Orthonormal sampling functions

Transcription

1 Appl. Comput. Hrmon. Anl. (006) Letter to the Editor Orthonorml smpling functions N. Kiblinger,,, W.R. Mdych b Fculty of Mthemtics, University of Vienn, Nordbergstrße 5, 090 Vienn, Austri b Deprtment of Mthemtics, University of Connecticut, Storrs, CT , USA Avilble online 7 April 006 Communicted by Chrles K. Chui on 5 November 005 Abstrct We investigte functions φ(x) whose trnsltes {φ(x k)}, wherek runs through the integer lttice Z, provide system of orthonorml smpling functions. The crdinl sine, whose importnt role in the smpling theory of bndlimited functions is well documented, is the clssic exmple. For the bndlimited cse we provide complete chrcteriztion of such functions φ nd give exmples with rpid decy including construction which is symmetric. We lso nlyze the generl cse of rbitrry smpling rte, >0, which leds to some unexpected observtions. 006 Elsevier Inc. All rights reserved. Keywords: Shift-orthonorml; Smpling function; Interpoltion; Crdinl sine; Shift-invrint spce; Wvelets; Multiresolution nlysis; Scling function. Introduction We introduce the relevnt notions for our study by strting from bsic model for encoding discrete-time dt into continuous-time functions. It is stndrd technique in the frmework of shift-invrint systems. Given genertor function φ in the spce L (R) of squre-integrble functions, the dt of complex numbers c k is trnsformed into function f on R by the synthesis mpping (c k ) f(x)= c k φ(x k), x R. It is well known if φ is the crdinl sine, sin πx sinc(x) = πx, x R, then the encoding hs the following useful fetures: * Corresponding uthor. E-mil ddresses: norbert.kiblinger@univie.c.t (N. Kiblinger), mdych@uconn.edu (W.R. Mdych). The first uthor ws supported by the Austrin Science Fund FWF grnt P /$ see front mtter 006 Elsevier Inc. All rights reserved. doi:0.06/j.ch

2 N. Kiblinger, W.R. Mdych / Appl. Comput. Hrmon. Anl. (006) (i) The synthesis mpping is n isometry from l (Z) into L (R). (ii) The dt cn be reconstructed in convenient wy by smpling f ( f(k) ). (iii) The rnge of the synthesis mpping consists of bndlimited functions, i.e., functions with compctly supported Fourier trnsform. These properties correspond to the following conditions on the genertor φ L (R). Byδ k,0 we denote the Kronecker delt. (i) φ is shift-orthonorml, i.e., φ(x)φ(x k)dx = δ k,0, k Z. (ii) φ is smpling function (fundmentl function of interpoltion, crdinl function), i.e., it is continuous nd φ(k)= δ k,0, k Z. (iii) φ is bndlimited. The crdinl sine is in mny wys the prototypicl genertor. However, the poor decy properties of the sinc re often disdvntgeous. One of our objectives, inspired by query posed by H.G. Feichtinger, is to obtin genertors with better decy, thus providing functions tht my replce the sinc-function in vrious pplictions. It turns out tht constrints on the bndwidth impose limittions to such properties s rpid decy or symmetry. These limittions together with wht is possible re summrized in Section s Theorem. Section lso contins chrcteriztion of clss of bndlimited shift-orthonorml smpling functions together with exmples tht indicte wht is possible regrding decy. The cse of more generl smpling rtes, i.e., replcing the integer lttice Z by scled lttice Z with some positive number other thn one, is usully delt with by simply scling the genertor; but not without introducing n dditionl normlizing fctor. Thus the existence nd nture of genertors φ which re both Z-smpling nd Z-shift-orthonorml with ny given different from one is not immeditely trnsprent. See the introductory prgrphs of Section 3. Nevertheless we cn give complete description of the sitution in this generl cse, see Theorem in Section 3. Section 4 is devoted to the proof of this theorem, which requires severl technicl lemms. The conditions (i), (ii), (iii) on function φ rise nturlly in wvelet nd smpling theory, for exmple see [5,6] for reltively recently published texts on these subjects which lso contin further references. Indeed, genertors of shift-invrint systems, wvelets, nd scling functions with combintions of these fetures hve been nlyzed to some extent in the literture, see, e.g., [ 4,7 0]. Some of our observtions re direct consequences of estblished mchinery while others my hve been noted erlier. In the text we indicte which observtions, to our knowledge, hve been recorded by other uthors nd provide explicit references.. Shift-orthonorml smpling functions It is useful to study the notions introduced bove in the Fourier domin. We use the following normliztion of the Fourier trnsform, for integrble φ, ˆφ(ξ) = φ(x)e πiξx dx, ξ R. As is well known, for exmple see [5, Lemm 7.5], function φ L (R) is shift-orthonorml if nd only if ˆφ(ξ k) = for.e. ξ R. ()

3 406 N. Kiblinger, W.R. Mdych / Appl. Comput. Hrmon. Anl. (006) A function φ L (R) with integrble Fourier trnsform is smpling function if nd only if ˆφ(ξ k) = for.e. ξ R. () We note tht the ssumption tht ˆφ be integrble is lwys stisfied for bndlimited φ L (R). Our first min result is the following clssifiction, including the exmples below, for bndlimited shift-orthonorml smpling functions. A continuous function φ is sid to hve rpid decy if lim x x N φ(x)= 0 for ll N =,,... Theorem. Suppose φ L (R) is shift-orthonorml smpling function such tht [ supp ˆφ Ω, Ω ], Ω >0. (i) Ω<. No such function exists. (ii) Ω =. The sinc-function is the unique exmple. (iii) <Ω 3. There exist rel-vlued exmples with rpid decy. If φ is integrble it cnnot be symmetric. (iv) Ω>3. There exist symmetric rel-vlued exmples with rpid decy. We note tht Theorem cn be viewed s the specil cse = of Theorem, which is formulted in Section 3 nd proved in Section 4. The detils of the generl result re rther involved while those for the specil cse re quite trnsprent s indicted below. Proof. Items (i) nd (ii) re direct consequences of chrcteriztions () nd (). Note tht we use the Fourier trnsform normlized in such wy tht sinc = [ /,/]. (iii) The existence is verified by Exmple. The impliction in the second sttement is shown s follows. Suppose there exists such function φ which is symmetric. Since φ is integrble, ˆφ is continuous nd we conclude tht ˆφ ( ± 3 ) = 0. Thus, by (), () the complex numbers λ = ˆφ ( ) nd λ = ˆφ ( ) stisfy λ + λ = nd λ + λ =. Next, φ is symmetric, hence so is ˆφ nd, therefore, λ := λ = λ. Hence, λ = λ =, which is impossible. Contrdiction. (iv) The existence is verified by Exmple. Bndlimited shift-orthonorml smpling functions with bndwidth constrint supp ˆφ [, ] re completely chrcterized in terms of their Fourier trnsform by the following lemm. The result is reformultion of [6, Lemm 0.5]. Lemm. Let 0 ε / nd suppose tht φ L (R) nd supp ˆφ [ ε, + ε]. (i) Then φ is smpling function if nd only if, ξ ε, ˆφ(ξ) = ± u( ξ ± ), ξ ± <ε, 0, ξ + ε, for some squre-integrble function u : [ ε, ε] C. (ii) A smpling function φ so chrcterized is in ddition shift-orthonorml if nd only if u stisfies u(ξ) =, for ll ξ [ ε, ε]. Proof. Both (i) nd (ii) follow from the fct tht ˆφ enjoys reltions () nd (). For exmple, if φ is smpling function then () implies ˆφ ( r + ) + ˆφ ( r ) = for.e. 0 <r<ε. We conclude tht ˆφ ( r ± ) = ± u for some u = u(r) C. If in ddition φ is shift-orthonorml, then by () there lso holds ˆφ ( r + ) ( + ˆφ r ) =. Thus + u + u = which is equivlent to u =.

4 N. Kiblinger, W.R. Mdych / Appl. Comput. Hrmon. Anl. (006) We stte two cses where Lemm yields n explicit form of φ. First,foru(ξ) = ξ/ε ± i (ξ/ε), we obtin the integrble rel-vlued shift-orthonorml smpling functions ( sin πεx φ(x)= ± π ) sin πx πεx J (πεx), x R, (3) πx where J denotes the Bessel function of the first kind of order one. The decy is φ(x)= O( x 3/ ),s x. Using u(ξ) = sin(πξ/ε) ± i cos(πξ/ε) yields the integrble rel-vlued shift-orthonorml smpling function cos πεx sin πx φ(x)= = π ( 4εx πx sinc εx ) sinc(x), x R. (4) Its decy is better thn the previous exmple, φ(x) = O( x ),s x. This exmple ws first obtined in the wvelet context [4, Exmple.3], lso discussed in [6, pp. 5, 38] nd [7,0]. In fct, for ε 6, (3) nd (4) re scling functions, i.e., they stisfy the dydic refinement eqution of wvelet theory: ( ) x φ = c k φ(x k), x R, with [ c k = φ(k/). Indeed we note tht Lemm implies, if shift-orthonorml smpling function φ stisfies supp ˆφ 3, ] 3, then φ is refinble. With suitble u we cn improve the smoothness of ˆφ nd thus the decy properties of φ. To this end, observe tht u cn lwys be written ( ) ( ) π π u(ξ) = e πiνε(ξ)/ = sin ν ε(ξ) + i cos ν ε(ξ), ξ [ ε, ε], where ν ε (ξ) = ν(ξ/ε), for some mesurble function ν : [, ] R. By clever choice of ν we cn hve tht ˆφ is C, so tht φ is of rpid decy. The next lemm describes the function ν tht we will use in Exmple. Lemm. Define ν : [, ] [, ] by { ( ( tnh tn π ξ )), <ξ<, ν(ξ) = ±, ξ =±. Then ν is C on [, ] nd ν (k) (±) = 0 for k =,,... Proof. Note tht ll derivtives of tnh(ξ) vnish t infinity s cn be seen by writing tnh(ξ) = ( exp( ξ))/( + exp( ξ)) nd using L Hospitl s rule. Thus the derivtives of ν t ξ =± re obtined by elementry clculus, in the sme wy s one shows tht ll the derivtives of exp( / ξ ) re zero t zero. By mking use of Lemm we construct bndlimited shift-orthonorml smpling function φ tht hs rpid decy. Exmple. Given 0 <ε, define φ L (R) by, ξ ε, ˆφ(ξ) = ± sin( ( )) π ν ε ξ ± ± i cos ( ( )) π ν ε ξ ±, ξ ± <ε, 0, ξ + ε, where ν(ξ) = tnh ( tn ( π ξ )) nd ν ε (ξ) = ν(ξ/ε). Then φ is rel-vlued shift-orthonorml smpling function with rpid decy such tht supp ˆφ = [ ε, + ε]. While the construction of this exmple does not give rise to n elementry formul in time domin, the function φ cn be redily implemented by the given formul in the Fourier domin. The specific cse of φ in Exmple with ε = 0. is illustrted in Fig.. Note the lck of symmetry in this function.

5 408 N. Kiblinger, W.R. Mdych / Appl. Comput. Hrmon. Anl. (006) Fig.. A bndlimited shift-orthonorml smpling function with rpid decy (Exmple with ε = 0.). Fig.. A symmetric bndlimited shift-orthonorml smpling function with rpid decy (Exmple with ε = 0.). It cn be deduced from Lemm tht shift-orthonorml smpling function φ with supp ˆφ [, ] cnnot be symmetric when ˆφ is continuous, indeed, item (iii) of Theorem implies tht this is true for bndwidths s lrge s 3. However, in (iv) we stte tht symmetric exmples with rpid decy do exist if the bndwidth is llowed to increse further. The next exmple verifies this observtion. Exmple. Given 0 <ε, define φ L (R) by, ξ ε, cos( ( )) αν ε ξ ± ± 3 sin( ( )) αν ε ξ ±, ξ ± <ε, ˆφ(ξ) = 0, ξ ± ε, cos( ( )) αν ε ξ ± 3, ξ ± 3 <ε, 0, ξ 3 + ε, where α = rccos 3, ν(ξ) = tnh ( tn ( π ξ )) nd ν ε (ξ) = ν ( ξ ε ). Then φ is symmetric rel-vlued shift-orthonorml smpling function with rpid decy such tht supp ˆφ [ 3 ε, 3 + ε]. This exmple is one of the min results of our nlysis. By Theorem (iii) it cnnot be improved in terms of smller bndwidth. The specific cse with ε = 0. is illustrted in Fig.. Remrk. A shift-orthonorml smpling function φ which enjoys the property tht ˆφ(ξ) = in neighborhood of the origin lso stisfies ˆφ(ξ) = 0 in neighborhood of the nonzero integers. If φ decys rpidly these properties imply tht φ stisfies severl fetures tht re importnt in pproximtion theory. For exmple, φ reproduces lgebric polynomils of ny order, tht is, for ll N = 0,,,..., x N = k N φ(x k), x R. (5) In prticulr, both Exmples nd stisfy this property. 3. Generl shift prmeters Let >0 nd φ L (R).

6 N. Kiblinger, W.R. Mdych / Appl. Comput. Hrmon. Anl. (006) (i) φ is Z-shift-orthonorml if φ(x)φ(x k)dx = δ k,0, k Z. (ii) φ is n Z-smpling function if it is continuous nd φ(k)= δ k,0, k Z. As is well known, ech of these notions cn be relted to the corresponding specil cse = by diltion: φ(x) is shift-orthonorml / φ(x/) is Z-shift-orthonorml, φ(x) is smpling function φ(x/) is n Z-smpling function. Note tht the bove diltions do not led directly to functions φ which re both Z-shift-orthonorml nd Z-smpling functions when. While the fctor of / my seem like minor inconvenience it does rise the interesting question of whether this fctor is indeed necessry, tht is, whether there exist functions φ which re both shift-orthonorml nd smpling functions when. Surprisingly the nswer to this question is yes, see Theorem below, but the development is not quite s slick s in the specil cse =, see Section 4. Our interest in this question ws prompted by the rticle [9]. We stte the chrcteriztions in the Fourier domin for the cse of generl >0. A function φ L (R) is Z-shift-orthonorml if nd only if ( ˆφ ξ k ) = for.e. ξ R. (6) A function φ L (R) with integrble Fourier trnsform is n Z-smpling function if nd only if ( ˆφ ξ k ) = for.e. ξ R. We formulte our min result for this generl setting of rbitrry shift-prmeters >0, clssifiction of bndlimited Z-shift-orthonorml Z-smpling functions in terms of the bndwidth. By we denote the lest integer greter thn or equl to. (7) Theorem. Given >0,let { for <, N = for, { 3 for <, nd M = for. Suppose φ L (R) is n Z-shift-orthonorml Z-smpling function such tht [ supp ˆφ Ω, Ω ], Ω >0. (i) Ω<N/. No such function exists. (ii) Ω = N/. There exist symmetric rel-vlued exmples. The function is unique if nd only if is n integer, where φ = sinc. There do not exist integrble exmples. (iii) N/ <Ω M/. There exist rel-vlued exmples with rpid decy. If φ is integrble it cnnot be symmetric. (iv) Ω>M/. There exist symmetric rel-vlued exmples with rpid decy. Remrk. The cse (iii) is void for, since in this cse M nd N coincide. We illustrte (ii) by constructing n exmple, for generl >0.

7 40 N. Kiblinger, W.R. Mdych / Appl. Comput. Hrmon. Anl. (006) Exmple 3. Given >0, let n = mx(, ), nd define φ L (R) by n + n ( ) n n, ξ n, ˆφ (ξ) = n n n ( ) n, n < ξ n, 0, ξ > n. Then φ is symmetric rel-vlued Z-shift-orthonorml Z-smpling function such tht supp ˆφ [ N, N ] with N s in Theorem (ii). Indeed, for we hve supp ˆφ = [ n, n ], while = yields supp ˆφ = [, ]. We note, when is n integer, then φ = sinc. We lso mention tht φ sinc in L (R), s. 4. Preliminry lemms nd proof of Theorem We provide the lemms used below for proving Theorem. The first is result tht llows us to nlyze bndlimited Z-shift-orthonorml Z-smpling functions by investigting clsses of vectors in C n.given>0 nd n N, we define Sn Cn by } n n Sn {λ = C n : λ k = λ k =. k= k= Lemm 3. Let >0. Given bndlimited function φ L (R), with supp ˆφ [ n ε, n ] + ε, n N, 0 ε<, we define v ξ C n nd w ξ C n+,forξ R,by ( ( v ξ = ˆφ n ) ( + ξ, ˆφ n 3 ) ( )) n + ξ,..., ˆφ + ξ nd ( w ξ = ˆφ ( n ) ( + ξ, ˆφ n ) ( )) n + ξ,..., ˆφ + ξ, respectively. Then φ is n Z-shift-orthonorml Z-smpling function if nd only if [ v ξ Sn for.e. ξ + ε, ] ε nd w ξ Sn+ for.e. ξ ( ε, ε). The equivlence follows from (6) nd (7). We omit the detils. In the sequel we nlyze properties of the set S n. Lemm 4. Given >0 nd n =,,..., the following hold: (i) Let <n.ifn =, then Sn =.Ifn =, 3,..., then S n is the (n )-sphere in Cn with center p = ( n,..., n) C n nd rdius r = ( n), oriented long the hyperplne through p tht is orthogonl to the position vector of p. (ii) For = n, we hve Sn ={(,...,) Cn }. (iii) For >n, we hve Sn =. Proof. (i), (ii), (iii) Cse I: n. The eqution n k= λ k = defines the (n )-hyperplne in C n through p nd orthogonl to the position vector of p. The eqution n k= λ k = defines the (n )-sphere of rdius R = round the origin in C n.thesetsn is thus the intersection of the hyperplne with this sphere, it yields the (n )- sphere described in the lemm. The rdius r is obtined by r + p = R. Finlly, we identify the cses = n, where the sphere degenertes to single point, nd >n, where the intersection is empty.

8 N. Kiblinger, W.R. Mdych / Appl. Comput. Hrmon. Anl. (006) Cse II: n =. While S ={}, we hve for ny other tht S =. For esy referencing, we summrize when Sn is non-empty. Lemm 5. (i) Let <. Then Sn if nd only if n. (ii) Let. Then Sn if nd only if n. We will need to know when S n contins symmetric vectors. A vector v Cn is clled symmetric if (v,v,...,v n ) = (v n,v n,...,v ). Lemm 6. (i) Let <,. Then Sn contins symmetric vector if nd only if n 3. (ii) Let =. Then Sn contins symmetric vector if nd only if n = or n 3. (iii) Let. Then Sn contins symmetric vector if nd only if n. Proof. (i) Cse I: n =. Then Sn = by Lemm 5. Cse II: n =. Suppose there exists symmetric vector v = (λ, λ) S. Then λ S/. However, since, we hve by Lemm 5 tht S / =. Contrdiction. Cse III: n 3. Then Sn contins, for exmple, the symmetric vector v = (λ,μ,...,μ,λ)defined by λ = ( n n ± ) nd μ = ( n n n ). n n n (ii) For n =, we hve S ={}. For n =,see(i),cseii. For n 3, see (i), Cse III. (iii) For n<, Sn = by Lemm 5. For n =,wehvesn ={(,...,)} by Lemm 4. For n>, see (i), Cse III. Lemm 7. Let >0 nd n =,,..., be such tht Sn. Then S n contins Hermiten vector, i.e., vector v of the form (v,v,...,v n ) = ( v n, v n,..., v ). Proof. We only need to consider the cse n = nd <, since for ll other cses we find by inspecting the proof of Lemm 6 tht Sn indeed contins rel-vlued symmetric vectors, such re Hermiten in prticulr. For n = nd <, let s = ( ) (. Then not just the rel-vlued vector + s, s) belongs to S but lso the complex vector v = ( + is, is), which is Hermiten. Proof of Theorem. First, note tht the numbers N nd M re defined in such wy tht N = min { n N: Sn } nd (8) M = min { n N: both Sn nd S n+ contin symmetric vectors} ; (9) s cn be verified by Lemm 5 nd Lemm 6, respectively. (i) Suppose such φ exists. By Lemm 3 there exist vectors v = v ξ for.e. ξ [ + ε, ε], tht belong to SN. Since ˆφ is supported on subset of [ N,N ] of smller mesure, we hve tht t lest one (in fct, mny) v ξ hs zero component. This implies tht the vector of length N obtined by removing the zero entry of v ξ belongs to SN. However, S N is empty by (8). Contrdiction. (ii) A. Existence: By (8) SN is non-empty. Therefore, Lemm 3 with ε = 0 provides vlid construction of such function φ. B. Uniqueness: In view of item A bove, uniqueness holds if nd only if SN consists of single vector. By Lemm 4 this is the cse if nd only if = N is n integer.

9 4 N. Kiblinger, W.R. Mdych / Appl. Comput. Hrmon. Anl. (006) C. Non-integrbility: Cse I: If =, then N = nd the sttement follows since the only smpling function φ L (R) with supp ˆφ [, ] is the crdinl sine (Theorem (ii)). Cse II: If, then we hve N. Suppose there exists such function φ with continuous ˆφ. By Lemm 3 the vector ( ( v := ˆφ N ) (, ˆφ N ) ( )) N,..., ˆφ C N belongs to SN+. The continuity of ˆφ nd the constrint on its support imply ˆφ ( ± N ) = 0. Tht is, the first nd lst entries of v vnish. Thus we conclude tht the vector of length N tht is left when deleting these two entries belongs to SN. However, by (8) S N is empty. Contrdiction. (iii) A. Existence: Since SN, there exists v S N. We note tht the ugmented vectors w = (v, 0) C N+ nd w = (0,v) C N+ belong to SN+. According to Lemm 4 S N+ is lwys connected set, hence we find continuous mpping w : [ ε, ε] SN+ such tht w( ε) = w nd w(ε) = w. Now define φ with supp ˆφ [ N ε, N + ε] by ( ˆφ ( N ) ( + ξ, ˆφ N ) ( )) N + ξ,..., ˆφ + ξ = w(ξ) for ξ ( ε, ε) nd ( ( ˆφ N ) ( + ξ, ˆφ N 3 ) ( )) [ N + ξ,..., ˆφ + ξ v for ξ + ε, ] ε. Then by Lemm 3 we hve tht φ is Z-shift-orthonorml nd Z-smpling. By Lemm 7 we cn ssume tht v is Hermiten vector see Lemm 7 for this notion nd construct w in such wy tht w( ξ)= w(ξ). From this property we obtin tht ˆφ stisfies ˆφ( ξ)= ˆφ(ξ), ξ R, nd hence φ is rel-vlued. Without further detils, we finlly note tht by choosing w ppropritely we lso obtin tht ˆφ is not just continuous but indeed C function. B. Non-symmetry if integrble: Suppose there exists such function φ which is symmetric. Cse I: If >, then M = nd we hve tht ˆφ cnnot be continuous. Contrdiction. Cse II: If <, then M = 3. Therefore, since ˆφ is continuous we hve ˆφ ( ± 3 ) = 0. Hence, since φ is symmetric we hve tht ˆφ is symmetric nd we obtin λ := ˆφ ( ) = ˆφ ( ). Then from Lemm 3 we obtin tht (λ, λ) S, i.e., S contins symmetric vector, in contrdiction to Lemm 6(ii). Cse III: If =, then M =. Therefore, since ˆφ is continuous we hve ˆφ ( ± ) = 0. Hence, λ := ˆφ(0) stisfies λ S or λ = λ =, which is impossible. Contrdiction. (iv) A. Existence: Such function is constructed s described bove with the following dditionl properties. First, the vector v is chosen to be symmetric. Second, the function w is chosen to be symmetric function. This is possible by letting w(0) = w 0, where w 0 is symmetric vector in SN. Note tht the proof of (iii)-a provides construction. References [] T.N.T. Goodmn, C.A. Micchelli, Orthonorml crdinl functions, in: C.K. Chui, L.B. Montefusco, L. Puccio (Eds.), Wvelets: Theory, Algorithms, nd Applictions, Acdemic Press, Sn Diego, 994, pp [] G. Gripenberg, On subdivision interpoltion schemes, SIAM J. Mth. Anl. 7 () (996) [3] R.M. Lewis, Crdinl interpolting multiresolutions, J. Approx. Theory 76 () (994) [4] Y. Liu, G.G. Wlter, A clss of bnd-limited crdinl wvelets, Adv. Mth. (Chin) 6 (6) (997) [5] D.F. Wlnut, An Introduction to Wvelet Anlysis, Birkhäuser, Boston, MA, 00. [6] G.G. Wlter, X. Shen, Wvelets nd Other Orthogonl Systems, second ed., Chpmn & Hll/CRC, Boc Rton, FL, 00. [7] G.G. Wlter, J. Zhng, Orthonorml wvelets with simple closed-form expressions, IEEE Trns. Signl Process. 46 (8) (998) [8] X.-G. Xi, Z. Zhng, On smpling theorem, wvelets, nd wvelet trnsforms, IEEE Trns. Signl Process. 4 () (993) [9] A.I. Zyed, B-splines nd orthonorml sets in Pley Wiener spce, in: W.O. Bry, Č.V. Stnojević (Eds.), Anlysis of Divergence, Birkhäuser, Boston, MA, 999, pp [0] A.I. Zyed, G.G. Wlter, Wvelets in closed forms, in: L. Debnth (Ed.), Wvelet Trnsforms nd Time Frequency Signl Anlysis, Birkhäuser, Boston, MA, 00, pp. 43.