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Shape preserving Hermite subdivision scheme constructed from quadratic polynomial. (English) Zbl 1499.65037

Summary: In this paper, a Hermite interpolatory subdivision scheme is constructed from the quadratic polynomial. We show that the scheme converges and preserve monotonicity and convexity. In general, the approximation order of the scheme is two. However, third order approximation is achieved by functions having more smoothness. Numerical examples are given to illustrate the effectiveness of our scheme.

MSC:

65D07 Numerical computation using splines
65D17 Computer-aided design (modeling of curves and surfaces)
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[1] Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision, vol. 453. American Mathematical Society (1991) · Zbl 0741.41009
[2] Conti, C.; Cotronei, M.; Sauer, T., Factorization of hermite subdivision operators preserving exponentials and polynomials, Adv. Comput. Math., 42, 5, 1055-1079 (2016) · Zbl 1358.65015 · doi:10.1007/s10444-016-9453-4
[3] Conti, C.; Cotronei, M.; Sauer, T., Convergence of level-dependent hermite subdivision schemes, Appl. Numer. Math., 116, 119-128 (2017) · Zbl 1372.65049 · doi:10.1016/j.apnum.2017.02.011
[4] Conti, C.; Hüning, S., An algebraic approach to polynomial reproduction of hermite subdivision schemes, J. Comput. Appl. Math., 349, 302-315 (2019) · Zbl 1510.65035 · doi:10.1016/j.cam.2018.08.009
[5] Conti, C.; Merrien, J-L; Romani, L., Dual hermite subdivision schemes of de RHAM-type, BIT Numer. Math., 54, 4, 955-977 (2014) · Zbl 1335.65022 · doi:10.1007/s10543-014-0495-z
[6] Conti, C.; Romani, L.; Unser, M., Ellipse-preserving hermite interpolation and subdivision, J. Math. Anal. Appl., 426, 1, 211-227 (2015) · Zbl 1309.41003 · doi:10.1016/j.jmaa.2015.01.017
[7] Cotronei, M.; Moosmüller, C.; Sauer, T.; Sissouno, N., Level-dependent interpolatory hermite subdivision schemes and wavelets, Constr. Approx., 50, 2, 341-366 (2019) · Zbl 07103916 · doi:10.1007/s00365-018-9444-4
[8] Dubuc, S., Scalar and hermite subdivision schemes, Appl. Comput. Harmon. Anal., 21, 3, 376-394 (2006) · Zbl 1111.65020 · doi:10.1016/j.acha.2006.04.007
[9] Dubuc, S., Merrien, J.-L.: A 4-point hermite subdivision scheme. Mathematical Methods for Curves and Surfaces. Innov. Appl. Math, pp. 113-122 (2001) · Zbl 0989.65006
[10] Dubuc, S.; Merrien, J-L, Convergent vector and hermite subdivision schemes, Constr. Approx., 23, 1, 1-22 (2005) · Zbl 1087.65017 · doi:10.1007/s00365-005-0602-0
[11] Dubuc, S.; Merrien, J-L, Hermite subdivision schemes and Taylor polynomials, Constr. Approx., 29, 2, 219-245 (2009) · Zbl 1180.65024 · doi:10.1007/s00365-008-9011-5
[12] Dyn, N.; Levin, D., Analysis of hermite-interpolatory subdivision schemes, Spline Funct. Theory Wavelets, 18, 105-113 (1999) · Zbl 0967.42024 · doi:10.1090/crmp/018/11
[13] Dyn, N.; Levin, D., Subdivision schemes in geometric modelling, Acta Numer., 11, 73-144 (2002) · Zbl 1105.65310 · doi:10.1017/S0962492902000028
[14] Han, B.; Thomas, Yu; Xue, Y., Noninterpolatory hermite subdivision schemes, Math. Comput., 74, 251, 1345-1367 (2005) · Zbl 1066.42024 · doi:10.1090/S0025-5718-04-01704-1
[15] Hüning, S., Polynomial reproduction of hermite subdivision schemes of any order, Math. Comput. Simul., 176, 195-205 (2020) · Zbl 1510.65036 · doi:10.1016/j.matcom.2019.12.010
[16] Jena, H., Jena, M.K.: Construction of trigonometric box splines and the associated non-stationary subdivision schemes. Int. J. Appl. Comput. Math. 7(129) (2021) · Zbl 1499.65042
[17] Jeong, B.; Yoon, J., Construction of hermite subdivision schemes reproducing polynomials, J. Math. Anal. Appl., 451, 1, 565-582 (2017) · Zbl 1361.65010 · doi:10.1016/j.jmaa.2017.02.014
[18] Jeong, B.; Yoon, J., Analysis of non-stationary hermite subdivision schemes reproducing exponential polynomials, J. Comput. Appl. Math., 349, 452-469 (2019) · Zbl 1503.65032 · doi:10.1016/j.cam.2018.07.050
[19] Kocić, LM; Milovanović, GV, Shape preserving approximations by polynomials and splines, Comput. Math. Appl., 33, 11, 59-97 (1997) · Zbl 0911.65006 · doi:10.1016/S0898-1221(97)00087-4
[20] Kuijt, F., van Damme, R.M.J.: Hermite-interpolatory subdivision schemes. (1998) · Zbl 0918.41002
[21] Kuijt, F.; Van Damme, R., Shape preserving interpolatory subdivision schemes for nonuniform data, J. Approx. Theory, 114, 1, 1-32 (2002) · Zbl 1003.41006 · doi:10.1006/jath.2001.3628
[22] Merrien, J-L, A family of hermite interpolants by bisection algorithms, Numer. Algorithms, 2, 2, 187-200 (1992) · Zbl 0754.65011 · doi:10.1007/BF02145385
[23] Merrien, J-L; Sablonniere, P., Monotone and convex c1 hermite interpolants generated by a subdivision scheme, Constr. Approx., 19, 2, 279-298 (2003) · Zbl 1018.41012 · doi:10.1007/s00365-002-0512-3
[24] Merrien, J-L; Sauer, T., From hermite to stationary subdivision schemes in one and several variables, Adv. Comput. Math., 36, 4, 547-579 (2012) · Zbl 1251.41014 · doi:10.1007/s10444-011-9190-7
[25] Merrien, J-L; Sauer, T., Extended hermite subdivision schemes, J. Comput. Appl. Math., 317, 343-361 (2017) · Zbl 1357.41030 · doi:10.1016/j.cam.2016.12.002
[26] Merrien, J-L; Sauer, T., Generalized Taylor operators and polynomial chains for hermite subdivision schemes, Numer. Math., 142, 1, 167-203 (2019) · Zbl 07049390 · doi:10.1007/s00211-018-0996-9
[27] Moosmüller, C., A note on spectral properties of hermite subdivision operators, Comput. Aided Geom. Des., 69, 1-10 (2019) · Zbl 1505.65132 · doi:10.1016/j.cagd.2018.12.002
[28] Moosmüller, C.; Dyn, N., Increasing the smoothness of vector and hermite subdivision schemes, IMA J. Numer. Anal., 39, 2, 579-606 (2019) · Zbl 1464.65019 · doi:10.1093/imanum/dry010
[29] Moosmüller, C., Hüning, S., Conti, C.: Stirling numbers and gregory coefficients for the factorization of hermite subdivision operators. IMA J. Numer. Anal · Zbl 1509.65014
[30] Schumaker, L., Spline Functions: Basic Theory (2007), Cambridge: Cambridge University Press, Cambridge · Zbl 1123.41008 · doi:10.1017/CBO9780511618994
[31] Schumaker, LI, On shape preserving quadratic spline interpolation, SIAM J. Numer. Anal., 20, 4, 854-864 (1983) · Zbl 0521.65009 · doi:10.1137/0720057
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