The interpolation functional
polynomial: the analogue of the Taylor formula

Ya. O. Baranetskij; I. I. Demkiv; M. I. Kopach; A. F. Obshta

doi:10.15330/ms.50.2.198-203

The paper deals with a functional Newton type polynomial, which has two properties: the first one is that interpolation nodes are continual, that is, they depend on continuous parameters, and the second one is the invariance of the interpolation formulas with respect to polynomials of the corresponding degree. The first property is provided by the substitution rule, the fulfillment of which for a given functional is a sufficient condition for the possibility of the interpolation in the space of piecewise continuous functions on [0; 1] with a finite number of points of discontinuity of the first kind. On the basis of Newton’s interpolation formulas, using the multiplicity of nodes by means of a passage to the limit, an interpolation functional Taylor type polynomial, which has the above-mentioned properties, is constructed.

1. M. Andersson, M. Passare, Complex Kergin Interpolation, J. Approx. Theory, 64 (1991), №2, 214–225.

2. I.I. Demkiv, An interpolation functional third-degree polynomial that does not use substitution rules, J. Math. Sci., 180 (2012), №1, 34–50.

3. A.D. Egorov, P.I. Sobolevsky, L.A. Yanovich, Functional Integrals: Approximate Evaluation and Applications, Dordrecht: Kluwer Academic Publishers, 1993, 419 p.

4. L. Filipsson, Kergin interpolation in Banach spaces, J. Approx. Theory, 127 (2004), №1, 108–123.

5. P. Kergin, A natural interpolation of $C^k$ functions, J. Approx. Theory, 29 (1980), №4, 278–293.

6. V.L. Makarov, I.I. Demkiv, Abstract Interpolation by Continued Thiele-Type Fractions, Cybern. Syst. Anal., 54 (2018), №1, 122–129.

7. V.L. Makarov, I.I. Demkiv, Interpolating integral continued fraction of the Thiele type, J. Math. Sci., 220 (2017), №1, 50–58.

8. V.L. Makarov, I.I. Demkiv, B.R. Mykhalchuk, Necessary and sufficient conditions for the existence of the functional interpolation polynomial on the continual set of interpolation nodes, Dopov. Nats. Akad. Nauk Ukr., 2003, №7, 7–12. (in Ukrainian)

9. V.L. Makarov, V.V. Khlobystov, The Newton-type interpolational formula for the nonlinear operators and it’s application, Conference of numerical methods and applications. Sofia, 22-27 August 1988, 272– 283.

10. C.A. Micchelli, P. Milman, A formula for Kergin interpolation in $R^k$, J. Approx. Theory, 29 (1980), №4, 294–296.

11. C.A. Micchelli, A constructive approach to Kergin interpolation in $R^k$: multivariate b-splines and lagrange interpolation, Rocky Mountain J. Math., 10 (1980), №3, 485–497.

12. W.A. Porter, Data interpolation: causality structure and system identification, Inf. and Contr., 29 (1975), №3, 217–233.

13. P.M. Prenter, Lagrange and Hermite interpolation in Banach spaces, J. Appr. Theory, 4 (1971), №4, 419–432.

	The interpolation functional polynomial: the analogue of the Taylor formula
Author	Ya. O. Baranetskij¹, I. I. Demkiv², M. I. Kopach³, A. F. Obshta⁴ kopachm2009@gmail.com³ 1) Lviv Polytechnic National University, Lviv, Ukraine; 2) Lviv Polytechnic National University, Lviv, Ukraine; 3) Vasyl Stefanyk Precarpathian National University Ivano-Frankivsk, Ukraine; 4) Lviv Polytechnic National University, Lviv, Ukraine
Abstract	The paper deals with a functional Newton type polynomial, which has two properties: the first one is that interpolation nodes are continual, that is, they depend on continuous parameters, and the second one is the invariance of the interpolation formulas with respect to polynomials of the corresponding degree. The first property is provided by the substitution rule, the fulfillment of which for a given functional is a sufficient condition for the possibility of the interpolation in the space of piecewise continuous functions on [0; 1] with a finite number of points of discontinuity of the first kind. On the basis of Newton’s interpolation formulas, using the multiplicity of nodes by means of a passage to the limit, an interpolation functional Taylor type polynomial, which has the above-mentioned properties, is constructed.
Keywords	Taylor formula; Newton’s interpolation formula; interpolation node; functional polynomial
DOI	doi:10.15330/ms.50.2.198-203
Reference	1. M. Andersson, M. Passare, Complex Kergin Interpolation, J. Approx. Theory, 64 (1991), №2, 214–225. 2. I.I. Demkiv, An interpolation functional third-degree polynomial that does not use substitution rules, J. Math. Sci., 180 (2012), №1, 34–50. 3. A.D. Egorov, P.I. Sobolevsky, L.A. Yanovich, Functional Integrals: Approximate Evaluation and Applications, Dordrecht: Kluwer Academic Publishers, 1993, 419 p. 4. L. Filipsson, Kergin interpolation in Banach spaces, J. Approx. Theory, 127 (2004), №1, 108–123. 5. P. Kergin, A natural interpolation of $C^k$ functions, J. Approx. Theory, 29 (1980), №4, 278–293. 6. V.L. Makarov, I.I. Demkiv, Abstract Interpolation by Continued Thiele-Type Fractions, Cybern. Syst. Anal., 54 (2018), №1, 122–129. 7. V.L. Makarov, I.I. Demkiv, Interpolating integral continued fraction of the Thiele type, J. Math. Sci., 220 (2017), №1, 50–58. 8. V.L. Makarov, I.I. Demkiv, B.R. Mykhalchuk, Necessary and sufficient conditions for the existence of the functional interpolation polynomial on the continual set of interpolation nodes, Dopov. Nats. Akad. Nauk Ukr., 2003, №7, 7–12. (in Ukrainian) 9. V.L. Makarov, V.V. Khlobystov, The Newton-type interpolational formula for the nonlinear operators and it’s application, Conference of numerical methods and applications. Sofia, 22-27 August 1988, 272– 283. 10. C.A. Micchelli, P. Milman, A formula for Kergin interpolation in $R^k$, J. Approx. Theory, 29 (1980), №4, 294–296. 11. C.A. Micchelli, A constructive approach to Kergin interpolation in $R^k$: multivariate b-splines and lagrange interpolation, Rocky Mountain J. Math., 10 (1980), №3, 485–497. 12. W.A. Porter, Data interpolation: causality structure and system identification, Inf. and Contr., 29 (1975), №3, 217–233. 13. P.M. Prenter, Lagrange and Hermite interpolation in Banach spaces, J. Appr. Theory, 4 (1971), №4, 419–432.
Pages	198-203
Volume	50
Issue	2
Year	2018
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

The interpolation functional polynomial: the analogue of the Taylor formula