Available at ISSN: Vol. 4, Issue 2 (December 2009) pp (Previously Vol. 4, No.

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1 Avilble t ISSN: Vol. 4, Issue December 009 pp Previously Vol. 4, No. Applictions nd Applied Mthemtics: An Interntionl Journl AAM On -ry Subdivision for Curve Design III. m-point nd m + -Point Interpoltory Schemes Jin-o Lin Deprtment of Mthemtics Pririe View A&M University Pririe View, TX USA JiLin@pvmu.edu Received: September 30, 009; Accepted: Jnury 7, 00 Abstrct In this pper, we investigte both the m-point -ry for ny nd m + -point -ry for ny odd 3 interpoltory subdivision schemes for curve design. These schemes include the extended fmily of the clssicl 4- nd 6-point interpoltory -ry schemes nd the fmily of the 3- nd 5-point -ry interpoltory schemes, both hving been estblished in our previous ppers Lin [9 nd Lin [0. Keywords: Subdivision; Curve design; Sttionry; Refinble functions; -ry MSC 000 #: 4H50; 7A4; 65D7; 68U07. Introduction THIS is continution of both Lin [9 nd Lin [0. The former ws for extending the clssicl 4- nd 6-point binry interpoltory subdivision schemes for curve design in Dyn, et l. [5 nd Weissmn [3 to -ry interpoltory schemes for ny 3, nd the ltter ws for the 3- nd 5-point -ry interpoltory schemes for ny odd

2 AAM: Intern. J., Vol. 4, Issue December 009, [Previously Vol. 4, No. 435 Recll tht function φ L R is sid to be refinble with diltion fctor Z +, which is, if φ stisfies the two-scle eqution φx Z p φx, with p } Z l R being φ s two-scle sequence. The simplest setting of is. cf., e.g., Chui [ nd Dubechies [4. A refinble φ is clled scling function, if, in ddition, the fmily φ : Z} genertes Riesz bsis of V φ 0 L R, mening there re constnts 0 < A B < such tht A n n φ n B n, n } n Z l R, n Z n Z where V φ 0 is the L -closure of ll liner combintions of the integer trnsltes of φ, nmely, n Z V φ 0 Clos L spn φ : Z}. By ting the Fourier trnsforms of both sides, φω P e iω/ ω φ, 3 P z p z, 4 Z where P is clled the two-scle symbol of φ. The two-scle sequence p } Z is normlly finitely supported, mening it hs finitely mny nonzero entries. Such scling function is sid to hve polynomil preservtion of order d +, denoted by φ PP d, if π d spn V φ 0, 5 where π d is the collection of ll polynomils of degree d. It is well-nown tht 5 is equivlent to the two-scle symbol P in 4 with the fctor + z + + z d+. A subdivision scheme corresponding to such scling function is given by λ n+ +l j p j+l λ n j, l 0,..., ; n Z +, 6 where λ 0, Z, re the initil control points. When, it is clled binry scheme, nd when 3, it is clled ternry scheme. For the generic, it is simply clled n -ry scheme. The min objective of this note is to extend both results in Lin [9 nd Lin [0 to m-point -ry for ny nd m + -point -ry for ny odd 3 interpoltory subdivision schemes, respectively. Our results re stted in Section s Theorem nd Theorem, with their proofs given in Section 3. Another short wy of proving Theorems & is given in Section 4. Two explicit exmples, s direct consequences of our elegnt formultions, re demonstrted in Section 5.. Min Results

3 436 Jin-o Lin Let φ m nd φ m+ be the scling functions with diltion fctor, which correspond the mnd m +-point interpoltory subdivision schemes for curve design. The smoothness of φ m nd φ m+ for ll diltion fctors is importnt for the convergence of their corresponding interpoltory subdivision schemes. We do not go to detils here. However, for some previous studies of scling functions their smoothness with different diltion fctors 3, the reders re referred to, e.g., Chui & Lin [3, Hn [6, Belogy & Wng [, Shui, Bo, & Zhng [, Peng & Wng [. For φ m, we hve the following: Theorem. The scling function φ m PP m with the smllest support, is determined by the two-scle symbol P m of the form where P m z mp 0 for > m, nd mp mp Z z m m p z m l! m + l! m z m S m z, 7 z m l ξ m+l ξ + η, l, l +,..., l + ; l 0,..., m, 8 nd S m is polynomil of exct degree m, stisfying S m nd S m z z m S m /z. η It then follows from 6 nd 8 tht the m-point -ry interpoltory subdivision scheme is λ n+ λ n+ +l λ n, m mp j+l λ n j m +j, l,..., ; n Z +, 9 for ny diltion fctor. For φ m+ PP m+, we hve the following. Theorem. The scling function φ m+ PP m+ with the smllest support, is determined from the two-scle symbol P m+ of the form P m+ z Z m+p z z m / z m+ S mz, 0 z

4 AAM: Intern. J., Vol. 4, Issue December 009, [Previously Vol. 4, No. 437 where Z + is odd, 3, m+ p m+ p 0 for > m + /, nd m+ p m l! m + l! m 0,..., /, when l 0; m l m+l ξ + η, l /,..., l + /, when l,..., m, nd S m is polynomil of exct degree m, stisfying S m nd S m z z m S m /z. ξ η Correspondingly, it follows from 6 nd tht the m + -point -ry interpoltory subdivision scheme is λ n+ λ n+ +l λ n m, j m m+ p j+l λ n +j, l,..., /; n Z Proofs of Min Results Proof of Theorem. Anlogous to the proof in Lin [9, n -ry m-point scheme needs t most m weights, i.e., the two-scle sequence mp j } Z of φ m hs t most m consecutive nontrivil entries. For PP m, the two-scle symbol P m must hve the form P m z z m z m S m z z for some polynomil S m π m of exct degree m stisfying S m nd S m z z m S m /z. It suffices to show tht S m is uniquely determined under the interpoltory condition P m w l z, z, 3 l0 where w l } l0 re the distinct roots of z, nmely, lπi w l exp, l 0,...,. First, with S m z m s z, we define m g j z j S m z z s m z j0 l0 m + l z l, m

5 438 Jin-o Lin so tht g j j m + j s, j Z +. 4 j Second, by defining g j 0 for j < 0, [ P m z m z m z m S m z z m m m m z m z g l z l l Z m z m z m l g j +l z j l0 j Z Z m z m l g j +l z j m+. j Z l0 Z Third, the interpoltory condition 3 now leds to which is equivlent to j+m m m j Z m Z g j+m z j, g j+m m δ j,0, j m +,, m. 5 Fourth, solving the lower tringulr system 5, we hve m + j g j m, j,, m, 6 m nd the coefficients s 0,..., s m of S m cn now be evluted from 4 nd 6, nmely, m m + j m s m + j m, j,,m. 7 m Solving 7 for s 0,..., s m nd computing g j } j Z+ by 4, we hve g l l + m! m l +, l Z +. 8 Finlly, by using 7, we rrive t the vlues of mp in 8. This completes the proof of Theorem. Proof of Theorem.

6 AAM: Intern. J., Vol. 4, Issue December 009, [Previously Vol. 4, No. 439 Denote by P m+ the two-scle symbol of φ m+. Completely nlogous to the proof of Theorem, both P m+ PP m+ nd φ m+ hving symmetric two-scle sequence led to P m+ z z m / z m+ S mz z m+p } m+ / m / for to-be-determined polynomil S m of exct degree m, stisfying S m nd S m z z m S m /z, where 3 is positive odd integer. Agin, by writing S m z m s z, introducing g j z j S m z z m+, j0 g j j m + j s, j Z +. 9 j nd defining g j 0 for j < 0, we hve / P m+ z m z m + l g j +l+ / z j m. l / j Z Z The interpoltory property yields m + g j+m + / m δ j,0, j m,,m, which leds to Z Solving the liner system m m + j s m m + j g j+ / m, j 0,, m. 0 m m + j for s 0,..., s m nd computing g j } j Z+ for generl j by 9, we hve g l l m + l l s m m! m m m, j 0,,m, l +, l Z +. Finlly, m+p } m+ / m / is obtined s listed in. 4. Another Short Wy of Proving Theorem & Theorem The proofs of our two theorems in Section 3 were direct nd cler-cut. We show in this section nother short or mybe not so-short wy, by using the Tylor expnsions of the expression

7 440 Jin-o Lin m z. For, see Dubechies [4. For 3, see Chui & Lin [3. For generic z nd multiwvelet setting, see Lin [8. First, for subdivision scheme generted from scling function φ to be symmetric, the two-scle symbol P of φ hs to be reciprocl. For this purpose, Pz needs to be function of z + z. m z For this reson, the fctor, requirement for φ PP z m, hs to be function of z + z. This is equivlent to the positive integer m hs to be even. Hence, if m is even then the diltion fctor or smpling rte cn be ny positive integer. However, if m is odd, hs to be even. Tht is why for m + -point subdivision scheme, hs to be odd. To get redy for the new short proofs of the two theorems, we estblish the following: Lemm. For ny positive even integer, + z + + z [ z + + z z z + z For ny positive odd integer, + z + + z z / + [ z + / + z + 5/ z 3/ + z 3/+. 4 The proof of Lemm is strightforwrd. We re now in the position of the new proofs of Theorems &. Another Short Proof of Theorem. When the polynomil preservtion is of even order, we cn write P m z m p z z m n z m S nz, 5 z Z for some reciprocl polynomil S n. Then the interpoltory condition 3 yields [ m z n S n z z + z + + z m z w m n l S n w l z. 6 w l z l It follows from Lemm tht for either even or odd, the first term in 6 is function of z z m nd the lst terms in 6 hve the fctor. Hence, the minimum z z

8 AAM: Intern. J., Vol. 4, Issue December 009, [Previously Vol. 4, No. 44 degree of n is m, nd z m S m z is the m th order Tylor polynomil of m z m 7 + z + + z z in terms of the vrible t. More precisely, by using Lemm, when is even, z [ m m z m z + z + + z + z + + z + z z [ nd when is odd, m [ z m z / + z + + z + z + + z + z z [ + / + 5/ + z + z ++ m m, 8 z 3/ + z 3/+ m.9 In other words, the exct expressions of P m z re formed from either the Tylor polynomils of 8 for even or the Tylor polynomils of 9 for odd. This completes the short proof of Theorem. Another Short Proof of Theorem. When the polynomil preservtion is of odd order, the diltion fctor hs to be odd s well, nd we cn write P m+ z m+p z Z + z + + z z m+ / n m+ S nz, 30 for some reciprocl polynomil S n. Agin, the interpoltory condition 3 leds to [ z n z / m+ S n z + z + + z m z w m+/ n l S n w l z. 3 w l z l Therefore, n m, nd z m S m z is the m + st order Tylor polynomil of z / m+ 3 + z + + z

9 44 Jin-o Lin in terms of the vrible t For convenience, we fix z. This completes the short proof of Theorem. z t z z z z in the sequel, so tht the Tylor expnsions re in terms of t. For exmple, it is esy to see, by using t in 33, tht for, 3, 4, nd 5, the expressions we need for Tylor expnsions in t re z + z t ; z + z + z t ; z + z + z 3 z 3 5z + z + z + z 3 + z 4 t 6 0 8t + t 33 ; 36 t 5 5 t Demonstrtion by Two Explicit Exmples We demonstrte our elegnt formultions in this section by giving two explicit exmples s direct consequences of Theorems &. Exmple. 8-point binry interpoltory: m 4,. It follows from 8, 7, nd 8 tht g 0,,g 6 } 5 6, 0, 9 6 s 0,,s 6 } 5 6, 5, 3 6 8p 7, 8p 5,, 8p 7 } 49, 0, 33, 0, 6 6 } ;, 3, 3 6, 5, 5 6 } ; 5 048, , , 5 048, 5 048, , , so tht, by using 9, the 8-point binry interpoltory scheme is given by λ n+ λ n, λ n+ + 3 j 4 8p j+ λ n +j 38 },

10 AAM: Intern. J., Vol. 4, Issue December 009, [Previously Vol. 4, No. 443 [ 5 λ n λn λ n 3 + λn λ n + λn λ n + λn +, n Z The subdivision scheme in cn lso be estblished by 6, with m 4 nd, or by using the cubic Tylor polynomil expnsion of 34, with t in 33. More explicitly, it follows from tht z 3 S 6 z t z + z 40 [ 8 + z P 8 z z 3 S z 4 6 z [ z 3 + z 4 z + z, 4 which lso yields the subdivision scheme in Exmple. 7-point ternry interpoltory: m 3. Agin, it follows from,, nd tht g 0,,g 6 } 8 9, 0, 35 9, 44 } 9, 0, 9 9, 5 ; 9 s 0,,s 6 } 8 9, 96 9, 553 9, 779 9, 553 9, 96 } 9, 8 ; p 0, 3 7 p 9,, 3 7 p 0} 8 656, 0, , 80, 0, 87 87, , 0, 87 87, ,, , , 0, 80 87, 35 8, 0, }., 0, , Hence, it follows from tht the 7-point ternry interpoltory scheme is given by 3 λ n p 3j λ n +j λ n+ 3 j [ 35λ n 3 336λn + 00λn λn 050λ n λn + 8λn +3, 4 λ n, 43 λ n j 3 3 7p 3j+ λ n +j

11 444 Jin-o Lin [ 8λ n λn 050λn λn +00λ n + 336λn λn +3, n Z Similr to 40 4 in Exmple, 4 44 cn lso be estblished by pplying 30, with m 3, or, with t in 33, by using the cubic Tylor polynomil expnsion of 35. More explicitly, it follows from t z z z 3 S 6 z 3 3 tht + z + z 3 7 P 7 z z 3 S 6 z 3z + z + z z z, 3z 3 which lso genertes the subdivision scheme in Acnowledgment The uthor would lie to thn the nonymous referees for their vluble comments tht helped improve the presenttion of the pper. References [ Belogy, E. nd Wng, Y Compctly supported orthogonl symmetric scling functions, Appl. Comput. Hrmonic Anl. 7, [ Chui, C. K. 99. An Introduction to Wvelets, Acdemic Press, Boston. [3 Chui, C. K. nd Lin, J.-A Construction of compctly supported symmetric nd ntisymmetric orthonorml wvelets with scle 3, Appl. Comput. Hrmonic Anl., -5. [4 Dubechies, I. 99. Ten Lectures on Wvelets, CBMS-NSF Regionl Conference Series in Applied Mth., Vol. 6, SIAM, Phildelphi. [5 Dyn, N., Levin, D., nd Gregory, J A 4-point interpoltory subdivision scheme for curve design, Computer Aided Geometric Design 4, [6 Hn, B orthonorml scling functions nd wvelets with diltion fctor 4, Appl. Comput. Hrmonic Anl. 8, -47. [7 Hssn, M. F. nd Dodgson, N. A. 00. Ternry nd three-point univrite subdivision schemes, in: Curve nd Surfce Fitting: Sint-Mlo 00, A. Cohen, J.-L. Merrien, nd L. L. Schumer eds., Nshboro Press, Brentwood, TN, [8 Lin, J.-A. 00. Polynomil identities of Bezout type, in: Trends in Approximtion Theory, K. Kopotun, T. Lyche, nd M. Nemtu eds., Vnderbilt University Press, Nshville, TN, [9 Lin, J.-A On -ry subdivision for curve design: I. 4-point nd 6-point interpoltory schemes, Appliction & Applied Mth. 3, 8 9. [0 Lin, J.-A On -ry subdivision for curve design: II. 3-point nd 5-point interpoltory schemes, Appliction & Applied Mth. 3, [ Peng, L. nd Wng, Y Construction of compctly supported orthonorml wvelets with beutiful structure, Science in Chin Series F: Informtion Sciences 47, [ Shui, P.-L., Bo, Z., nd Zhng, X.-D. 00. M-bnd compctly supported orthogonl symmetric interpolting scling functions, IEEE Trns. Signl Proc., Vol. 49, Issue 8, pp [3 Weissmn, A A 6-point interpoltory subdivision scheme for curve design, Msters Thesis, Tel-Aviv University.