1

11 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 11 B-spine of order n can aso be defined via its Fourier transform by 1 e iξ n B n := iξ The B-spine of order n is a piecewise poynomia of degree n 1 in C n 1 ε R, supported on [0, n], and has refinement mask 1 + e iξ n Since Lξ is bounded, Lξ is actuay the refinement mask of a refinabe distribution Therefore, φ is the convoution of the B-spine B n with the distribution The reguarity of φ comes from the B-spine factor whie the distribution factor takes away the reguarity But the distribution component aso provides some desirabe properties for φ, such as interpoatory properties, orthogonaity of its shifts and approximation order of certain quasi-interpoants The decay of φ can be characterized by â as stated in the foowing theorem The proof of this theorem can be found in [10] Note that in the foowing theorem, we write â in the form of 1 + e iξ nlξ âξ = = cos n ξ/ Lξ, ξ [ π, π] Theorem 31 Let a be the refinement mask of the refinabe function φ of the form Suppose that âξ = cos n ξ/ Lξ, ξ [ π, π] 31 Lξ L π 3 for ξ π 3, LξLξ L π 3 for π 3 ξ π Then φξ C1 + ξ n+κ, with κ = og L π 3 / og, and this decay is optima This theorem aows us to estimate the decay of the Fourier transform of a refinabe function via its refinement mask Since 1 φ = φ, the decay rate of 1 φ is haf of that of φ Thus we can focus on the anaysis of the decay of the Fourier transforms of pseudo-spines of Type II Based on 1 of Lemma, we wi show that P m, y, defined in, satisfies 31 This wi ead directy to the estimate of the reguarity of pseudo-spines Note that the corresponding resut for = m 1 was proven in [] which ed to the optima estimates for the decay of the Fourier transforms of the orthogona and interpoatory refinabe functions Here, the more genera resut for pseudo-spines is obtained by a simper proof than the origina one of [] and [10] Proposition 3 Let P m, y be defined as in, where, m are nonnegative integers with m 1 Then P m, y P m,, for y [0, ], 3 33 P m, yp m, y1 y P m,, for y [ 3, 1]

12 1 BIN DONG AND ZUOWEI SHEN Proof It is cear that 3 is true Indeed, by using 1 of Lemma we have that P m, y is monotonicay increasing on [0, 3 ] in fact, it is monotonicay increasing on 0, Hence, we focus on the proof of 33 Throughout this proof, we et m be fixed Let 3 W m, y := P m, yp m, y1 y P m, Then, the inequaity 33 is equivaent to 3 W m, y 0 for a y [ 3, 1] In order to show 3, we show, instead, 35 W m,+1 y W m, y 0, for a y [ 3, 1], = 0, 1,, m Note that since for = 0, P m,0 y = 1 for a y [0, 1], 3 is obviousy true for = 0 Hence, 3 foows from 35 and 33 foows from 3 We now compute W m,+1 y W m, y By 1 of Lemma, one obtains +1 m m 1 + W m,+1 y W m, y = y y1 y =0 =0 m 1 + m 1 + y y1 y =0 =0 3 P 3 m,+1 Spitting the sum +1 =0 W m,+1 y W m, y = +P m, y, one obtains m m 1 + y y1 y =0 =0 m + +1 m y +1 y1 y + 1 =0 m 1 + m 1 + y y1 y =0 m 1+ =0 +Pm, 3 P 3 m,+1 Combining the first and the third term, one obtains m + W m,+1 y W m, y = y1 y =0 +1 m y y1 +1 y +P m, =0 3 P 3 m,+1 m 1 + y

13 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 13 Since W 3 m,+1 3 Wm, = 0 0 = 0, it suffices to show that Wm,+1 y W m, y monotonicay decreases on [ 3, 1], which is equivaent to showing that m m 1 + Gy := y1 y y +1 + y +1 y1 y =0 monotonicay decreases on [ 3, 1] For this purpose, we obtain G as foows: m 1 + G y = + 1 8yy1 y y =0 1 m + +y1 y y + 1 =0 +1 m y y1 y =0 +y +1 8y =0 =0 m + + 1y1 y + 1 Appying 1 of Lemma 1 to the second and the fourth term above, one obtains m 1 + G y = + 1 8yy1 y y +y1 y +1 =0 =0 m 1 + m + y y y1 y +1 m 1 + m + m y =0 m y y1 y m 1 + +y +1 8y =0 y1 y m + y1 y Since + 1 m+ +1 = m + m 1+ by 1 of Lemma 1, we have m + m y y1 y +1 m + y y1 y +1 = Hence, G y = m yy1 y y + m + y1 y +1 y = y y1 y + m + 8yy +1 y1 y

14 1 BIN DONG AND ZUOWEI SHEN Puing the common factor y y1 y out from each term of the above summation, one obtains m 1 + G y = y y1 y + 1 8yy1 y =0 + m + y1 y y + m + 8yy +1 For 0 m, consider g, y := + 1 8y y1 y + m + y1 y y + m + 8yy +1 = + 1y1 y y1 y 8y + + 1y 1 8y +m + 1 y1 y +1 8y y +1 The inequaity y1 y y and 8y for y [3/, 1] show that g, y 0 and G y 0 on this interva Remark 33 It is cear that W m,0 y = 0, y [ 3, 1], because P m,0 = 1 It was aso proven by [] that W m,m 1 y 0, y [ 3, 1], which is equivaent to 33 The decreasing of W m, y, for y [ 3, 1], as increases shown above indicates some difficuties to prove 33 directy for an arbitrary, 0 < < m 1, since it has a smaer margin than the case when = m 1 In fact, to some extent, the proof of 33 for the case when = m 1 reies on a numerica check for m 1 see [10] Inequaity 33 for the case = m 1 as proven in [] aso see [10] is one of the cornerstones of the waveet theory, because it immediatey eads to the optima estimate of the decay of the Fourier transforms hence, an estimate of the reguarity of both interpoatory and orthogona refinabe functions We take a different approach here by proving that W m, y, y [ 3, 1], decreases as increases As a resut, we obtain 33 for a 0 m 1 by the fact that W m,0 y = 0, y [ 3, 1] This shows that introducing the concepts of the pseudospines gives a better understanding and a more compete picture of the proof of 33 and aso, we hope, enriches the theory of waveets Note that the proof of 33 for a 0 m 1 given here does not rey on any numerica computation and is simper than the origina proof of [] and [10] With this proposition, one obtains the reguarity of pseudo-spines by appying Theorem 31 Theorem 3 Let φ be the pseudo-spine of Type II with order m, Then φξ C 1 + ξ m+κ, where κ = ogp m, 3 / og Consequenty, φ C α ε with α = m κ 1 Furthermore, et 1 φ be the pseudo-spine of Type I with order m, Then 1 φξ C 1 + ξ m+ κ Consequenty, 1 φ C α1 ε with α 1 = m κ 1

15 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 15 Proof Since P m, sin ξ = m + sin ξ/ cos ξ/, =0 the refinement mask of the pseudo-spine of Type II with order m, is m + âξ = cos m ξ/ sin ξ/ cos ξ/ =0 = cosξ/ m Pm, sin ξ/ Hence, Lξ in Theorem 31 is exacty P m, sin ξ/ here Let y = sin ξ/ Appying 3 of Proposition P m, y P m,, y [0, ] we have Lξ = P m, sin ξ/ 3 = P m, y P m, = Pm, sin π 3 for ξ π 3 Note that Lξ = P m, sin ξ = P m, sin ξ/1 sin ξ/ = P m, y1 y Appying 33 of Proposition 3 P m, yp m, y1 y 3, 3 P m, y [, 1], we have LξLξ = P m, sin ξ/p m, sin ξ/1 sin ξ/ Hence, by Theorem 31, φ satisfies = P m, yp m, y1 y 3 P m, = Pm, sin π 3 π, for 3 φξ C 1 + ξ m+κ, ξ π where κ = ogp m, 3 / og This eads to φ C α ε, where α = m κ 1 Since the decay of 1 φ is exacty haf of φ, we have 1 φξ C 1 + ξ m+ κ, consequenty, 1 φ C α 1 ε, where α = m κ 1 Tabe 1 gives the decay rates β m, of the Fourier transform of pseudo-spines of Type II with order m,, for m 8 and 1 m 1 The reguarity exponent of the corresponding pseudo-spine is, at east, α = β m, 1 ε The decay rate of the Fourier transform of the pseudo-spine of Type I with the same order is β m, and its reguarity exponent α 1 is α 1 Therefore, the tabe shows that for either type of the pseudo-spines and fixed order m, the decay rate of their Fourier transform decreases as increases, whie for fixed, it increases as m increases This is true indeed as shown in the foowing proposition

16 16 BIN DONG AND ZUOWEI SHEN Tabe 1 Decay rates β m, = m κ of pseudo-spines of Type II with order m,, for m 8 and 1 m 1 m, = 1 = = 3 = = 5 = 6 = 7 m = m = m = m = m = m = m = Proposition 35 Let β m, = m κ with κ = og P m, 3 / og as given in Theorem 3 and 0 m 1 Then: 1 For fixed m, β m, decreases as increases For fixed, β m, increases as m increases 3 When = m 1, β m, increases as m increases Consequenty, the decay rate β,1 = is the smaest among a β m,, with m and 0 m 1 Proof Part 1 foows directy from 1 of Lemma, which shows that P 3 m, increases as increases for fixed m For part, note that β m, = m og P 3 m, og Consider β m, = m og P 3 m, og = m 1 P 3 = m, m P 3 m, Hence, part is equivaent to the fact that I m := m 3 P m, decreases as m increases for fixed, which is equivaent to showing that for fixed 0 m 1, 37 I m+1 I m < 0 Note that I m+1 I m = m 1 3 P m+1, m 3 P m, m + m 1 + = m 1 3 =0 Inequaity 37 foows from the fact that for 0 m 1, m + 38 = m + m 1 + = 1 + m 1 + m 1 + m m < This concudes the proof of part

17 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 17 For part 3, using a simiar argument as in the proof of part, one can derive that it is equivaent to showing that J m := m 3 P m,m 1 decreases as m increases, which, in turn, is equivaent to showing that 39 J m+1 J m < 0 for m 1 Note that, simiar to the proof of part, we have Let m m + J m+1 J m = m 1 M := =0 m m + =0 3 3 m 1 m 1 =0 =0 m 1 + m Then, 39 is equivaent to M < 0 for m 1 It is easy to check that M < 0, when m = 1 We consider now the case when m First, we note that M = = m 1 =0 m 1 =1 m + m m 1 3 =0 m m 1 + =0 m m m 3 m 3 + m m 3 m, where the ast identity foows from 1 of Lemma 1 Substituting for 1 in the first term, one obtains that 310 M = 3 m =0 m + m =0 Spitting the second term in 310, one obtains m m m 3 m, 311 M = 3 + m =0 m m m m 3 m 3 m m 1 m 1 + =0 m For the ast two terms of 311, we have m 3 m m 3 m = m m 1 m m m = 3 < 0 m m m m 1 m m 1 m m 1

18 18 BIN DONG AND ZUOWEI SHEN Therefore, M < 3 < = m =0 m =0 m =0 m + m + m + m 3 3 =0 m =0 m 1 + m 1 + m Appying 38, one obtains, for 0 m, m + = 1 + m 1 + m 1 + m < 3 Therefore, we concude that M < 0 and part 3 foows Finay, note that the decay rate of the Fourier transform of the pseudo-spine of Type I with order, 1 is β m, Hence, it foows from parts 1-3 that the decay rate of an arbitrary pseudo-spine of either type with order m,, m >, 0 m 1 is higher than Asymptotica Anaysis Proposition 35 reveas that the decay rates of the Fourier transforms of pseudo-spines of either type increase as m increases for fixed and decrease as increases for fixed m In this subsection, we give an asymptotica anaysis of the decay rate which, in turn, gives an asymptotica anaysis of the reguarity of 1 φ and φ as the order m, Theorem 36 Let 1 φ and φ be the pseudo-spines of Type I and II respectivey with order m, Fix = λm, 0 λ 1, where λm denotes the argest integer which is smaer than or equa to λm Then, we have 1 φξ C1 + ξ µ m and φξ C1 + ξ µm, where µ = og 1+λ λ+1 λ 3 λ og, asymptoticay for arge m This means that the asymptotic rate of the pseudo-spine of Type I and Type II are µ and µ respectivey Proof As the estimate of the Type I foows immediatey from that of Type II, we ony give the estimate for pseudo-spines of Type II We first prove the foowing fact: 31 x P m, x y P m, y, for 0 < x y 1 Indeed, assertion 1 of Lemma gives for 0 < x y 1, m 1 + m 1 + x P m, x = x y = y P m, y =0 The key step to compute the asymptotic rate is to estimate the upper and ower bound of P 3 m, in terms of m and For this, et x = 3 and y = 1 in 31 Then we obtain P m, P m, 1 = 3 m + =0

19 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 19 Next, et x = 1 and y = 3 in 31, we obtain Since one obtains P m, 1 = =0 P m, 3 3 Pm, 1 m + = P m, =0 m + =0 m + Putting 313 and 31 together, we obtain the foowing estimates of P m, 3, 3 m + P m, 3 3 m + =0, For m 1, we have m + m + m =0 Hence, m + P m, 3 m 3 m + Next, we wi use this estimate to anayze the decay of φ with order m, as m goes to infinity The upper bound of P 3 m, in 315 gives m og P 3 m, m og m 3 m+ og og We estimate the right hand side of the above inequaity asymptoticay for arge m, to obtain the asymptotica ower bound of m og P m, og 3 For this, we first reca the Stiring approximation, ie m! πe m+ 1 og m m see eg [15], where a m b m means that am b m 1, m By Stiring approximation, we have 316 og m! og πe m+ 1 og m m m og m m Appying 316, one obtains m + og = ogm +! og m! og! m + ogm + m + m og m m og m + ogm + m og m og

20 0 BIN DONG AND ZUOWEI SHEN Thus, m og m 3 m+ og = m og m + og 3 + og m+ og m m og m ogm + og m m og og By the assumption, = λm, 0 λ 1 Hence, when m is sufficienty arge, m λ and therefore, m og m 3 m+ m og 1 + λ 3+3λ λ λ og og og 1+λ = m λ+1 λ 3 λ og Now we obtain the asymptotica ower bound of m og P m, og, ie asymptoticay, for arge m with = λm, 317 m og P 3 m, og 1+λ m λ+1 λ 3 λ og og Next, we use the eft hand side of 315 to obtain the asymptotica upper bound of m og P m, og 3 First note that 315 gives m og P 3 m, m og 3 + og m+ og og Appying arguments simiar to the estimate of the ower bound by using 316, we wi obtain the foowing m og 3 + og m+ m m og m ogm + og m m og og og og 1+λ m λ+1 λ 3 λ og This eads to the asymptotica ower bound of m og P m, og for arge m with = λm, og m 318 m og P 3 m, og λ λ+1 λ 3 λ og, ie asymptoticay, Combining 317 and 318, we concude that for arge m, the asymptotica upper and ower bounds coincide and equa to 319 m og P 3 m, og 1+λ m λ+1 λ 3 λ = µm og og Therefore the equation 319 gives that, fixing = λm and asymptoticay, for arge m, we have φξ C1 + ξ µm and 1 φξ C1 + ξ µ m, where µ = og 1+λ λ+1 λ 3 λ og

21 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 1 Remark 37 The above theorem shows that, asymptoticay for arge m, the smoothness of the pseudo-spines of type I and II increases at a rate µ/ and µ respectivey The proof of Theorem 36 aso eads to the foowing two observations: 1 Consider pseudo-spines of Type II with order m, m p, where p is a fixed positive integer independent of m The asymptotic rate is og 3 og 0150 Indeed, when = m p, λ m = m p m 1 for sufficienty arge m Simiary, for pseudo-spines of Type I with order m, m p, the corresponding asymptotic rate is 1 og 3 og 0075 Assume that is fixed for a m The asymptotic rates of pseudo-spines of Type I and II with order m, are 1 and respectivey This is simpy because, for the fixed integer, λ m 0 for sufficienty arge m Tabe Asymptoticay for arge m, the smoothness of φ increases at rate µ, which is given in the foowing tabe with some choices of m = 0 = m 10 = m 8 = m 6 = m = m = m 1 µ Exampe 38 In Tabe, we give µ, the asymptotica rate of pseudo-spines of Type II with order m, λm, as m goes to infinity and the parameter λ = 1 10, 1 8, 1 6, 1, 1, 1 The asymptotic rate µ 0 for pseudo-spines of Type I with the same order is ust µ 0 = µ 33 Approximation order We foow [1] to give a brief discussion of the approximation order of P n through Q n where P n is given by 18 with the underying refinabe function φ and Q n is given by 19 with the underying tight frameets Ψ Characterizations of approximation order of Q n were given in Theorem 8 of [1] Furthermore, Lemma of [1] says that P n = Q n on L R when the tight frameets Ψ are obtained via the unitary extension principe see Section for the UEP from the MRA generated by the same refinabe function φ The foowing theorem is a specia case of Theorem 8 in [1] with the understanding P n = Q n Theorem 39 Let φ be a pseudo-spine of order m, with refinement mask a Let P n be the operator as defined in 18 with φ as the underying refinabe function Then the approximation order of the operator P n is min{m, m 1 }, with m 1 the order of the zero of 1 â at the origin With this, we have the foowing: Theorem 310 Let m and be nonnegative integers satisfying m 1 1 Let 1 φ be the pseudo-spine of Type I with order m, and 1 â be its refinement mask Then the corresponding operator P n provides approximation order min{m, + } Let φ be the pseudo-spine of Type II with order m, and â be its refinement mask Then the corresponding operator P n provides approximation order +

22 BIN DONG AND ZUOWEI SHEN Proof It was shown in [1] that 1 1 â = O + Therefore, Theorem 39 gives the rest of the proof of 1 For, we compute the order of zeros of 1 â at the origin We rewrite 1 â as 1 â = 1 R m,sin ξ/, where R m, y was defined in 3 It is obvious that for ξ = 0, 1 Rm, sin ξ/ = 0 Reca that the derivative of R m, y was given by of Lemma, ie m R m,y = m + y 1 y m 1 Appying 30 to take the first derivative of 1 Rm, sin ξ/ with respect to ξ, one obtains 1 Rm,sin ξ/ = R m,sin ξ/r m,sin ξ/sin ξ/ = m + 1 R m, sin ξ/ m + sin ξ/ cos m ξ/ sin ξ/ = m + 1 m + R m, sin ξ/ sin +1 ξ/ cos m 1 ξ/ Since R m, sin ξ/ and cos m 1 ξ/ is equa to 1 when ξ = 0, and since sin +1 ξ/ has zero of order + 1 at ξ = 0, we concude that 1 âξ = 1 R m,sin ξ/ = O ξ + Then Theorem 39 shows that the approximation order of P n with the pseudospine of Type II as the underying refinabe function is min{m, + } = +, for 0 m 1 Remark 311 The above resut says that when m 1, the approximation order of a pseudo-spine of Type I with order m, and one of Type II with the same order are the same, athough the support of the Type I is haf of that of Type II When > m 1, the approximation order of Type I is m and Type II is + > m The reguarity of the Type II is about two times that of the Type I with the same order Furthermore, one can obtain symmetric short Riesz Waveets and tight frameets from pseudo-spines of Type II, as we wi see in the ast section Riesz Waveets in Frameets In this section, we focus on the structure of the tight frame systems constructed from pseudo-spines by appying the unitary extension principe [] We show that in amost a pseudo-spine tight frame systems constructed both in [1] and the symmetric tight frame systems constructed by a pseudo-spine of Type II in this section, there is one frameet whose diations and shifts aready form a Riesz basis for L R

23 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 3 1 Riesz Waveets For a given ψ, define the waveet system Xψ := {ψ n,k = n/ ψ n k : n, k Z} We ca Xψ a Besse system if for some C 1 > 0, and for every f L R, f, g C 1 f L R g Xψ A Besse system Xψ is a Riesz basis if there exists C > 0 such that, C {c n,k } Z c n,k ψ n,k, for a {c n,k } Z n,k Z L R and the span of {ψ n,k : n, k Z} is dense in L R The function ψ is caed Riesz waveet if Xψ forms a Riesz basis for L R and Xψ is aso caed the Riesz waveet system As a pseudo-spines are compacty supported, refinabe and in L R, the sequence of spaces V n n Z defined via 17 forms an MRA Since the obective here is to construct Riesz waveets, one needs to start with stabe refinabe functions Indeed, it was shown in Proposition 11 and Lemma of [13] that a pseudospines are stabe in fact, we proved in [13] that the shifts of them are ineary independent, which is stronger than stabe For a given stabe refinabe function φ L R, the key step in the construction of the Riesz waveet ψ is to seect some desirabe sequence b, caed a waveet mask The waveet ψ is then defined by b and the corresponding refinabe function φ as ψ := k Z bkφ k, It can be written equivaenty in the Fourier domain as ψξ = bξ/ φξ/ When {φ k : k Z} forms an orthonorma basis for V 0 φ, eg pseudo-spine of Type I with order m, m 1, define φ is a 1 ψ := k Z bkφ k with bk = 1 k 1 a1 k, k Z, or equivaenty, bξ = e iξâξ + π Then the corresponding waveet system Xψ with the pseudo-spine of Type I with order m, m 1 being the underying refinabe function forms an orthonorma basis for L R We are interested to know whether the function ψ defined in 1 is a Riesz waveet, when the refinabe function φ is chosen to be the pseudo-spines with other orders In fact, it was shown in [17] that it is true, when φ is a B-spine, ie a pseudo-spine with order m, 0 or when φ is a pseudo-spine of Type II with order m, m 1 In the rest of this subsection we wi show that for a pseudo-spines, the waveet defined by 1 is a Riesz waveet To prove this, we use the foowing theorem which is the specia case of Theorem 1 of [17] When both refinement masks are finitey supported, a simiar resut was aready obtained before in [5, 6, 9]

24 BIN DONG AND ZUOWEI SHEN Theorem 1 Let a be a finitey supported refinement mask of a refinabe function φ L R with â0 = 1 and âπ = 0, such that â can be factorized into the form âξ = 1 + e iξ n Lξ = cosn ξ/ Lξ, ξ [ π, π], where L is the Fourier series of a finitey supported sequence with Lπ 0 Suppose that âξ + âξ + π 0, ξ [ π, π] Define ψξ := e iξ âξ + π φξ and 3 Lξ := Lξ âξ + âξ + π Assume that ρ L := Lξ L R < n 1 and ρ L := Lξ L R < n 1, Then Xψ is a Riesz basis for L R As we wi show, the key step in the appication of the above theorem is to estimate the upper bound of Lξ and Lξ Reca that the refinement masks of pseudo-spines of Type I and II are, for ξ [ π, π], 1 m âξ := cos m ξ/ sin ξ/ cos ξ/ and 6 âξ := cos m ξ/ =0 m + sin ξ/ cos ξ/ =0 Hence, the corresponding L function in for pseudo-spines of Type I is 1 m + 1 Lξ = sin ξ/ cos ξ/, =0 and for pseudo-spines of Type II is m + Lξ = sin ξ/ cos ξ/ =0 Denoting y = sin ξ/, we have 7 1 â = 1 y m P m, y and 1, â = 1 y m P m, y, 8 1 L = P m, y 1, L = P m, y Furthermore, we have and 1 âξ + 1 âξ + π = R m, y + R m, 1 y âξ + âξ + π = R m,y + R m,1 y,

25 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 5 with y = sin ξ/ Hence, 9 1 L = P m, y 1 R m, y + R m, 1 y P m, y and L = Rm, y + R m, 1 y The estimation of 1 L L R and L L R are based on the foowing resut: Proposition Let m and be given nonnegative integers with m 1 and 1 L and L be defined in 9 Then, 1 1 L L R = sup y [0,1] P m, y 1 R m, y+r m, 1 y < m 1 L L R = sup y [0,1] P m, y R m, y+r m, 1 y < m 1 Proof Note that from 1 of Lemma, m + 10 P m, y = y 1 y = =0 m 1 + y, y [0, 1], hence both P m, y 1 and P m, y attain their maximum on [0, 1] at the point 1 and the maximum vaues are: 1 P m, 1 1 m + m + = and P m, 1 = By 3 of Lemma, one obtains 1 L L R = sup y [0,1] m + Appying 3 of Lemma 1, ie =0 P m, y 1 R m, y + R m, 1 y 1 m+ 1 m+ =0 m+ max y [0,1] R m, y + R m, 1 y m+ 11 m+ 1, =0 one obtains 1 L L R m 1 < m 1 The proof of is simiar to that of 1 Indeed, by of Lemma P m, y L L R = sup y [0,1] Rm, y + R m, 1 y m + 1 max y [0,1] Rm, y + R m, 1 y Appying 11 again, we have = m+ 1 m+ =0 m+ L L R m 1 < m 1

26 6 BIN DONG AND ZUOWEI SHEN Theorem 3 Let k φ, k = 1, be the pseudo-spine of Type I and II with order m, The refinement masks k a, k = 1,, are given in 13 and 1 Define 1 k ψξ := e iξ kâξ + π k φξ, k = 1,, then X k ψ forms a Riesz basis for L R Proof To appy Theorem 1, we first note that 1 âξ + 1 âξ + π = R m, sin ξ/ + R m, cos ξ/ 0 and âξ + âξ + π = R m,sin ξ/ + R m,cos ξ/ 0, for a ξ [ π, π], where R m, is defined in 3 by 3 and of Lemma Next, one needs to check whether 13 ρ 1 L = 1 L L R < m 1, ρ L = L L R < m 1, 1 ρ 1 L = 1 L L R < m 1 and ρ L = L L R < m 1, hod Inequaities in 1 foows from Proposition For 13, we note that for both k = 1 and k =, we have k âξ + k âξ + π 1 for a ξ R Hence, This concudes the proof k Lξ k Lξ for a ξ R In [1], three constructions of tight frameets were given for pseudo-spines of Type I The number of frameets is either two or three Interested readers may consut [1] Section 31 for detais We observe that in a the three constructions, one of the frameets ψ 1 is defined by ψ 1 := b 1 ξ/ φξ/, where b1 := e iξ â + π and â is the refinement mask of a pseudo-spine It was shown in Theorem 3 of this paper that Xψ 1 forms a Riesz basis for L R This impies that a pseudospine tight frame systems constructed in [1] aready have one of the subsystems form a Riesz basis for L R We further remark that it was observed in [17] that the same phenomenon occurs for the tight spine frame systems constructed in [1] This, together with our new finding here, gives insight into the redundant structure of tight frame systems given in [1]

27 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 7 Symmetric Frameets In this subsection, we give a construction of symmetric tight frameets from pseudo-spines of Type II by using the unitary extension principe of [] The constructions of symmetric tight frameets from a symmetric refinabe functions by using the unitary extension principe have been discussed in [1, 7, 16] We note that the constructions of tight frameets given in [1] can aso be appied to pseudo-spines of Type II However, the constructions there cannot guarantee that a tight frameets are symmetric, even though pseudo-spines of Type II are symmetric Therefore, in this subsection, we make use of the symmetry of the pseudo-spines of Type II to obtain symmetric tight frameets Furthermore, the resuts of the previous subsection revea that the Construction beow aso has one frameet ψ such that Xψ itsef aready forms a Riesz basis for L R The construction here is based on the unitary extension principe UEP of [] We give a brief discussion here whie the more genera version and comprehensive discussions of the UEP can be found in [1] and [] Let â be the refinement mask of φ L R with â0 = 1 and et b, = 1,,, r, be waveet masks If â and b are trigonometric poynomias that satisfy r { 1, ν = 0 15 âξâξ + ν + b ξ b ξ + ν = 0, ν = π, =1 for a ξ [ π, π], and Ψ := {ψ 1, ψ,, ψ r } L R are given by ψ ξ := b ξ φξ, = 1,,, r, then the UEP asserts that XΨ is a tight frame for L R When appying constructions in [1] on pseudo-spines of Type II to obtain a set of three tight frameets, ony the first frameet is symmetric To overcome this, one can appy Construction 3 in [16] to convert these frameets to a set of five symmetric or antisymmetric tight frameets It was further shown in [16] that the Construction 3 eads to new tight frameets from the same MRA as the od tight frameets whenever the od ones are derived from the MRA generated by a symmetric refinabe function We forgo the idea of giving the detais of this construction and eave it to readers by consuting [16], because next we wi give a different approach that eads to a symmetric tight frame system with ony three generators The ideas of this construction are based on those of [7] and one of the constructions of [1] Note that the construction here is generic and can be appied to any symmetric refinabe function whose mask is a trigonometric poynomia and satisfies 16 â + â + π 1 Construction Let φ L R be a compacty supported refinabe function with its trigonometric poynomia refinement mask â satisfying â0 = 1 and 16 Moreover we assume that φ, hence its refinement mask â, is symmetric about the origin Let T T = 1 â â + π and A :=, where T is obtained via the Feér-Riesz emma Define b1 ξ := e iξ âξ + π, b ξ := Aξ + e iξ A ξ and b 3 ξ := e iξ b ξ + π

28 8 BIN DONG AND ZUOWEI SHEN Let Ψ := {ψ 1, ψ, ψ 3 }, where 17 ψ ξ := b ξ/ φξ/, = 1,, 3 Then XΨ is a tight frame for L R Moreover, ψ 1 is symmetric about 1, ψ is symmetric about 1 and ψ 3 is antisymmetric about 1 We aso note that since ψ 1 is defined exacty the same as 1 in Theorem 3, Xψ 1 forms a Riesz basis for L R when φ is a pseudo-spine Furthermore, since b and b 3 have zeros at both 0 and π, one can check easiy that neither the shifts of ψ nor those of ψ 3 can form a Riesz system Hence, Xψ and Xψ 3 cannot form a Riesz basis for L R Proof In order to verify that XΨ is a tight frame for L R, one needs to show that the masks {â, b 1, b, b 3 } satisfy 15 Note that Hence, ââ + π + b1 = e iξ â + π and b 3 = e iξ b + π, 3 b b + π = ââ + π ââ + π + b b + π b b + π = 0 =1 Next, we show that 18 â + Since it remains to show that 3 b = 1 =1 â + b 1 = â + â + π, b + b 3 = 1 â â + π = T Since Aξ = 1 T ξ = 1 1 âξ âξ + π and since T is π-periodic, the spectra factorization which is based on the Feér- Riesz emma eads to the function Aξ aso to be π-periodic Furthermore, the Fourier coefficients of Aξ are rea Hence, we have 19 Aξ = Aξ + π, and Aξ = A ξ, for a ξ R Since b ξ = Aξ + e iξ A ξ and b 3 ξ = e iξ b + π = e iξ A ξ Aξ, appying 19, one obtains b ξ = Aξ + e iξ A ξ Aξ + e iξ A ξ and b 3 ξ = = Aξ + A ξ + e iξ AξA ξ + e iξ A ξaξ = Aξ + e iξ AξA ξ + e iξ A ξaξ e iξ A ξ Aξ e iξ A ξ Aξ = Aξ + A ξ e iξ AξA ξ e iξ A ξaξ = Aξ e iξ AξA ξ e iξ A ξaξ

29 PSEUDO-SPLINES, WAVELETS AND FRAMELETS 9 Hence, b ξ + b 3 ξ = Aξ = T ξ, which gives 18 and thus concudes that the masks {â, b 1, b, b 3 } satisfy 15 Therefore, XΨ is indeed a tight frame for L R by the unitary extension principe Now we show that ψ 1 is symmetric about 1 whie ψ is symmetric about 1 and ψ 3 is antisymmetric about 1 It is we known that a function f L R, is symmetric about the point γ 1 R if and ony if which is equivaent to fx = fγ 1 x ae, 0 fξ = e iγ 1ξ f ξ ae Simiary, a function f L R is antisymmetric about the point γ R if and ony if fx = fγ x ae, which is equivaent to 1 fξ = e iγ ξ f ξ ae By the definition of b 1 and the fact that â is symmetric about the origin and πperiodic, one obtains b1 ξ = e iξ âξ + π = e iξ e iξ â ξ + π = e iξ b1 ξ Since φ is symmetric about the origin, then by 0 one obtains φξ = φ ξ, for a ξ R Therefore, ψ 1 ξ = b 1 ξ/ φξ/ = e iξ b1 ξ/ φ ξ/ = e iξ ψ1 ξ, which, by 0, means that ψ 1 is symmetric about 1 Simiary by the definition of b, one obtains b ξ = Aξ + e iξ A ξ = e iξ A ξ + e iξ Aξ = e iξ b ξ Appying and the definition of ψ, one obtains, ψ ξ = b ξ/ φξ/ = e i ξ b ξ/ φ ξ/ = e i ξ ψ ξ, which, by 0, means that ψ is symmetric about 1 Simiary, we can show that ψ 3 is antisymmetric about 1 The approximation order provided by a tight frame XΨ can be characterized by the approximation order of the corresponding operator Q n see [1], which is defined in 19 We have shown in Section 33 that, for the operator P n defined in 18, we have Q n f = P n f, for f L R, provided that Ψ is derived from the UEP and the underying MRA is generated by the same φ as that defines P n Therefore, by Theorem 310, if we start from the pseudo-spine of Type II with order m, in Construction, the tight frame system XΨ provides approximation order + In the end, we give one exampe of antisymmetric tight frameets constructed from Construction using pseudo-spines of Type II with order 3, 1

30 BIN DONG AND ZUOWEI SHEN a b c d Figure 1 a is the pseudo-spine of Type II with order 3, 1 and b-d are the corresponding antisymmetric tight frameets Exampe 5 Let â to be the mask of the pseudo-spine of Type II with order 3, 1 ie âξ = cos 6 ξ/ sin ξ/ We define where A = 1 b1 ξ := e iξ âξ + π = e iξ sin 6 ξ/ cos ξ/, b ξ := Aξ + e iξ A ξ and b 3 ξ := e iξ A ξ Aξ, e iξ e iξ e iξ e iξ The graphs of Ψ are given by b-d in Figure 1 approximation order The tight frame system has References 1 A F Abdenour and I W Seesnick, Symmetric neary shift-invariant tight frame waveets, IEEE Trans Signa Process, Vo 53, Issue 1, 005, C de Boor, A Practica Guide to Spines, Springer-Verag, New York, C de Boor, R DeVore and A Ron, On the construction of mutivariate prewaveets, Constr Approx , A Cohen and J P Conze, Réguarité des bases d ondeettes et mesures ergodiques, Rev Math Iberoamer 8 199, A Cohen and I Daubechies, A stabiity criterion for biorthogona waveet bases and their reated subband coding scheme, Duke Math J , A Cohen, I Daubechies and J C Feauveau, Biorthogona bases of compacty supported waveets, Comm Pure App Math 5 199, C K Chui and W He, Compacty supported tight frames associated with refinabe functions, App Comput Harmon Ana , C C Chen and K M Koh, Principes and Techniques in Combinatorics, Word Scientific, Singapore, A Cohen, Biorthogona Waveets in Waveets: A Tutoria in Theory and Appications, ed by C K Chui, Academic Press, San Diego, I Daubechies, Ten Lectures on Waveets, in: CBMS Conf Series in App Math, vo 61, SIAM, Phiadephia, I Daubechies, Orthonorma bases of compacty supported waveets, Comm Pure App Math , I Daubechies, B Han, A Ron, and Z Shen, Frameets: MRA-based constructions of waveet frames, App Comput Harmon Ana , B Dong and Z Shen, Linear independence of pseudo-spines, Proc Amer Math Soc, to appear