This paper presents formulas for constructing quadratic minimal splines, which explicitly depend on the components of a generating vector function. Formulas for various approximation functionals for minimal splines used as coefficients in local approximation methods are obtained. Examples of special cases of approximation schemes, known as quasi-interpolation, are provided. Results of numerical experiments on approximating a circular arc by minimal splines are considered.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Yu. S. Volkov and Yu. N. Subbotin, “Fifty years of Schoenberg’s problem on the convergence of spline interpolation,” Tr. Inst. Mat. Mekh., 20, No. 1, 52–67 (2014).
Yu. S. Volkov and V. T. Shevaldin, “Shape-preserving conditions for interpolation with quadratic splines,” Tr. Inst. Mat. Mekh., 18, No. 4, 145–152 (2012).
S. B. Stechkin and Yu. N. Subbotin, Splines in Computational Mathematics [in Russian], Moscow (1976).
Yu. S. Zavyalov, B. I. Kvasov, and V. L. Miroshnichenko, Methods of Spline Functions [in Russian], Moscow (1980).
L. L. Schumaker, Spline Functions: Basic Theory, John Wiley and Sons, Inc., New York (1981).
Yu. K. Demyanovich, Local Approximation on a Manifold and Minimal Splines [in Russian], St. Petersburg Univ. Press, St. Petersburg (1994).
C. de Boor, A Practical Guide to Splines (Revised edition), Springer-Verlag, New York (2001).
B. I. Kvasov, Methods of Shape Preserving Spline Approximation [in Russian], Fizmatlit, Moscow (2006).
I. G. Burova and Yu. K. Demyanovich, Minimal Splines and Applications [in Russian], St. Petersburg Univ. Press, St. Petersburg (2010).
A. I. Grebennikov, Spline Methods and Solutions of Ill-Posed Problems in Approximation Theory [in Russian], Moscow Univ. Press, Moscow (1983).
T. Lyche and L. L. Shumaker, “Quasi-interpolants based on trigonometric splines,” J. Approx. Theor., 95, 280–309 (1998).
P. Sablonniere, “Quadratic spline quasi-interpolants on bounded domains of ℝd, d = 1, 2, 3,” Rend. Sem. Mat., 61, No. 3, 229–246 (2003).
T. Lyche and K. Mörken, Spline Methods. Draft, Centre Math. Applications University of Oslo (2005).
P. Constantini, C. Manni, F. Pelosi, and M. Lucia Sampoli, “Quasi-interpolation in isogeometric analysis based on generalized B-splines,” Comput Aided Geom. Design, 27, No. 8, 655–668 (2010).
A. A. Makarov, “Biorthogonal systems of functionals and decomposition matrices for minimal splines,” Ukr. Mat. Visn., 9, No. 2, 219–236 (2012).
Y. Jiang and Y. Xu, “B-spline quasi-interpolation on sparse grids,” J. Complexity, 27, No. 5, 466–488 (2014).
W. Gao and Z.Wu, “A quasi-interpolation scheme for periodic data based on multiquadric trigonometric B-splines,” J. Comput. Appl. Math., 271, 20–30 (2014).
M. Li, L. Chen, and Q. Ma, “A meshfree quasi-interpolation method for solving Burger’s equation,” Comput. Meth. Eng. Sci., 2014, No. 3, 1–8 (2014).
R. G. Yu, R. H. Wang, and C. G. Zhu, “A numerical method for solving KdV equation with multilevel B-spline quasi-interpolation,” Appl. Anal., 92, No. 8, 1682–1690 (2013).
C. Dagnino, S. Remogna, and P. Sablonniere, “On the solution of Fredholm integral equations based on spline quasi-interpolating projectors,” BIT Numer. Math., 54, No. 4, 979–1008 (2014).
M. Derakhshan and M. Zarebnia, “On the numerical treatment and analysis of twodimensional Fredholm integral equations using quasi-interpolant,” Comp. Appl. Math., 39, 106 (2020).
P. Sablonnière, D. Shibih, and T. Mohamed, “Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell–Sabin partitions,” BIT Numer. Math., 53, No. 1, 175–192 (2013).
L. Lu, “On polynomial approximation of circular arcs and helices,” Comput. Math. Appl., 63, 1192–1196 (2012).
G. Jaklič, “Uniform approximation of a circle by a parametric polynomial curve,” Computer Aided Geom. Design, 41, 36–46 (2016).
A. Rababah, “The best uniform cubic approximation of circular arcs with high accuracy,” Commun. Math. Appl., 7, No. 1, 37–46 (2016).
C. Apprich, A. Dietrich, K. Höllig, and E. Nava-Yazdani, “Cubic spline approximation of a circle with maximal smoothness and accuracy,” Computer Aided Geom. Design, 56, 1–3 (2017).
A. A. Makarov, “On an example of circular arc approximation by quadratic minimal splines,” Poincaré Anal. Appl., 2018, No. 2, 103–107 (2018).
Yu. K. Demyanovich, “Smoothness of spline spaces and wavelets decompositions,” Dokl. RAN, 401, No. 4, 1–4 (2005).
Yu. K. Dem’yanovich and A. A. Makarov, “Necessary and sufficient nonnegativity conditions for second-order coordinate trigonometric splines,” Vest. St.Petersburg Univ., Ser. 1, 4(62), No. 1, 9–16 (2017).
O. Kosogorov and A. Makarov, “On some piecewise quadratic spline functions,” Lect. Notes Comput. Sci., 10187, 448–455 (2017).
A. A. Makarov, “Construction of splines of maximal smoothness,” Probl. Mat. Anal., 60, 25–38 (2011).
A. A. Makarov, “On functionals dual to minimal splines,” Zap. Nauchn. Semin. POMI, 453, 198–218 (2016); English transl., J. Math. Sci., 224, No. 6, 942–955 (2017).
E. K. Kulikov and A. A. Makarov, “On de Boor–Fix type functionals for minimal splines,” in: Topics in Classical and Modern Analysis (Applied and Numerical Harmonic Analysis) (2019), pp. 211–225.
E. K. Kulikov and A. A. Makarov, “On approximation by hyperbolic splines,” Zap. Nauchn. Semin. POMI, 472, 179–194 (2018); English transl., J. Math. Sci., 240, No. 6, 822–832 (2019).
E. K. Kulikov and A. A. Makarov, “On approximate solution of a singular perturbation boundary-value problem,” Diff. Uravn. Prots. Upr., No. 1, 91–102 (2020).
E. Kulikov and A. Makarov, “On biorthogonal approximation of solutions of some boundary value problems on Shishkin mesh,” AIP Conference Proceedings, 2302, 110005 (2020).
E. K. Kulikov and A. A. Makarov, “Quadratic minimal splines with multiple nodes,” Zap. Nauchn. Semin. POMI, 482, 220–230 (2019); English transl., J. Math. Sci., 249, No. 2, 256–262 (2020).
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 504, 2021, pp. 136–156.
Translated by the authors.
Rights and permissions
About this article
Cite this article
Kulikov, E.K., Makarov, A.A. Construction of Approximation Functionals for Minimal Splines. J Math Sci 262, 84–98 (2022). https://doi.org/10.1007/s10958-022-05801-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-05801-3