Abstract
In this paper we discuss methods for investigating the convergence of multivariate vector subdivision schemes and the regularity of the associated limit functions. Specifically, we consider difference vector subdivision schemes whose restricted contractivity determines the convergence of the original scheme and describes the connection between the regularity of the limit functions of the difference subdivision scheme and the original subdivision scheme.
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References
A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93(453) (1991).
M. Charina and C. Conti, Regularity of multivariate vector subdivision schemes, Preprint, Dipartimento di Energetica, Università di Firenze (2002).
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.
D. Cox, J. Little and D. O’Shea, Ideals, Varieties and Algorithms, 2nd ed. (Springer, Berlin, 1996).
W. Dahmen and C.A. Micchelli, Biorthogonal wavelet expansion, Constr. Approx. 13 (1997) 294–328.
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.
N. Dyn and D. Levin, Subdivision schemes in geometric modelling, Acta Num. (2002) 1–72.
N. Dyn and D. Levin, Matrix subdivision – analysis by factorization, in: Approximation Theory, ed. B.D. Bojanov (DARBA, Sofia, 2002) pp. 187–211.
B. Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, J. Approx. Theory 124 (2003) 44–88.
B. Han and R.Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998) 1177–1199.
R.Q. Jia, Subdivision schemes in Lp spaces, Adv. Comput. Math. 3 (1995) 309–341.
R.Q. Jia, S.D. Riemenschneider and D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533–1563.
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.
Q.T. Jiang, Multivariate matrix refinable functions with arbitrary matrix, Trans. Amer. Math. Soc. 351 (1999) 2407–2438.
Q.T. Jiang and Z. Shen, On the existence and weak stability of matrix refinable functions, Constr. Approx. 15 (1999) 337–353.
C.A. Micchelli and T. Sauer, Regularity of multiwavelets, Adv. Comput. Math. 7 (1997) 455–545.
C.A. Micchelli and T. Sauer, On vector subdivision, Math. Z. 229 (1998) 621–674.
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.
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.
T. Sauer, Polynomial interpolation, ideals and approximation order of refinable functions, Proc. Amer. Math. Soc. 130 (2002) 3335–3347.
Z. Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998) 235–250.
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Charina, M., Conti, C. & Sauer, T. Regularity of multivariate vector subdivision schemes. Numer Algor 39, 97–113 (2005). https://doi.org/10.1007/s11075-004-3623-z
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DOI: https://doi.org/10.1007/s11075-004-3623-z