A.-R.K. Ramazanov, V.G. Magomedova. Coconvex interpolation by splines with three-point rational interpolants

For discrete functions $f(x)$ defined on arbitrary grid nodes $\Delta: a=x_0<x_1<\dots<x_N=b$ $(N\geqslant 3)$, we study the issues of preserving the (upward or downward) convexity and coconvexity with a change of convexity direction by rational spline-functions $R_{N,1}(x)=R_{N,1}(x,f,\Delta,g(t))=(R_i(x)(x-x_{i-1})+R_{i-1}(x)(x_i-x))/(x_i-x_{i-1})$, where $x\in [x_{i-1},x_i]$ $(i=1,2,\dots,N)$, $R_i(x)=\alpha_i+\beta_i(x-x_i)+\gamma_i/(x-g_i(t))$ $(i=1,2,\dots,N-1)$, and $R_i(x_j)=f(x_j)$ $(j=i-1,i,i+1)$. The location of the pole $g_i(t)$ with respect to the nodes $x_{i-1}$ and $x_i$ is defined by the parameter $t$. We assume that $R_0(x)\equiv R_1(x)$ and $R_N(x)\equiv R_{N-1}(x)$. For these spines we derive the conditions $1/2<|q_i|<2$ of convexity preservation, where $q_i=f(x_{i-2},x_{i-1},x_i)/f(x_{i-1},x_i,x_{i+1})$ for $i=2,3,\dots,N-1$.

Keywords: interpolation spline, rational spline, coconvex interpolation, shape-preserving interpolation.

The paper was received by the Editorial Office on February 28, 2018.

Abdul-Rashid Kehrimanovich Ramazanov, Dr. Phys.-Math., Prof., Dagestan State University, the Republic of Dagestan, Makhachkala, 367002 Russia; Dagestan Scientific Center RAN, the Republic of Dagestan, Makhachkala, 367025 Russia, e-mail: ar-ramazanov@rambler.ru

Vazipat Gusenovna Magomedova, Cand. Sci. (Phys.-Math.), Dagestan State University, the Republic of Dagestan, Makhachkala, 367002 Russia, e-mail: vazipat@rambler.ru

REFERENCES

1.   Schweikert D.G. An interpolation curve using a spline in tension. J. Math. Phys., 1966, vol. 45, iss. 1–4, pp. 312–317. doi: 10.1002/sapm1966451312 .

2.   Miroshnichenko V.L. Convex and monotone spline interpolation. Constuctive Theory of Function: Proc. Int. Conf., (Varna, 1984), Sofia: Publ. House of Bulgarian Acad. Sci., 1984, pp. 610–620.

3.   Miroschnichenko V.L. Sufficient conditions for monotonicity and convexity of cubic splines of class $C^2$. Sib. Adv. Math., 1992, vol. 2, no. 4, pp. 147–163.

4.   Kvasov B.I. Metody izogeometricheskoi approksimatsii splainami [Methods of Shape-Preserving Spline Approximaton]. Moscow: Fizmatlit Publ., 2006, 360 p. ISBN: 5-9221-0733-X .

5.    Volkov Yu.S., Bogdanov V.V., Miroschnichenko V.L., Shevaldin V.T. Shape-preserving interpolation by cubic splines. Math. Notes, 2010, vol. 88, no. 6, pp. 798–805. doi: 10.1134/S0001434610110209 .

6.   Schaback R. Spezielle rationale Splinefunktionen. J. Approx.Theory, 1973, vol. 7, no. 2. pp. 281–292. doi: 10.1016/0021-9045(73)90072-5 .

7.   Spath H. Spline algorithms for curves and surfaces. Winnipeg: Utilitas Mathematica Publ. Inc., 1974, 198 p. ISBN: 0919628990 .

8.   Hussain M.Z., Sarfraz M., Shaikh T.S. Shape preserving rational cubic spline for positive and convex data. Egyptian Informatics J., 2011, vol. 12, pp. 231–236. doi: 10.1016/j.eij.2011.10.002 .

9.   Edeo A., Gofeb G., Tefera T. Shape preserving $C^2$ rational cubic spline interpolation. American Sci. Research J. Engineering, Technology and Sciences, 2015, vol. 12, no. 1. pp. 110–122.

10.   Ramazanov A.-R.K., Magomedova V.G. Splines for rational interpolants. Dagestan. Elektron. Mat. Izv., 2015, no. 4, pp. 21–30 (in Russian).

11.   Ramazanov A.-R.K., Magomedova V.G. Splines for three-point rational interpolants. Tr. Matem. centra im. N.I. Lobachevskogo. Kazan, 2017, vol. 54, pp. 304–306 (in Russian).

12.   Ramazanov A.-R.K., Magomedova V.G. Splines for three-point rational interpolants with autonomous poles. Dagestan. Elektron. Mat. Izv., 2017, no. 7, pp. 16–28 (in Russian).

13.   Ramazanov A.-R.K., Magomedova V.G. Unconditionally convergent interpolational rational splines. Math. Notes, 2018, vol. 103, no. 4, pp. 588–599 (in Russian). doi: 10.4213/mzm11201 .