Optimal properties for deficient quartic splines of Marsden type


  • Alexandru Bica University of Oradea, Romania
  • Diana Curilă-Popescu University of Oradea, Romania
  • Mircea Curilă University of Oradea, Romania


In this work, we obtain an improved error estimate in the interpolation with the Hermite \(C^{2}\)-smooth deficient complete quartic spline that has the distribution of nodes following the Marsden type scheme and investigate the possibilities to compute the derivatives on the knots such that the obtained spline \(S\in C^{1}[a,b]\) has minimal curvature and minimal \(L^{2}\)-norm of \(S^{\prime }\) and \(S^{\prime \prime \prime }\). In each case, the interpolation error estimate is performed in terms of the modulus of continuity.%MCEPASTEBIN%


J.H. Ahlberg, E.H. Nilson, J.L., Walsh, The Theory of Splines and Their Applications, Academic Press, New York, London (1967).

D. Bini, M. Capovani, A class of cubic splines obtained through minimum conditions, Math. Comp. 46 (173), (1986), pp. 191–202, https://doi.org/10.1090/s0025-5718-1986-0815840-5

P. Blaga, G. Micula, Natural spline functions of even degree. Studia Univ. Babes-Bolyai Cluj-Napoca Mathematica 38(2), (1993), pp. 31–40.

S. Dubey, Y. P. Dubey, Convergence of C2 deficient quartic spline interpolation. Adv. Comput. Sciences & Technol.10(4), (2017), pp. 519–527

M.N. El Tarazi, S. Sallam, On quartic splines with application to quadratures. Computing 38, (1987), pp. 355–361, https://doi.org/10.1007/bf02278713

A. Ghizzetti, Interpolazione con splines verificanti una opportuna condizione. Calcolo 20, (1983), pp. 53–65, https://doi.org/10.1007/bf02575892

T.A. Grandine, T.A. Hogan, A parametric quartic spline interpolant to position,tangent and curvature. Computing 72, (2004), pp. 65–78, https://doi.org/10.1007/978-3-7091-0587-0_6

X. Han, X. Guo, Cubic Hermite interpolation with minimal derivative oscillation. J. Comput. Appl. Math. 331, (2018), pp. 82–87, https://doi.org/10.1016/j.cam.2017.09.049

G.W. Howell, Error bounds for polynomial and spline interpolation, PhD Thesis, University of Florida (1986).

G. Howell, A. K. Varma, Best error bounds for quartic spline interpolation. J. Approx. Theory 58, (1989), pp. 58–67, https://doi.org/10.1016/0021-9045(89)90008-7

G.W. Howell, Derivative error bounds for Lagrange interpolation: An extension of Cauchy’s bound for the error of Lagrange interpolation. J. Approx. Theory 64, (1991), pp. 164–173, https://doi.org/10.1016/0021-9045(91)90015-3

A.A. Karaballi, S. Sallam, Lacunary interpolation by quartic splines on uniform meshes. J. Comput. Appl. Math. 80, (1997), pp. 97–104, https://doi.org/10.1016/s0377-0427(97)00015-0

J. Kobza, Cubic splines with minimal norm. Appl. Math. 47, (2002), pp. 285–295, https://doi.org/10.1023/a:1021749621862

J. Kobza, P. Zencak, Some algorithms for quartic smoothing splines. Acta Univ. Palacki. Olomuc, Fac. rer. nat., Mathematica 36, 79-94 (1997).

J. Kobza, Quartic splines with minimal norms. Acta Univ. Palacki. Olomuc, Fac. rer.nat., Mathematica 40, (2001), pp. 103–124

Z. Liu, New Sharp Error bounds for some corrected quadrature formulae. Sarajevo J. Math. 9(21), (2013), pp. 37–45, https://doi.org/10.5644/sjm.09.1.03

M. Marsden, Quadratic spline interpolation. Bull. Amer. Math. Soc., 80(5), (1974), pp. 903–906, https://doi.org/10.1090/s0002-9904-1974-13566-4

Gh. Micula, E. Santi, M.G. Cimoroni, A class of even degree splines obtained through a minimum condition. Studia Univ. ”Babes-Bolyai” Mathematica 48(3), (2003), pp. 93–104.

R. Plato, Concise numerical mathematics, in: Graduate Studies in Mathematics, vol. 57, AMS Providence, Rhode Island (2003)

S.S. Rana, Y. P. Dubey, Best error bounds for deficient quartic spline interpolation. Indian J. Pure Appl. Math. 30(4), (1999), pp. 385–393

S.S. Rana, R. Gupta, Deficient discrete quartic spline interpolation. Rocky Mountain J. Math. 35 (4), (2005), pp. 1369–1379, https://doi.org/10.1216/rmjm/1181069690

S. Sallam, M.N. Anwar, Shape preserving quartic C2-spline interpolation with constrained curve length and curvature. J. Information Comput. Science2(4), (2005), pp. 775–781.

. Ujevic, A. J. Roberts, A corrected quadrature formula and applications. ANZIAM J. 45, (2004), pp. 41–56

Yu. S. Volkov, Best error bounds for the derivative of a quartic interpolation spline. Mat. Trud. 1(2), (1998), pp. 68–78 (in Russian)

Yu. S. Volkov, Interpolation by splines of even degree according to Subbotin and Marsden. Ukrainian Math. J.66 (7), (2014), pp. 994–1012, https://doi.org/10.1007/s11253-014-0990-z

Y. Zhu, Crational quartic/cubic spline interpolant with shape constraints, https://doi.org/10.1007/s00025-018-0883-9




How to Cite

Bica, A., Curilă-Popescu, D., & Curilă, M. (2021). Optimal properties for deficient quartic splines of Marsden type. J. Numer. Anal. Approx. Theory, 49(2), 113-130. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1228