Analysis and construction of a family of refinable functions based on generalized Bernstein polynomials

Transcription

1 Cheng and Yang Journa of Inequates and Appcatons 6 6:66 DOI.86/s R E S E A R C H Open Access Anayss and constructon of a famy of refnabe functons based on generazed Bernsten poynomas Tng Cheng and Xaoyuan Yang * * Correspondence: xaoyuanyang@vp.6.com Department of Mathematcs, LMIB of the Mnstry of Educaton, Behang Unversty, Beng, Chna Abstract In ths paper, we construct a new famy of refnabe functons from generazed Bernsten poynomas, whch ncude pseudo-spnes of Type II. A comprehensve anayss of the refnabe functons s carred out. We then prove the convergence of cascade agorthms assocated wth the new mass and construct Resz waveets whose daton and transaton form a Resz bass for L R. Stabty of the subdvson schemes, reguarty and approxmaton orders are obtaned. We aso ustrate the symmetry of the correspondng refnabe functons. MSC: 4C4; 4A5; 46B5 Keywords: refnabe functons; generazed Bernsten poynomas; cascade agorthms; Resz waveet bases Introducton Durng the ast decades, the study of refnabe functons has attracted consderabe attenton. Ths w be foowed by a descrpton of the deveopment of refnabe functons. Intay, B-spnes appeared as a speca famy of refnabe functons, whch satsfed the scang equaton n []. After that, pseudo-spnes of Type I were ntroduced by Daubeches et a. [] andseesnc[], aong wth the appearance of pseudo-spnes of Type II n [4]. Especay, the propertes of them, such as stabty, reguarty, approxmaton orders etc., were fuy anayzed n [, 4, 5]. Later, there appeared other refnabe functons that had been dscovered, for nstance, dua pseudo-spnes [6, 7], pseudo-box spnes [8], Batte-Lemare refnabe functons [9 ], Butterworth refnabe functons [, ], pseudo-butterworth refnabe functons [4], generazed pseudo-butterworth refnabe functons [5], and so on. Notce that amost a contrbutons above focus on the extensons based upon pseudo-spnes. However, we dscover that pseudo-butterworth refnabe functons, as an extenson of pseudo-spnes, ac compact support, athough they have exponenta decay to compensate the ac. It s natura to consder a new extenson of pseudo-spnes, whch has compact support and exponenta decay, n partcuar, mass of new refnabe functons derved from generazed Bernsten poynomas [6]by substtuton and summaton. 6 Cheng and Yang. Ths artce s dstrbuted under the terms of the Creatve Commons Attrbuton 4. Internatona Lcense whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgna authors and the source, provde a n to the Creatve Commons cense, and ndcate f changes were made.

2 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page of 9 We start wth generazed Bernsten poynomas, defned as n tt + α t +[ ]α t t + α t +[n ]α S n t, + α + α + [n ]α where α. New mass wth order m,, α for gven nonnegatve ntegers m,,satsfyng < m 5,and α < m+ 7, are defned by substtutng t sn ω, n m + n and the summaton of + terms of them as foows: τ ω: m + ω sn + α / m+ m+ cos ω + α + α. They ncude amost a mass of pseudo-spnes of Type II [4]whenα.Furthermore, the propertes of the new refnabe functons correspondng to the mass are addressed. Convergence of cascade agorthms s mpemented, whch guarantees the exstence of refnabe functons. At the same tme, we construct ther Resz waveets whose daton and transaton form a Resz bass for L R. Fnay, stabty of the subdvson schemes, reguarty, approxmaton orders, and symmetry are anayzed. The remander of ths paper s organzed as foows: Secton coects some notatons. Secton eaborates the convergence of cascade agorthms based on the mass. Secton 4 constructs a Resz bass for L R. Secton 5 anayzes the stabty of the subdvson schemes. Secton 6 shows the reguarty and the nfuences of the parameters m,, α on the decay rate. Secton 7 gves the approxmaton order of the new refnabe functons. Secton 8 ustrates the symmetry of the refnabe functons. Premnares For the convenence of the reader, we revew some defntons and propertes as regards refnement mass n ths secton. By L R, we denote a the functons f xsatsfyng f x L R : R f x dx < and by Z the set of a sequences c defned on Z such that c Z : c <. Z A compacty supported functon φ L R s refnabe f t satsfes the refnement equaton φ Z τφ,

3 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page of 9 where τ, caed the refnement mas of φ, s a fntey supported sequence. The Fourer transform of φ s φξ φte ξt dt, ξ R. R For a gven fntey supported sequence c, ts correspondng Laurent poynoma s defned by cz: Z cz, forz C\{}. The correspondng trgonometrc poynoma s ĉξ c e ξ, ξ R. Wth the above, the refnement equaton can be wrtten n terms of ts Fourer transform as ˆφξ ˆτξ/ ˆφξ/, ξ R. 4 We ca ˆτ the refnement mas for convenence, too. By the teraton of equaton 4, the correspondng refnabe functon φ can be wrtten n terms of ts Fourer transform as ˆφω: ˆτ ω. 5 Defne T : R/[πZ]anddefneL, R as the subspace of a f L Rsuchthat f L, R : ˆf ω +π <, L T Z where L T denotes the space of a π-perodc measurabe functons wth fnte essenta upper bound. The notaton T and the space L, Rwerentroducedn[7]and[8], respectvey. AsystemXψ{ψ n, n/ ψ n :n, R} s a Resz bass f there exst < C C < such that, for a sequences c Z, C c Z c[n, ]ψ n, C c Z n, Z L R hods and the span of {ψ n, : n, Z} s dense for L R. The functon ψ s caed a Resz waveet f the XψformaReszbassforL R, whch s aso caed a Resz waveet system. Afunctonφ L R s caed stabe f there exst two postve constants A, B,such that for any sequence c Z A c Z cφ B c Z. L R Z

4 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 4 of 9 It s we nown that ths condton s equvaent to the exstence of two other postve constants A, B,suchthat A ˆφξ +π B, Z for amost a ξ R [9]. The upper bound aways hods f φ has compact support and the ower bound s equvaent to ˆφξ +π, 6 Z for a ξ R,wheredenotes the zero sequence. In partcuar, we consder the stabty of the compacty supported centered B-spnes, B m ξe m ξ snξ/ m. ξ/ Snce t s obvous that there s C >,suchthat B m ξ > C for a ξ [ π, π], we have [ B m ξ, B m ξ ] ξ B m ξ + B m ξ +π B m ξ C. Hence a B-spnes are stabe. We use Z P n : f Z f, φ n, φ n, to approxmate f L R. A functon φ satsfes the Strang-Fx condton of order m f ˆφ, D ˆφπ, Z \{}, < m. Under certan condtons on φ e.g., f t s compacty supported and ˆφ, the Strang- Fx condton s equa to the requrement that τˆ has a zero of order m at each of the ponts n {, π}\. In [], f φ satsfes the Strang-Fx condton of order m and the correspondng mar τˆ satsfes ˆ τ O m at the orgn, then the approxmaton order s mn{m, m }. Inthefoowng, wewadoptsomeofthenotatonsfrom [8]. The transton operator Tâ for π-perodc functons â and f can be defned as [Tâf ]w: âω/ f ω/ + âω/ + π f ω/ + π, ω R. For τ R, a quantty s defned by ρ τ â, :m sup ω τ /n T ṋ a sn n The notaton ρâsdefnedby. L T ρâ:nf { ρ τ â, : âω + π snω/ τ L Tandτ }.

5 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 5 of 9 Afunctonf beongs to the Höder cass C β T wthβ >,ff s a π-perodc contnuous functon such that f s n tmes contnuousy dfferentabe and there exsts a postve number C satsfyng f n x f n y C x y β n for a x, y T,wheren s the argest nteger such that n β. Convergence of cascade agorthms based on the mass In ths secton, a demonstraton of the convergence of cascade agorthms n the space L, R s gven. For notatona smpcty, we w ntroduce the foowng three defntons: B ω: G ω: T ω: ω sn + α m + ω sn m+ + α m + ω sn / m+ + α. ω / m+ cos + α + α, m+ + α m+ cos ω cos ω / m+ + α + α + α, We w provde three emmas about the reatons of the quanttes ρ τ â, assocated wth mass and a condton of the convergence of cascade agorthms whch are necessary for the foowng theorem. Lemma [8], Theorem 4. Let âbeaπ-perodc measurabe functon such that â C β T wth â and β >.If âω +e ω τ Âω a.e. ω R for some τ such that Âω L T, then ρ τ â, nf T ṋ a n L T m T ṋ a n L T ρ Â,. n N n Lemma [8], Theorem 4. Let â and ĉ be π-perodc measurabe functons such that âω ĉω for amost every ω R. Then ρ τ â, ρ τ ĉ,, τ R. Lemma [8], Theorem. Let â C β T wth â and β >.If ρâ<,then the cascade agorthm assocated wth the mas âconvergesnthespacel, R. A usefu condton of provng the convergence of cascade agorthms s descrbed n the foowng emma.

6 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 6 of 9 Lemma 4 For two postve ntegers, m, < m 5,f then α < max B ω ω T m + 7, 7 m+,,,...,. 8 Proof For,,...,,tsobvousthat B ω cos ω +m + α sn ω +α B + ω. We cam that B ω B + ω cos ω +m + α sn ω +α >. 9 Snce < m 5,for,,...,,wehave < m +. There are two cases to consder: Case I: Suppose that cosω. By 7and, t s easy to see that Then α >> cos ω cosω m +. ω +m + α > sn + α. Ths mpes Condton 9. Case II: Supposethat cosω<.inthe same way,we get α > cosω m +, for,,...,.then hods. Ths concudes the cam 9. By usng 7, one gets Then Thus 4m + m + α < 4m + m + m + 7 m + α m + α < + α + m + α +m + α < 4. m + m + α < m + m + m + 4..

7 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 7 of 9 Smary, one has m + α +m + α <. For any x,notcethat x > 4 ++x and B ω, whch s a contnuous functon on [ π, π] and s dfferentabe on π, π, has themaxmumvaueatω π. The reason as foows: the equaton [B ω] has three zeros, at ω,±π. Snce[B ω] >,B s the mnmum of B ω on[ π, π]. Thus B ±πsthemaxmumofb ωon[ π, π]. Therefore, appyng 9,,, 4, and ω B ω sn m+ m+ α m+ / m+ ω / m+ cos + α + α + α α + +α m + α m+ +m + α 4 m+, m + α + α + m + α we get the nequaty 8. Theorem If we et τ ω be the mas, then the cascade agorthm assocated wth the mas τ ω converges n the space L, R. Proof For cos ω +e ω,onehas τ ω 4 +e ω 4 Tω +e ω 4 Tω. Appyng B ω m+ ω / m+ cos + α + α + α m+ / m+ + α

8 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 8 of 9 and Lemma 4,weobtan max T ω max ω T ω T < max ω T < max ω T B ω+ Brngng Lemma and Lemma together yeds m + B ω m + + max B ω ω [ π,π] m+ + m+. 5 ρ τ ω ρ 4 τ ω, ρ Tω, <. Thus, by Lemma, the cascade agorthm assocated wth the mas τ ωconvergesn the space L, R. 4 Resz waveets In ths secton, we sha construct Resz waveets based on the mass. The foowng two emmas anayze recurrence reatons of τ ω, whch are usefu for the constructon of Resz waveets. Lemma 5 If et τ ω be the mas, then τ ω satsfes and τ m+,,α ωτ m+,,α ω+ τ m+,,α ωτ ω cos ω +mα +m + α sn ω +α +m + α G ω 6 G ω. 7 Proof By the summaton of + terms of generazed Bernsten poynomas satsfyng the recurrence [6] S n+ t we can derve that t +n α S m++ t +nα t + α S n t+ +nα t +m + α S m+ t +m + α + S n t, t + α S m+ t. 8 +m + α

9 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 9 of 9 Substtutng t sn ω n8, one gets ω S m++ sn cos ω +m + α ω S m+ sn +m + α + sn ω + α ω S m+ sn +m + α ω S m+ sn + cos ω +mα +m + α G ω. 9 By usng 9, one obtans ω S m++ sn ω S m+ sn + + S m+ ω sn cos ω +mα +m + α ω S m+ sn ω sn S m+ sn ω +α +m + α G ω G ω. Thus, the condtons 6 and7 are demonstrated. Lemma 6 If et τ ω be themas,then we derve τ ω + τ ω + π >, ω T. Proof When,onehasm >6, and τ m,,α ω τ m,,α ω + π m + m cos ω m + ω + sn m ω m sn ++mα / m + α ω cos m ω cos ω m cos ++mα m + α / m + α + α / m + α + α, ω / m sn + α + α.

10 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page of 9 It foows from the above two equates of τ m,,α ω and τ m,,α ω + π that τ m,,α ω + τ m,,α ω + π m sn ω ++mα m ω cos +mα / m + α + α m cos ω + ++mα m ω / m sn + α + α +mα + m sn ω m +mα + + m cos ω m +mα sn ω +α cos ω +α + m sn ω m sn ω + +mα +α + + m cos ω m cos ω + +mα +α + m sn ω +mα + + m cos ω +mα ω msn ω mcos. The frst nequaty foows from the genera formua of the Bernou nequaty. Substtutng y sn ω n, et f y + ym my + + +mα We cam that f y> ym. ym +mα. Frst of a, f y s derved as foows: f y + ym m +mα +mα + + +m + ym +mα ym +mα m my + + ym +mα +mα ym ym my m

11 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page of 9 Addtonay, { + ym +mα + m f y ym +mα [ m my my +mα m + ym +mα m [ m y m y +mα m + ]} [ m my my +mα +mα m + ym ] +mα + m + ym [ m my +mα +mα m + ym ] +mα + + ym my m +mα +mα ] ym. +mα m m [ ] m + y m y +mα +mα m ym + +mα [ ] ym m m + m + y +mα +mα m ym + +mα + +. ym +mα Combnng f andf >yeds Here, { } f y mn f, f, f. f + + m y m +mα m +mα m + + /m +α m m +mα m >. Let gx + /x +α x. The condton foows from the fact that gxsncreasngon[5,+ ]. It s obvousy true for f > and f >.Thsconcudesthecam. Assume hodswhen. Consder the case.byusng6 n Lemma 5,wehave τ m+,,α ω τ m+,,α ω + τ m+,,α ω cos ω +mα +m + α ω S m+ sn cos ω +m α + +m + α ω S m+ sn.

12 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page of 9 Addng that τ ω + π τ m,,α ω + π + τ m,,α ω + π sn ω +m α +m + α ω S m+ cos sn ω +m α + +m + α Combnng together the above equates so that ω S m+ cos. τ ω + τ ω + π τ m,,α ω + τ m,,α ω + π + f ω+f ω + π, where f ω τ m,,α ω cos ω +m α ω S m+ sn +m + α cos ω +m α ω + S m+ sn. +m + α The condton foowsfromthefact that τ m,,α ω + τ m,,α ω + π >. Ths concudes the proof of Lemma 6. Condtons for constructng a Resz waveet bass are gven by the foowng emma. Lemma 7 [8], Theorem. Let â C β T wth â and β >.Let φ be the refnabe functon correspondng to the refnement mas â. Denote ˆbωe ω âω + π. Defne a waveet ψπ:ˆbω ˆ ˆφω. Then the shfts of φ are stabe n L R and φ generates a Resz waveet bass n L R f and ony f ˆb and dω:âωˆbω + π âω + πˆbω for a ω R. ρâ<and ρˆã<, where ˆãω:ˆbω + π/dω. Theorem Let φ be the refnabe functons from generazed Bernsten poynomas wth the refnement mas. Defne a waveet such that ˆ ω:e ω τ ω + π ˆφ ω. Then generates a Resz waveet bass n L R. Proof Note that ˆbωe ω τ ω + π, thus ˆb τ π,and dωτ ωˆbω + π τ ω + πˆbω τ ω e ω τ ω +π τ ω + πe ω τ ω + π e ω τ ω + τ ω + π, ω R.

13 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page of 9 Therefore, condton n Lemma 7 hods true. ρτ ω<hasbeenprovedntheorem.notngthat τ ω ˆbω + π dω we obtan τ ω e ω τ ω +π e ω τ ω + τ ω + π τ ω τ ω + τ ω + π, τ ω τ ω + τ ω + π +e ω 4 4 T ω τ ω + τ ω + π. Combnng 5and n Lemma 6 yeds max ω T T ω τ ω + τ ω + π <. By usng Lemma and Lemma,tsdervedthat ρ τ ω ρ 4 τ ω, ρ T ω τ ω + τ ω + π, <. Consequenty, condton n Lemma 7 hodstrue. 5 Stabty of the subdvson schemes In ths secton, we w prove the stabty of refnabe functons based on the mass. The foowng emma gves the recurrence reatons of T ω. Lemma 8 If we et τ ω be the mas, then T ω satsfes and T m+,,α ωt m+,,α ω+ cos ω +mα cos π + m + α T ±π +m + α T m,,α ±π+ G ω, for ω π, 4 m + B ±π. 5 Proof We consder nto two cases. Suppose that ω π. Equaton6 s apped n Lemma 4 so that T m+,,α ωt m+,,α cos ω ω+ +mα cos π + m + α G ω.

14 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 4 of 9 Suppose, on the other hand, that ω π; we chec the condton 5. Note that T ±π m + B ±π m + B ±π+ m + +m + α T m,,α ±π+ + m + m + B ±π. B ±π Ths estabshes the emma. Theorem Let τ ω be the mas, then the refnabe functons φ wth the mass τ ω are stabe. Proof We cam that the condton 6 hods. Indeed, the refnabe functons wth the mass τ ω, whch beong to L R asshownntheorem, havebeenproved.they are compacty supported for fnte refnement mass. In the foowng, we derve the nequaty τ ω m + sn ω > cos ω m+ + α + α + [m + ]α + α + α + [m + ]α m + ω ω m+ sn cos ω + α + α + [m + ]α cosm m + ω ω sn cos > ω + α + α + [m + ]α cosm. 6 Appyng the stabty of B-spnes yeds the exstence of two postve constants A, B, such that A B m ξ +π B, for amost a ξ R, Z where B m denotes the B-spne of order m. Itmpesthat B m ω has the fnte functona vaue. Combnng 6 wth7yeds φ ω τ ω > cos m ω + α + α + [m + ]α

15 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 5 of 9 Snce < ths eads to M m cos m ω M + + α + α + [m + ]α M m M + + α + α + [m + ]α M cos m ω m M + + α + α + [m + ]α cos m ω M cos m ω. + α + α + [m + ]α <, M m. 7 M + + α + α + [m + ]α Moreover, φ ω B m ω m M + M M. + α + α + [m + ]α Thus, there exst two postve constants A, B,suchthat A φ ξ +π B, for amost a ξ R. Z Ths concudes the theorem. 6 Reguarty Ths secton s devoted to an anayss of the reguarty of refnabe functons φ wth the mas τ ω defnedby. Our prmary goa s to obtan the ower bound of the reguarty exponents γ of refnabe functons φ by estmatng the decay rates β of ther Fourer transform. The reaton s expressed by γ β ε, for any sma enough ε >;see[]. Consequenty, φ C γ.next,wewgvean estmate of the decay rates β of the Fourer transform of refnabe functons φ wth the mas τ ω. By [, ], for any stabe, compacty supported refnabe functons φ n L R wth φ, the refnement mas τ must satsfy τ and τπ.thus,τ

16 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 6 of 9 can be factorzed as τωcos n ω Lω, where n s the maxma mutpcty of the zeros of τ at π and Lω s a trgonometrc poynoma wth L. Therefore, one obtans φω τ ω cos n ω L ω ω sn c n L ω, whch shows the decay of φ can be characterzed by τ as stated n the foowng theorem. Theorem 4 [], Lemma 7.7 Let τ be the refnement mas of the refnabe functon φ of the form τω cos n ω Lω, ω [ π, π]. If Lω π L for ω π, LωLω π L for π 8 ω π, then φω C + ω n+k wth K og L π / og, and ths decay s optma. By usng the foowng theorem, the decay rate of Fourer transforms of refnabe functons φ wth the mas τ ω s anayzed. Especay, the case α wasdemonstrated n [4], Theorem.4. We w gve the dscusson of the case < α < m+ 7. Theorem 5 Let φ be refnabe functons wth the mas τ ω. Then φ +K, C + ω / og s optma. As a resut, φ C γ, where γ β ε, for any sma enough ε >. where K og T π / og, and the decay rate β og T π Proof It s obvous that τ ω cos ω T ω, ω [ π, π]. We cam that T ω T π, for ω [, π ]. 9 Indeed, T ω s a contnuous functon on [ π, π ] and s dfferentabe on π, π. The maxmum vaue of T ω on[ π, π ] can be derved as foows: We fnd ω s the ony zero of equaton [T ω].snce[t ] >,T s the mnmum

17 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 7 of 9 of T ωon[ π, π ]. Consequenty, T T ± π. Here, T ωsaneven functon. Hence, 9 hods. Nextweshow T ωt ω π [ ] π T, for ω, π. Here, T ωt ω s a contnuous functon on ω [ π, π] and s dfferentabe on ω π, π. Snce the equaton [T ωt ω] hasnozeroson ω [ π, π], we are requred to compare T πt πwtht π T πt πt πt T π T 4π. As a resut of and T π T 4π T T π π T π, ony T π andt π need to be compared. Let m, α, ω be fxed. For,we cam that Here, π T m,,α > T m,,α π. T m,,α π 4 m ++mα / m 4 + α + m m and T π m / m m ++mα α + α. Settng gmt m,,α π T m,,α π, n whch α s fxed, then gm Notce that 4 m ++mα m / m 4 + α + m m / m m ++mα α + α. gm >.

18 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 8 of 9 Indeed, snce and < m 5,m 7. When m 7,onehas 6 / 7 g7 5/4 + 7 α 4 + α + α αα 6 / 7 + α >, for α < /7. Assumng that gm >,wenowprovegm>.let and g mt m,,α g mt π Then t s obvous that and Thus, g m g m π 4 m ++mα m m / m m ++mα α + α. / m 4 + α + α 4 +m α 4 m ++mα + mα 4 m ++m αg m m αm ++mα + mαm ++m α g m. Let gm g m g m g m 4 +m α 4 m ++mα + mα m ++m α 4 m αm ++mα g m + mαm ++m α 4 > g m g m +m α 4 m ++mα + mα m ++m α 4 m αm ++mα. + mαm ++m α 4 hα +m α 4 m ++mα +mα m ++m α 4 m ++mα. m ++m α m α +mα

19 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 9 of 9 h α s derved as foows: h α { m 4m +454m +97m 4 87m 6 +8m 7 α m 755m,8m +,754m 4 +,m 5,79m m 7 α + 8 4m +,984m,8m 96m 4 +,976m 5,968m 6 +,548m 7 α m 96m +,67m,584m 4 +4,m 5 4,5m 6 +,84m 7 α 4 + 5m,56m +,88m 56m 4 +,9m 5,86m 6 +,88m 7 α m 768m +5m 4 +5m 5 768m 6 +56m 7 α 6} / { 8 m + +mα +m +4 +mα + mα }. Note that for any fxed m >6,h α>onα. Thus, hα sncreasngonα. Ths together wth h > mpes that gm >.Thsconcudesthecam. Supposng T π > T π, we verfy π T m,+,α > T m,+,α π. It foows from Lemma 8 that sequvaentto T > π + +4mα +m + α G + +m + α T π+ Snce m + α π +m + α T > + π m + m + B m,+,α π. B m,+,α π, t s obvous that hods.hence,bytheorem4, φ satsfes φ +K, C + ω where K og T π / og, and the decay rate β og T π optma. As a resut, φ C γ,whereγ β ε >. / og s ε, for any sma enough Thefoowngtabegvesthedecayratesβ of the Fourer transform of refnabe functons φ wth the mas τ ω. To satsfy the constrant condton ofm,, α, we set α.ntabe. Tabe shows that for fxed, α, β ncreases as m ncreases and for fxed m, α, β decreases as ncreases. Ths s true ndeed as presented n the foowng proposton. Proposton 6. Let the decay rate β og T π / og beasgvenntheorem 5, then:

20 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page of 9 Tabe Decay rates β of refnabe functons wth the mas τ ω, for 7 m, < m 5,andα. m, m m m m m m m For fxed m, α, β decreases as ncreases. For fxed, α, β ncreases as m ncreases. Proof Appyng β og T π whch s equvaent to β Lemma 8,onehas Thus, Let T m+,,α π T m+,,α βm+,,α βm+,,α + π,, T π >,yeds β og T ogt π T π π ++4mα +m + α G 4+mα 4 + m + α G π.. Foowng the formua gven by π. I m+,,α βm+,,α. 4+mα For 4+m+α Sm+ 4 >,tshowsthatim+,,α > I m+,,α,whchsequvaenttosayng for fxed m, α, I ncreases as ncreases. Hence, for fxed m, α, β decreases as ncreases. Appyng Lemma 8 to the foowng expresson, one obtans T m+,,α π T π α G +m + α π. Furthermore, we have I m+,,α I α + m + α G π. For 4 4 +α +m+α G π >,wesmarygetim+,,α < I, whch means that for fxed, α, I decreases as m ncreases. Therefore, for fxed, α, β ncreases as m ncreases. 7 Approxmaton orders In ths secton, the approxmaton orders of refnabe functons wth the mas τ ωare anayzed n the foowng theorem.

21 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page of 9 Theorem 6 Let φ be refnabe functons wth the mas τ ω. Then φ provdes the approxmaton orders +. Proof In fact, the approxmaton orders of τ ω and τ ω are ndependent on α. For convenence, we set α.foowng [4],et R m, y m + y y m+, where y sn ω, then ω R m, sn O ω +. Snce m + R m, y m + y y m and sn ω sequatowhenω π, cosm ω has a zero of order 4m.Weconcude that R m, yo ω 4m +. For mn { +,4m } +, then φ provdes the approxmaton orders +. 8 Symmetry Symmetrc coeffcents of the mas are of great sgnfcance n mage processng. In the foowng, we w gve a symmetry proof and provde a graphca ustraton. Lemma 9 For m, Z +, α,,,...,, z e ω, we derve where z + z + α 4 + z + z m+ + α b, z + z, 4 m+ b, D,,,...,m +, D... m+ D... m+, m++ m++... m++ m+

22 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page of 9 and... f /+... m+... D f / +... m+, m++ f m++ / m m++ m+ where f x 4 x + α m+ Proof Set xzz + z,thenx [, ] and 4 x + α m+ + 4 x + α, x [, ]. + 4 x + α m+ b, x. 4 Fxng x,,..., m++,byusng4, we obtan. b, + b, + + m+ b, m+ f, b, + b, + + m+ b, m+ f,... m++ b, + m++ b, + + m++ m+ b, m+ f m++. 5 Snce the coeffcent determnant of 5... m+ D... m+, m++ m++... m++ m+ appyng Cramer s rue yeds b, D,,,...,m +, D where... f /+... m+... D f / +... m m++ f m++ / m m++ m+ Lemma For N, we have z + z / / z + z, f s an odd number, z + z + / z, ese. 6

23 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page of 9 Proof We consder nto two cases. Suppose that s an odd number. One gets z + z z / z + +/ z. Let,then +/ z / z. Thus, z + z / / / z + z + +/ / z + z. z z Suppose, on the other hand, that s an even number. It s cear that z + z z / z + /+ z + z. / Let,then /+ z / z. Therefore, z + z / / / z + z + /+ / z + z / z + z / z + z + / z. Ths concudes the proof. Theorem 7 Let τ ω be the mas,then the coeffcents of the mas are symmetrc.

24 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 4 of 9 Proof Snce sn ω 4 eω +e ω, cos ω + 4 eω +e ω, we set z e ω and obtan τ ω m + ω sn + α m+ ω cos + α / m+ + α m + m+ 4 e ω + e ω + α + e ω + e ω / m+ + α + α 4 m + 4 z + z + α Let a m+ + z + z / m+ + α + α. 4 m+ / m+ + α, by usng Lemma 9,onecanobtan τ ω m + 4 z + z + α m+ a m+ + z + z / m+ + α + α 4 m+ a b, z + z b, z + z, where {b, } m+ s n Lemma 9.Letc, τ ω m+ a b, z + z m+ c, z + z m+ c, m+ d z + z. z + z a b, and d c,,then

25 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 5 of 9 We consder nto two cases. Suppose that m + s an even number. Appyng Lemma yeds m+ ω d z + z τ m+ / d + m+/ + d m+ / + m+/ + z + + z + + z + z m+/ + d d + m+ h z + z + h z, z + + z + z + z m+/ + d d where h m+/ d, h / {,,...,m+ /} / {,,...,m+/} + d + d, ese., f s an odd number, 7 Suppose, on the other hand, that m + s an odd number. We have m+ ω d z + z τ m+ / d + + z + + z + m+ / + d m+ / + + m+ / z + z m+ / + d + d m+ h z + z + h z, z + + z + z + z m+ / + d d

26 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 6 of 9 where h m+ / h d, / {,,...,m+ /} / {,,...,m+ /} Ths estabshes the proof. + d +, f s an odd number, d, ese. 8 In the end, we provde two exampes of symmetrc refnabe functons from the mas τ ωform 8,,α.andτ ωform 6,,α.. Exampe Consder the refnabe functon wth the mas τ ω, where,m 8, α.,.e. τ 8,,. ω ω sn + h + h ± e ±ω. ω cos 9 In the foowng, Tabe shows the symmetrc coeffcents {h ± } / of the mar τ ω and Fgure gves the correspondng refnabe functon when,m 8,α.. The graph gves the symmetry and compacty supported refnabe functon correspondng to the mas τ 8,,. ω, whch has approxmaton order 6 and decay rate Exampe Consder the refnabe functon wth the mas τ ω, where m 6,, and α.,.e. τ 6,,. ω 9 ω sn + h + 9 h ± e ±ω. 8 ω cos / In the foowng, Tabe shows the symmetrc coeffcents {h ± } 9 of the mar τ ω and Fgure gves the correspondng refnabe functon when m 6,,andα.. Tabe Symmetrc coeffcents of the mas τ 8,,. ω h h

27 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 7 of 9 Fgure The refnabe functon wth τ 8,,. ω. Tabe Symmetrc coeffcents of the mas τ 6,,. ω h h The graph gves the symmetry and compacty supported refnabe functon correspondng to the mar τ 6,,. ω, whch has approxmaton order 8 and decay rate Resuts and dscusson In ths paper, a further famy of refnabe functons s constructed, whch ncudes pseudospnes of Type II, based on refnement mass, whch are generazed Bernsten poynomas. They possess besde a compact support aso a certan Höder reguarty Tabe, ths mpes a sutabe decay n Fourer doman and the approxmaton order n the correspondng shft-nvarant space. The nteger shfts of the refnabe functons form a Resz bass of ther span and the correspondng unvarate waveets can be constructed n the usua manner.

28 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 8 of 9 Fgure The refnabe functon wth τ 6,,. ω. Concusons We study new mass τ ω: m + ω sn + α / m+ + α, m+ cos ω + α whch ncude amost a mass of pseudo-spnes of Type II [4]whenα, to provde derved propertes. We obtan the convergence of cascade agorthms n Theorem, whch guarantees the exstence of refnabe functons. In Theorem, Resz waveets whose daton and transaton form a Resz bass for L R are constructed. Theorem anayzes the stabty of the subdvson schemes. Reguarty and the nfuences of parameters m,, α on the decay rate are showed n Secton 6. Secton 7 gves the approxmaton order of the new refnabe functons. Fnay, the symmetry of the refnabe functons, whch s of mportance, s ustrated n Theorem 7. Competng nterests The authors decare that they have no competng nterests. Authors contrbutons A authors contrbuted equay to ths artce. They read and approved the fna manuscrpt. Acnowedgements Ths research s supported by the Natona Natura Scence Foundaton of Chna No. 67 and Beng Natura Scence Foundaton No Receved: 5 November 5 Accepted: 8 June 6 References. Lawton, W, Lee, SL, Shen, Z: Characterzaton of compacty supported refnabe spnes. Adv. Comput. Math., Daubeches, I, Han, B, Ron, A, Shen, ZW: Frameets: MRA-based constructons of waveet frames. App. Comput. Harmon. Ana. 4, -46. Seesnc, I: Smooth waveet tght frames wth zero moments. App. Comput. Harmon. Ana., Dong, B, Shen, ZW: Pseudo-spnes, waveets and frameets. App. Comput. Harmon. Ana., Dong, B, Shen, ZW: Lnear ndependence of pseudo-spnes. Proc. Am. Math. Soc. 4, Dyn, N, Hormann, K, Sabn, MA, Shen, Z: Poynoma reproducton by symmetrc subdvson schemes. J. Approx. Theory 55, 8-4 8

29 Chengand YangJourna of Inequates and Appcatons 6 6:66 Page 9 of 9 7. Dong, B, Dyn, N, Hormann, K: Propertes of dua pseudo-spnes. App. Comput. Harmon. Ana. 9, 4-8. L, S, Shen, Y: Pseudo box spnes. App. Comput. Harmon. Ana. 6, Km, HO, Km, RY, Ku, JS: On asymptotc behavor of Batte-Lemare scang functons and waveets. App. Math. Lett., La, MJ: On the computaton of Batte-Lemare s waveets. Math. Comput. 6, Jeng, FL, Jyh, CJ, Chy, H: Computaton of Batte-Lemare waveets usng an FFT-based agorthm. J. Sc. Comput., Fan, A, Sun, Q: Reguarty of Butterworth refnabe functon. Asan J. Math. 5, Kam,H,Km,RY:SoboevexponentsofButterworthrefnabefunctons.App.Math.Lett., Km, HO, Km, RY, Km, SS: Pseudo-Butterworth refnabe functons. Curr. Dev. Theory App. Waveets 4, Luo, ZX, Q, WF, Zhou, MM, Zhang, JL: Generazed pseudo-butterworth refnabe functons. J. Fourer Ana. App. 9, Boyer, RP, The, LC: Generazed Bernsten poynomas and symmetrc functons. Adv. App. Math. 8, Shen, Y, L, S: Cascade agorthm assocated wth Höder contnuous mass. App. Math. Lett., Han, B: Refnabe functons and cascade agorthms n weghted spaces wth Höder contnuous mass. SIAM J. Math. Ana. 4, Ja, RQ, Wang, JZ: Stabty and near ndependence assocated wth waveet decompostons. Proc. Am. Math. Soc. 7, Daubeches, I: Ten Lectures on Waveets. CBMS-NSF Regona Conference Seres n Apped Mathematcs, vo. 6. SIAM, Phadepha 99. Jang, Q, Shen, ZW: On exstence and wea stabty of matrx refnabe functons. Constr. Approx. 5,