Regularity of multivariate vector subdivision schemes. (English) Zbl 1071.65019
Subdivision schemes are computational means for generating recursively discrete functions defined on denser and denser grids in \({\mathbb R}^s\). At each step of the subdivision recursion the new values are obtained simply by local averaging of the previously computed values on the coarser grid. The averaging coefficients form the so-called refinement mask. If they are real numbers the scheme is said to be a scalar subdivision scheme acting on scalar sequences. If the averaging coefficients are matrices, the scheme is called either a matrix subdivision scheme or a vector subdivision scheme as it now maps vector valued sequences to vector valued sequences. Vector subdivision schemes play an important role in the convergence and regularity analysis even of scalar multivariated subdivisions schemes, in the analysis of hermite type subdivision schemes, and in the context of multi-wavelets.
The authors give a characterization of the convergence of multivariated subdivision schemes and derive sufficient conditions for a refinable function to possess a certain order of differentiability. This is done classifying the subdivision schemes with respect to the dimension of a certain finite dimensional subspace depending on the matrices that form the mask of the subdivision scheme as introduced by C.A. Micchelli and T. Sauer [Adv. Comput. Math 7, 455-545 (1997; Zbl 0902.65095); Math Z, 229, 621-674 (1998; Zbl 0930.65007)]. Based on this dimension, the authors use a suitable difference operator to pass to a difference scheme whose mask consists of larger matrices. The conditions on the mask of the original scheme ensuring the existence of the difference scheme allow for an algebraic interpretation that generalizes the ”zero at \(-1\)” property from the univariate case. The concept of the restricted spectral radius, introduced and investigated by the authors, enable to characterize the convergence of the original subdivision scheme in terms of the spectral properties of the difference scheme. Finally, the authors show that the convergence of the difference scheme implies that the original scheme is convergent to a smoother limit function.
The authors give a characterization of the convergence of multivariated subdivision schemes and derive sufficient conditions for a refinable function to possess a certain order of differentiability. This is done classifying the subdivision schemes with respect to the dimension of a certain finite dimensional subspace depending on the matrices that form the mask of the subdivision scheme as introduced by C.A. Micchelli and T. Sauer [Adv. Comput. Math 7, 455-545 (1997; Zbl 0902.65095); Math Z, 229, 621-674 (1998; Zbl 0930.65007)]. Based on this dimension, the authors use a suitable difference operator to pass to a difference scheme whose mask consists of larger matrices. The conditions on the mask of the original scheme ensuring the existence of the difference scheme allow for an algebraic interpretation that generalizes the ”zero at \(-1\)” property from the univariate case. The concept of the restricted spectral radius, introduced and investigated by the authors, enable to characterize the convergence of the original subdivision scheme in terms of the spectral properties of the difference scheme. Finally, the authors show that the convergence of the difference scheme implies that the original scheme is convergent to a smoother limit function.
Reviewer: Sonia Pérez Díaz (Madrid)
MSC:
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
Keywords:
matrix subdivision scheme; vector subdivision scheme; refinement mask; multi-wavelets; convergence; difference operator; difference scheme; restricted spectral radiusReferences:
| [1] | A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93(453) (1991). · Zbl 0741.41009 |
| [2] | M. Charina and C. Conti, Regularity of multivariate vector subdivision schemes, Preprint, Dipartimento di Energetica, Università di Firenze (2002). · Zbl 1071.65019 |
| [3] | D.R. Cheng, R.Q. Jia and S.D. Riemenschneider, Convergence of vector subdivision schemes in Sobolev spaces, Appl. Comput. Harmon. Anal. 12 (2002) 128-149. · Zbl 1006.65153 · doi:10.1006/acha.2001.0363 |
| [4] | D. Cox, J. Little and D. O?Shea, Ideals, Varieties and Algorithms, 2nd ed. (Springer, Berlin, 1996). |
| [5] | W. Dahmen and C.A. Micchelli, Biorthogonal wavelet expansion, Constr. Approx. 13 (1997) 294-328. · Zbl 0882.65148 · doi:10.1007/s003659900045 |
| [6] | N. Dyn, Subdivision schemes in CAGD, in: Advances in Numerical Analysis, Vol. II: Wavelets, Subdivision Algorithms and Radial Basis Functions, ed. W.A. Light (Oxford Univ. Press, Oxford, 1992) pp. 36-104. · Zbl 0760.65012 |
| [7] | N. Dyn and D. Levin, Subdivision schemes in geometric modelling, Acta Num. (2002) 1-72. · Zbl 1105.65310 |
| [8] | N. Dyn and D. Levin, Matrix subdivision ? analysis by factorization, in: Approximation Theory, ed. B.D. Bojanov (DARBA, Sofia, 2002) pp. 187-211. · Zbl 1041.42028 |
| [9] | B. Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, J. Approx. Theory 124 (2003) 44-88. · Zbl 1028.42019 · doi:10.1016/S0021-9045(03)00120-5 |
| [10] | B. Han and R.Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998) 1177-1199. · Zbl 0915.65143 · doi:10.1137/S0036141097294032 |
| [11] | R.Q. Jia, Subdivision schemes in Lp spaces, Adv. Comput. Math. 3 (1995) 309-341. · Zbl 0833.65148 · doi:10.1007/BF03028366 |
| [12] | R.Q. Jia, S.D. Riemenschneider and D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533-1563. · Zbl 0905.65139 · doi:10.1090/S0025-5718-98-00985-5 |
| [13] | R.Q. Jia, S.D. Riemenschneider and D.X. Zhou, Smoothness of multiple refinable functions and multiple wavelets, SIAM J. Matrix. Anal. Appl. 21(1) (1999) 1-28. · Zbl 0940.42021 · doi:10.1137/S089547989732383X |
| [14] | Q.T. Jiang, Multivariate matrix refinable functions with arbitrary matrix, Trans. Amer. Math. Soc. 351 (1999) 2407-2438. · Zbl 0931.42021 · doi:10.1090/S0002-9947-99-02449-6 |
| [15] | Q.T. Jiang and Z. Shen, On the existence and weak stability of matrix refinable functions, Constr. Approx. 15 (1999) 337-353. · Zbl 0932.42028 · doi:10.1007/s003659900111 |
| [16] | C.A. Micchelli and T. Sauer, Regularity of multiwavelets, Adv. Comput. Math. 7 (1997) 455-545. · Zbl 0902.65095 · doi:10.1023/A:1018971524949 |
| [17] | C.A. Micchelli and T. Sauer, On vector subdivision, Math. Z. 229 (1998) 621-674. · Zbl 0930.65007 · doi:10.1007/PL00004676 |
| [18] | H.M. Möller and T. Sauer, Multivariate refinable functions of high approximation order via quotient ideals of Laurent polynomials, Adv. Comput. Math. (2003), to appear. |
| [19] | T. Sauer, Stationary vector subdivision ? quotient ideals, differences and approximation power, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 229 (2002) 621-674. · Zbl 1057.42029 |
| [20] | T. Sauer, Polynomial interpolation, ideals and approximation order of refinable functions, Proc. Amer. Math. Soc. 130 (2002) 3335-3347. · Zbl 1025.42023 · doi:10.1090/S0002-9939-02-06678-9 |
| [21] | Z. Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998) 235-250. · Zbl 0913.42028 · doi:10.1137/S0036141096302688 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
