ON B-SPLINE FRAMELETS DERIVED FROM THE UNITARY EXTENSION PRINCIPLE

Transcription

1 SIAM J MATH ANAL Vo 0, No 0, pp c 0000 Society for Industria and Appied Matheatics ON B-SPLINE FRAMELETS DERIVED FROM THE UNITARY EXTENSION PRINCIPLE ZUOWEI SHEN AND ZHIQIANG XU Abstract The spine waveet tight fraes considered in [A Ron and Z Shen, J Funct Ana, 8 997, pp 08 7] have been used widey in frae based iage anaysis and restorations see, eg, survey artices [B Dong and Z Shen, MRA-based waveet fraes and appications, IASLecture Notes Series, Suer Progra on The Matheatics of Iage Processing, Par City Matheatics Institute, 00; Z Shen, in Proceedings of the Internationa Congress of Matheaticians, Vo IV, Hindustan Boo Agency, Hyderabad, India, 00, pp ] However, except for the properties of the tight frae and the approxiation order of the truncated frae series see Ron and Shen and I Daubechies et a, App Coput Haron Ana, 003,pp 6],thereare few other properties of this faiy of tight fraes that are currenty nown The ai of this paper is to present a few new properties of this faiy that wi provide soe reasons why it is efficient in iage anaysis and restorations In particuar, we first present a recurrence forua for coputing the generators of higher order spine waveet tight fraes fro ower order ones This sipifies the coputations of the exact vaues of the functions in appications Second, we represent each generator of spine waveet tight fraes as a certain order of derivative of soe univariate box spine that satisfies a few additiona properties This is a crucia property used in [J-F Cai et a, J Aer Math Soc, 5 0, pp ], where connections between tota variationa and waveet frae based approaches for iage restorations are estabished Finay, we further show that each generator of sufficienty high order spine waveet tight fraes is cose to a derivative of a propery scaed Gaussian function This eads to the resut that the waveet syste generated by finitey any consecutive derivatives of a propery scaed Gaussian function fors a frae whose frae bounds can be aost tight Key words fraes, B-spine fraeets, unitary extension principe AMS subject cassifications C5, C0, A5 DOI 037/ Introduction The ai of this paper is to investigate the faiy of the spine waveet tight fraes derived fro [0] We start with basic notions For given Ψ := {ψ,,ψ } L R, the waveet syste generated by Ψ is defined as XΨ := {ψ,n, := n/ ψ n : ; n, Z} The syste XΨ L R is caed a tight frae if f = g XΨ f,g g hods for a f L R If XΨ L R is a tight frae syste of L R generated by a utiresoution anaysis MRA, then its generators Ψ are caed fraeets Received by the editors Deceber 8, 0; accepted for pubication in revised for Noveber, 0; pubished eectronicay DATE Departent of Matheatics, Nationa University of Singapore, Singapore 9076, Singapore atzuows@nusedusg This author was supported by R fro the Nationa University of Singapore LSEC, Institute of Coputationa Matheatics, Acadey of Matheatics and Syste Science, Chinese Acadey of Sciences, Beijing 0009, China xuzq@secccaccn This author was supported by NSFC grant 7336 and by the Funds for Creative Research Groups of China grant 00

2 ZUOWEI SHEN AND ZHIQIANG XU The MRA starts fro a refinabe function ϕ Acopactysupportedfunctionϕ is refinabe if it satisfies a refineent equation ϕx = j Z a j ϕx j for soe sequence a Z The refineent equation can be written via its Fourier transfor as ϕ =â ϕ ae R We ca the sequence a the refineent as of ϕ and â therefineentsyboof ϕ Here,weuse f to denote the Fourier transfor of f L R, which is defined as f := fxexp ixdx For a refinabe function ϕ L R, et V 0 be the cosed shift invariant space generated by {ϕ : Z} and V j := {f j :f V 0 }, j Z It is nown that when ϕ is copacty supported, {V j } j Z fors an MRA Reca that an MRA is a faiy of cosed subspaces {V j } j Z of L R thatsatisfiesiv j V j+, ii j V j is dense in L R, and iii j V j = {0} see [] and [9] A specia faiy of refinabe functions is B-spines Let ϕ be the centered B-spine of order, which is defined in the Fourier doain by ˆϕ =e ij sinc, where 3 sincx := { sinx/x for x 0, for x =0, and j := { 0, is even,, is odd Then ϕ is refinabe with refineent sybo â =e ij cos Tight fraeets can be constructed by the unitary extension principe UEP of [0] fro a given utiresoution anaysis For a given B-spine ϕ of order, it was shown in [0] that the functions Ψ = {ψ : =,,} defined in the Fourier doain by ˆψ := i e ij cos / sin + / / for a tight waveet frae in L R, ie, Ψ is a fraeet set We ca Ψ the B- spine fraeet of order The B-spine fraeet is either syetric or antisyetric and has sa support for a given soothness order Siiary with B-spines, each B-spine fraeet has an anaytic for Since the pubication of the UEP of [0], there are any theoretica deveopents and appications of MRA based waveet fraes In particuar, the B-spine fraeets

3 ON B-SPLINE FRAMELETS DERIVED FROM UEP 3 Ψ derived fro the UEP in [0] are widey used in various appications, which incude iage inpainting in [5]; iage denoising in [8]; high and super resoution iage reconstruction in [0]; deburring and bind deburing in [7, 8, 6, 9]; and iage segentation in [] In a these appications, the tensor products of univariate B- spine fraeets constructed in [0] were used For sipicity of notation, we introduce the case d =,whichisusediniagerestoration Let, x, y := ψ xψ y, 0, ;x, y R, ψ where ψ 0 := φ for convenience We denote Ψ = {ψ, : 0, ;, 0, 0} Then, XΨ is a tight waveet frae for L R The interested reader shoud consut the survey artices [5, ] for detais There are a few theoretica and appied issues in our ind that otivate our adventure here One of the is that whie the function can be saped by its inner product with proper diation and shifts of underying refinabe functions, the question of how to sape its derivative propery so that the corresponding waveet coefficients can be viewed as proper sapes of its derivative is not copetey answered This is the crucia fact used in [], where a theory is deveoped to connect the tota variationa ethod and the fraeets based approach for iage restorations Resuts in section and section are otivated by this Another otivation is to consider whether there are siiar recurrence foruas for the spine fraeets of [0], so that we can copute vaues fast when needed This eads to soe resuts of section The paper is organized as foows Section deveops soe basic properties of B-spine fraeets In particuar, in subsection, we present recurrence foruas for B-spine fraeets ψ, in which the we-nown recurrence forua of B-spines can be viewed as a specia for of recurrence foruas of B-spine fraeets This gives a fast agorith for coputing the We further show that the B-spine fraeets can be derived fro the th derivative of soe univariate box spines in subsection This fact was used in [], where the approxiation of derivatives of a given function via its waveet frae representation is needed In section 3, we investigate the asyptotic property of B-spine fraeets ψ, =,, We first prove that the univariate box spines defined in section unifory converge to a scaed Gaussian function under a id condition, and we further show that ax ax x R ψ x G x n 5/ 3/, where G is the th derivative of soe scaed Gaussian function Gx See section 3 for the detaied definition This eads to the discovery that a waveet syste generated by a finite nuber of consecutive directives of scaed Gaussian function fors a frae whose bounds are aost tight Properties of B-spine fraeets In this section, we give a recurrence forua for the B-spine fraeets which coputes higher order fraeets fro ower order ones We aso show that one can represent the derivatives of higher order fraeets by ower order ones Furtherore, we derive another set of foruas that represents each fraeet as a derivative of a univariate box spine

4 ZUOWEI SHEN AND ZHIQIANG XU Recurrence foruas for B-spine fraeets Whie the recurrence foruas for B-spines and their derivatives are we-nown see [], the corresponding foruas for B-spine fraeets are not avaiabe yet This section is to estabish such foruas Let B := ϕ + j /, where ϕ is given in and j is defined 3 Reca the foowing we-nown recurrence forua of B-spines: 5 B + x = x + + B x x B x Based on 5, one can copute B-spines fast and easiy, which aes B-spines usefu The derivative of B-spines can be coputed in ters of ower order spines as foows: d 6 dx B +x =B x + B x The ai of this section is to give corresponding foruas for the B-spine fraeets ψ, =,, To state the foruas convenienty, we present the foruas for the function ψ := ψ + j Note that the Fourier transfor of ψ is 7 ψ cos =i sin+ We note that the foruas presented in this subsection are used to cacuate the function vaue and the derivative of ψ Whenis even, ψ ψ Whenis odd, one can obtain those of the function ψ by the haf-transation of ψ Hence, the foruas given in this subsection aso wor for ψ with a proper shift Next, we present the recurrence reations of fraeets ψ Theore Let N be given,, andthefraeet ψ derived fro B-spine of order be given via its Fourier transfor as 7 Then, we have + the foowing recurrence forua between ψ and ψ :for ψ + + x + + x = ψ x the recurrence forua between ψ+ + x =x + x ψ + ψ + and ψ x + x ψ is + + x ψ x ; ψ x x Proof We first prove 8, woring in the Fourier doain Note that d d ψ = i x ψ xe ix dx, which ipies that the Fourier transfor of function g x := x 0 ĝ =i + ψ x is ψ x cos sin + cos sin + +

5 ON B-SPLINE FRAMELETS DERIVED FROM UEP 5 Note that and x x ψ x + = g x + + ψ x = ψ x + ψ x g x A sipe anipuation shows that the Fourier transfor of the right-hand side of 8 becoes + i + exp ĝ + i exp +exp i ψ exp i ĝ + ψ + cos = i + sin++ = ψ+ + This proves 8 Siiary, the Fourier transfor of the right side of 9 is i exp ĝ + ψ +exp i ĝ ψ ĝ = i + sin+ + = ψ+ +, which proves 9 Furtherore, cobining 8 and 9, we have a recurrence agorith for efficienty coputing ψ, =,, When <,wecanuse8tocopute ψ by ψ ;wecanuse9tocopute ψ by ψ Hence, we finay can reduce the coputation of ψ to that of ψ Note that the function ψ is a Haar waveet with if x [ /, 0, ψ x = if x [0, /], 0 if x > / We next show the ethod for coputing ψ by a tabe In the foowing tabe, for the notation, weusetheforua8,whieforthenotation, weuse9: B B B 3 B ψ ψ ψ ψ 3 3 ψ ψ 3 3 ψ ψ ψ 3 ψ Finay, we copute the B-spine B 5 and corresponding fraeets see Figure by appying the ethod here Next, we give the recurrence forua for coputing the derivatives of ψ

6 6 ZUOWEI SHEN AND ZHIQIANG XU Fig The B 5 and corresponding fraeets Theore Let N be given,, andthefraeet derived fro B-spine of order be defined by its Fourier transfor as 7 When, we have d ψ x = ψ x + ψ x dx When =, wehave d dx ψ x = ψ x + + ψ x ψ ψ x Proof We prove here whie can be proved siiary A sipe cacuation shows that the Fourier transfor of the right side of is i cos sin + e i e i 3 =i + cos sin+ d Note that the Fourier transfor of dx i + ψ x is cos sin+ Cobining 3 and, we concude Rear Note that ψ 0 = B Ifwetae =0in8,therecurrencereation 8 is reduced to 5, which is the recurrence forua for B-spines Siiary, if we tae =0in,thenisreducedtothederivativeforuaofB-spines6

7 ON B-SPLINE FRAMELETS DERIVED FROM UEP 7 Representing ψ as the th derivative of a univariate box spine We first reca the definition of box spines The univariate box spine B Ξ associated with a atrix Ξ R is the distribution given by the rue see [3] 5 BxΞϕxdx = ϕξudu for a ϕ DR, R [, where DR is the test function space The box spine can be considered as a voue function of the section of unit cubes see [3, 5, ] If we tae Ξ =,,, R,thentheboxspineB Ξ is reduced to a B-spine of order Inthefoowing theore, we show that the B-spine fraeet can be considered as the higher order derivative of a box spine up to a constant Theore 3 Let N be given and Supposethatthefraeetψ is defined by its Fourier transfor in Set [ Ξ, :=,,, }{{},, Then 6 ψ x = } {{ } ] d dx B x j Ξ,, where j is defined in 3 The spine Bx j Ξ, is copacty supported, and its Fourier transfor does not vanish at the origin In particuar, ψ is the - order derivative of B j /,whereb is the B-spine of order, which is copacty supported and whose Fourier transfor does not vanish at the origin Proof This again is proved in the Fourier doain It foows fro the definition of box spines 5 that the Fourier transfor of the box spine B Ξ, is sin sin ˆBΞ, = Then the Fourier transfor of d dx B x j Ξ, can be coputed as e ij/ i ˆBΞ, = i = i = i = ˆψ, e ij/ sin sin sin e ij/ cos sin e ij/ cos sin+

8 8 ZUOWEI SHEN AND ZHIQIANG XU which proves 6 According to the definition of box spines, we have B x j =B x j Ξ, And hence, ψ is the -order derivative of B j / Rear Theore 3 shows that one can obtain the B-spine fraeet by cacuating the derivative of box spines, which provides a new path to construct spine fraeets We hope to construct utivariate spine fraeets by cacuating the derivative of soe reevant box spines in future wor Theore 3 shows that each spine fraeet defined in with vanishing oents of order up to is the th derivative of a univariate box spine whose support is the sae as the fraeet and whose Fourier transfor dose not vanish at the origin More iportanty, a genera forua for such a function is given here In [], a genera forua is absent, athough it shows the existence of such functions and gives the expicit for of such functions for the spine fraeets defined in with =, Such functions are used in [] to discretize differentia operators by using spine fraeets Indeed, et ϕ, x := Bx j Ξ, According to Theore 3 and the integration by parts, we can obtain that 7 f,ψ,n, = n/ n f + n,ϕ, which is used in [] to approxiate the nor of the differentia operator by the weighed nor of the waveet coefficients 3 The asyptotic property of B-spine fraeets 3 The asyptotic convergence of univariate box spines It is nown that up to a noraization, B-spines are cose to Gaussian function pointwise, as we as in L p sense with p<+ as the order of the spine cose to infinity see [3] Motivated by this, in this subsection we investigate the asyptotic convergence of univariate box spines, which is hepfu to understand the convergence of ψ with ψ being the -order derivative of a box spine up to a constant To state the resuts convenienty, throughout the rest of this paper we sha use the notation X a,b, Y to refer to the inequaity X C Y,wheretheconstantC ay depend on a, b,, but no other variabe In the next theore, we show that the noraized box spines converge unifory to a Gaussian function Theore For each N, et Ξ := [a,,a ] R, where a j > 0,j =,,LetB Ξ be the box spine associated with Ξ Assue that 8 Ξ = σ + ɛ with σ R afixedconstantandi ɛ =0,andassuethat 9 c ax j a j in j a j c,

9 ON B-SPLINE FRAMELETS DERIVED FROM UEP 9 where c and c are fixed positive constants independent of Then, 6 0 ax x πσ exp 6x σ BxΞ n 3 c,c + ɛ nɛ n In order to prove Theore, we need the foowing ea of the box spine B Ξ Lea 5 Under the conditions of Theore, ax f exp σ n c,c + ɛ nɛ, where f := ˆBΞ a j = sinc Proof Withoutossofgeneraity,wesupposethatforeachfixed Then 8 and 9 ipy that j= 0 <a a a c,c a a c,c We first consider the case π/a Note that sinc is a onotone decreasing function in [0,π]and sinc π for π Then, we have ax π/a a f = ax π/a sinc j j= ax π, π a sinc a c,c β, where β< is a positive constant And hence, when π/a, f exp σ f +exp σ c,c, which ipies We next consider the case where π/a Tayor expansion shows that when π/a, 3 n f = n sinc j= a j = Ξ + S,

10 0 ZUOWEI SHEN AND ZHIQIANG XU where S = Ξ Ξ is a unifory convergent series on π/a By 3, we now obtain that when π/a, f = sinc j= a j =exp σ exp ɛ exp S Hence, f exp σ exp S exp σ exp ɛ +exp σ exp S First, we prove that 5 exp σ exp ɛ ɛ nɛ and 6 exp σ exp S n Then, cobining, 5 and 6, we obtain Equations 5 and 6 reain to be proved We first prove that exp σ exp ɛ ɛ nɛ By Tayor expansion, when nɛ /σ, exp σ exp ɛ exp ɛ ɛ nɛ ; when nɛ /σ, exp σ exp ɛ exp σ ɛ This gives 5 We next prove 6 Note that when n /σ, 7 exp σ exp S exp S c,c n When n /σ π/a,wehave exp σ,

11 which ipies that 8 exp σ ON B-SPLINE FRAMELETS DERIVED FROM UEP Cobining 7 and 8, one derives 6 Proof of Theore Note that exp π σ exp S n 6 expixd = πσ exp f expixd = BxΞ π Then 6 ax x πσ exp 6x σ BxΞ exp σ f d = exp σ f n d σ + exp σ f d + + n σ π a π a π a π a n 3 where β =ax{ π n σ exp σ f d exp σ f d + ɛ nɛ n + πa, sinc a 6x σ, n + n β 3 + ɛ nɛ n, } < Here, we use to obtain that f exp σ d n 3 Note that a /, =,,, is a bounded sequence and exp σ n n for σ + ɛ nɛ n Using a siiar ethod as in the proof of 8, we have that n σ π a f exp σ d n

12 ZUOWEI SHEN AND ZHIQIANG XU To estiate f exp /d π a we use the facts of and π a f d π a exp π a π a f d + d π a a d exp /d, Theore ipies that the noraized box spine B Ξ, converges unifory to a Gaussian function Coroary 6 Suppose that Ξ, = [,,, /,,/] }{{}}{{} Then, for each fixed, thefunction B xξ, converges unifory to 6 π exp 6x,as Proof Bythedefinitionofboxspines,wehave B x Ξ Ξ,, = B x / Ξ, Note that for each fixed, / = Then,Theoreshowsthatthebox Ξ, spine Bx, and hence / B xξ,, converges unifory to 6 the Gaussian function π exp 6x Rear 3 A we-nown resut is that B x converges unifory to 6 π exp 6x with see [3, ] In fact, the resut can be considered as a particuar case of Coroary 6 Note that B xξ,0 = B x If we tae = 0 in Coroary 6, then we have that B xξ,0, and hence B x, converges unifory to 6 π exp 6x with 3 The asyptotic property of B-spine fraeets We observe fro Coroary 6 that by changing variabes, BxΞ, is cose to 6 π Reca that Theore 3 says that ψ x = / exp x d dx B x j Ξ,

13 ON B-SPLINE FRAMELETS DERIVED FROM UEP 3 These two observations ead us to consider the reation between ψ x andtheth derivative of a Gaussian function G, x, which is defined as x G, x := C, exp, where C, = 6 π / Let 9 G x := d dx G, x j, =,,, where j is given in 3, and G := {G,,G } Theore 7 Let N be given,, andthefraeetψ by its Fourier transfor in derived fro B-spines of order Then, be defined ax ax x R ψ x G x n 5/ 3/ In order to prove Theore 7, we need the foowing two eas n Lea 8 Let 0 ;thentheinequaity ax exp hods Proof Forconvenience,denote and F := exp := For each fixed [,] Z, thefunctionf is increasing on the interva [0, ], whie F is decreasing on [, And hence F arrives at the axiu vaue at According to the inequaity e, we have n F n 3

14 ZUOWEI SHEN AND ZHIQIANG XU Then, when we have that n F This ipies that whenever hods, one has n n 6 n 5 5, n 3 n n 6 n 5 5 n 3 n 6 n n 5 30 ax F F We next consider the case where 5 5 Using the inequaity, one gets that Therefore, when 5 n F n 5,onehasthat e n 3 n F n e n n, 3 which ipies that whenever 5 5,thefoowinghods: 3 ax F F We now turn to the case 5 Forthiscase,weappytheinequaity e to obtain that n F Hence, when 5, wehavethat n e e n n 3 3 ax F F We finay consider the case where enough, we have 0 n When 0 n n n 6 n 5 Note that when is arge, F reachtheaxiuvaue

15 ON B-SPLINE FRAMELETS DERIVED FROM UEP 5 at 0 := 0 n Thenasipecacuationshowsthatwhen 33 ax F F 0 n 0 n n 6 n 5, Cobining 30, 3, 3, and 33, we concude the proof Lea 9 For every R, theinequaity ax sinc / sinc exp n hods Proof For convenience, we ony provide the proof for the case where 0 The proof of the other case is siiar By Tayor expansion, when 0 3π,wehave sinc sinc =exp n Then, for 0 3π,wehave / 7/8 exp O sinc sinc exp / exp exp / exp, / 7/8 880 where the ast inequaity is obtained by Lea 8 Next, when 0 0 n, note that and exp F = 7/8 n 880 / exp is a bounded function Hence, for 0 0 n,wehave sinc / sinc exp n Finay, we consider the case when 3π Weassertthatwhen 3π,thefoowing inequaity hods: 8 e /8 3 ax sinc sinc 3π

16 6 ZUOWEI SHEN AND ZHIQIANG XU With this assertion, we have sinc / sinc exp sinc sinc + exp / sinc sinc + / exp Here, the ast inequaity is foowed by 3 and Lea 8 Equation 3 reains to be proved Note that when 3π, sinc sinc Then we ony need prove ax / 3π/ / 3π/ 3π/8 Appying the inequaity e,wehavethat ax This proves 3 Proof of Theore 7 Let and M := ax 3π/ 3π/8 8 e /8 3π 8 e /8 3π 3π/8 sinc sinc exp { } { } n n I := R : 0,I := R :0 3π, { I 3 := R : 3π } Appying Lea 9 and Lea 8, we concude that n 5/ Md, Md I 3/ I, respectivey By an arguent siiar to that eading to 3, we can obtain that there exists 0 <γ<suchthat I3 Md γ n 5/ 3/

17 This eads to ax ax x R ψ ON B-SPLINE FRAMELETS DERIVED FROM UEP 7 x G x Md = Md + Md + Md I I I 3 n 5/ 3/ Rear It was proved in [7] that for each fixed, uptoanoraization,a proper scaed ψ unifory converges to the -order derivative of a scaed Gaussian function with tending to infinity Our resut is in a different direction In fact, we show that for sufficienty arge fraeets ψ,,ψ,,ψ unifory in x and cose to derivatives of consecutive orders,, of a scaed Gaussian function whose scae depends on Gaussian frae Theore 7 eads us to consider whether a waveet syste generated by a finite nuber of consecutive derivatives of a propery scaed Gaussian function fors a frae of L R In this section, we show that the frae property of Xψ,,ψ canbetransferredtothatofxg,,g Here we reca the definition of G For each fixed N, weconsidertherescaedgaussian function where and where j is given in 3, and x G, x =C, exp, C, = 6 π /, G x = d dx G, x j, =,,, G = { } G,,G Before stating the foowing ain theore of this section, we reca the definitions of the frae and Besse sequence A faiy {f j } j J L R is caed a frae with bounds A and B if A f j J f,f j B f hods for a f L R If A = B, then{f j } j J is caed an A-tight frae Moreover, a faiy {f j } j J L R is caed a Besse sequence with a bound R if f,f j R f hods for a f L R j J

18 8 ZUOWEI SHEN AND ZHIQIANG XU Theore 0 Let XG be the waveet syste generated by functions G Then XG is a frae syste with frae bounds A and B for sufficienty arge Furtherore, the frae is cose to being tight when is sufficienty arge In fact, asyptoticay, we have i A = i B = We ca XG agaussian frae ToproveTheore0,weneedthefoowing theore, which is proved in [6], together with severa eas Theore see [6] Let {f j } j J be a frae of L R with bounds A and B Assue that {g j } j J L R is such that {f j g j } j J is a Besse sequence with a bound R<AThen{g j } j J is a frae with bounds A Let 35 φ R A and B + := ψ G,=,,, and Φ := {φ,,φ } R B Since XΨ is a tight frae with frae bound, to prove that XG is a frae, according to Theore, we ony need to show that XΦ is a Besse sequence with a bound R 0 An estiate of the Besse bound of a given a sequence is provided in [] that enabes us to estiate the Besse bound of XΦ seeaso [] Let 36 R := sup Z n Z = n n +π Then, for arbitrary f L R, the foowing inequaity hods: φ XΦ f,φ R f, ie, R is the Besse upper bound of the syste XΦ Next, we estiate R For this, we need the foowing eas Lea Let be the Fourier transfor of φ defined in 35 Thenthe foowing three estiates for hod: i ii iii ++, 0, ax ax R n

19 ON B-SPLINE FRAMELETS DERIVED FROM UEP 9 Proof First, a sipe coputation eads to ˆψ = Ĝ = sinc sinc exp For i, when 0, a sipe arguent shows that exp +, sinc sinc +, which ipies that ++ For ii, the Tayor expansion shows that when π, n sinc sinc = O6 Then, when, sinc sinc =exp, exp 78 exp 880 O6 which ipies that Finay, the concusion of iii can be obtained by Lea 9 directy Lea 3 Let R be given by 36 Then Proof Let R, := R, := Then, we have that sup R n5 n Z = sup and n, Z\{0} n Z = 37 R R, + R, i R =0 n n +π

20 0 ZUOWEI SHEN AND ZHIQIANG XU To estiate R, we consider R, and R,,respectiveyWefirstestiateR, For this, we rewrite R, = sup n = sup [S +S +S 3 ], where S := n 5 S 3 := = n Z = n og = n, S := n og <n<5 = By i in Lea, we obtain that for, S = n n 5 = + n n 5 = = n n 5 = 5 56 Using ii in Lea, when, S 3 = n = = n og = n og = n og 6 n og n og n n n = + n og = n n n n + n+ n, n Here, the ast inequaity uses the fact of { + } n+ n og, Z + being a bounded sequence and n og n = n og 56 n = og 8

21 ON B-SPLINE FRAMELETS DERIVED FROM UEP Moreover, by iii in Lea, we have S 3 = = og n 5 Cobining the resuts above, we obtain that n n5 38 R, = sup [S +S +S 3 ] n5 We next turn to R, = sup Z\{0} = n Z To state convenienty, we set Then βπ := sup = n Z R, Z\{0} n n +π n n +π βπ Set 0 := 0 When 0, using the arguent siiar to the one in the estiation of R, = β0, we can show that βπ for 0 We cai that when 0, 39 βπ And hence, 0 R, = 0 3 π / + βπ + 0 Cobining 38 and 0, we obtain that R = R, + R, n5, 0 3 π / which ipies the concusion Finay, we prove 39 A sipe observation is that βπ = β π Hence, we ony need consider the case where 0 Forconvenience,et β + π := β π := sup = n Z + sup = n<0 n n +π, n n +π

22 ZUOWEI SHEN AND ZHIQIANG XU Then βπ β + π+β π To estiate β + π, we furtherore set β + + π := sup β +π := = n Z + sup = n Z + n n +π, n n +π Then β + π =ax{β + +π,β +π} Noting that 0 =0,byLea, we have We next consider β + + π = sup = n Z + = n Z + = n Z + = β +π = n Z + n n +π + n +π + n +π 6 n +π 3 π / sup = n Z + = sup = n Z + n Z + 6 n/ π / n n +π n n π Asipeobservationisthatax{ n, n π} π 0 π Then using i in Lea, we obtain that Set n 0 := og π +5 Then, sup = 0 n n 0 n n π 0 n n 0 = 0 n n 0 n n π + π = 0 n n 0 6 π og 6 π π π + π =

23 We next consider ON B-SPLINE FRAMELETS DERIVED FROM UEP 3 sup = n 0+ n sup n n π By i in Lea, when n n 0 +, sup n + π, n π Hence, Therefore, sup = n n 0+ = n n 0+ 5 π n n 0+ + n π n n π + π n π + n π 0 π, 0 This eads to β + π og π 6 π + 0 π β + π =ax{β π,β + π} π / Using ii in Lea and a siiar anaysis as above, we can obtain that β π = sup = n<0 π Putting everything together, we have that βπ =β + π+β π This proves 39 Proof of Theore 0 Reca that Φ = {φ,,φ },φ n n +π 6 π + π 3 π / = ψ G, =,,, and that XΨ is a tight frae with frae bound, where Ψ = {ψ,,ψ } Lea 3 shows that XΦ is a Besse sequence with a bound R 0 Then

24 ZUOWEI SHEN AND ZHIQIANG XU Tabe The nuerica resuts of frae bounds of XG A B Fig The graphs of G x = 3/πx exp x eft and G x = 7/8πx exp 6x right Theore gives us that XG is a frae with frae bound A = R and B =+ R as sufficienty arge Furtherore, it can be cose to a tight frae, since i A = i B =,whichcopetestheproof As the spine fraeet case, the Gaussian functions used in Theore 0 to derive a waveet frae syste can be used to sape derivatives of functions, so that the nors of its various weighted waveet coefficients at each diation eve can be viewed as approxiations of corresponding nors of derivatives of functions Rear 5 Athough Theore 0 confirs the case where is sufficienty arge, the resut of Theore 0 sees to hod for sa can be as sa as For sa, cobining36andtheore,wecanestiatethefraeboundsofxg nuericay We ist the frae bound estiation of XG, 8, in Tabe, which ceary shows the frae property of XG for sa For exape, for =,XG is a frae with frae bounds A and B 900 Figure shows the graphs of the functions of G and G,respectivey Furtherore,oncea waveet syste is a frae in L R, with proper choosing of weights, the syste can becoe a frae in various proper spaces, such as Soboev or Besov spaces Rear 6 In the iterature, there is ony one exape of a certain order of the derivative of the Gaussian function whose diations and transations of a proper chosen attice for a frae in L R That is the Mexican hat function which is the second derivative of a Gaussian function see, eg, [] Here we provide a arge faiy of waveet frae systes derived fro various order of derivative of the Gaussian function Furtherore, the systes given here can have consecutive derivatives of soe variations of the Gaussian functions, which can be usefu when the approxiation of the different order of the derivatives of a function is needed Furtherore, when is sufficienty arge, the corresponding syste is aost a tight frae for L R

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