Many constructions of cubic splines are described in the literature. Most of the methods focus on cubic splines of defect 1, i.e., cubic splines that are continuous together with their first and second derivative. However, many applications do not require continuity of the second derivative. The Hermitian cubic spline is used for such problems. For the construction of a Hermitian spline we have to assume that both the values of the interpolant function and the values of its derivative on the grid are known. The derivative values are not always observable in practice, and they are accordingly replaced with difference derivatives, and so on. In the present article, we construct a C1 cubic spline so that its derivative has a minimum norm in L2 . The evaluation of the first derivative on a grid thus reduces to the minimization of the first-derivative norm over the sought values.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Yu. S. Zav’yalov, B. I. Kvasov, and V. L. Miroshnichenko, Spline Function Methods [in Russian], Nauka, Moscow (1980).
S. B. Stechnkin and Yu. N. Subbotin, Splines in Computational Mathematics [in Russian], Nauka, Moscow (1976).
J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York (1967).
C. De Boor, A Practical Guide to Splines, Springer, New York (1978).
V. I. Dmitriev, I. V. Dmitrieva, and J. G. Ingtem, “Integral spline function,” Comput. Math. and Model..
Yu. S. Volkov, “A general polynomial spline-interpolation problem,” Tr. Inst. Mat. Mekh. UrO RAN, No. 22, 114–125 (2016) [doi.https://doi.org/10.21538/0134-4889-2016-22-4-114-125].
Yu. S. Volkov and Yu. N. Subbotin, “50 years of Schoenberg’s problem on spline-interpolation convergence,” Tr. Inst. Mat. Mekh. UrO RAN, No. 20, 52–67 (2014).
J. Ingtem, “Minimal-norm-derivative spline function in interpolation and approximation,” Moscow Univ. Comput. Math. and Cyber., 32, No. 4, 201–213 (2008).
A. N. Tikhonov and V. Ya. Arsenin, Methods of Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1979).
D. A. Silaev, “Semi-local smoothing splines,” Tr. Sem. aim. I. G. Petrovskogo, Moscow State University, 29, 443–454 (2013); J. Math. Sci. (N. Y.), 197.
M. C. Mariani, K. Basu, “Spline interpolation techniques applied to the study of geophysical data,” Physica A: Statistical Mechanics and Its Applications, 428, 68–79 (2015).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Matematika i Informatika, No. 60, 2019, pp. 16–24.
Rights and permissions
About this article
Cite this article
Dmitriev, V.I., Ingtem, J.G. The Regularized Spline (R-Spline) Method for Function Approximation. Comput Math Model 30, 198–206 (2019). https://doi.org/10.1007/s10598-019-09447-w
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10598-019-09447-w