Operator interpolation formulas of Hermitе type with arbitrary multiplicity nodes based on identity transformations of function
https://doi.org/10.29235/1561-2430-2018-54-3-263-272
Abstract
The problem of construction and research of Hermite interpolation formulas with nodes of arbitrary multiplicity for operators given in functional spaces of one and two variables is considered. The construction of operator interpolation polynomials is based both on interpolation polynomials for scalar functions with respect to an arbitrary Chebyshev system and on identity transformations of functions. The reduced operator formulas contain the Stieltjes integrals and the Gateaux differentials of an interpolated operator and are invariant for a special class of operator polynomials of appropriate degree. For some of the obtained operator polynomials, an explicit representation of the interpolation error is found. Particular cases of Hermite formulas based both on the integral transformations of Hankel, Abel, Fourier and on the Fourier sine (cosine) transform are considered. The application of separate interpolation formulas is illustrated by examples. The presented results can be used in theoretical research as the basis for construction of approximate methods for solving integral, differential and other types of nonlinear operator equations.
About the Author
M. V. Ignatenko
Belarusian State University
Belarus
Belarus
Marina V. Ignatenko – Ph. D. (Physics and Mathematics), Associate Professor, Associate Professor of Web-Technologies and Computer Simulation Department.
4, Nezavisimosti Ave., 220030, Minsk.
References
1. Watson G. N. A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1944. 804 p.
2. Samko S. G., Kilbas A. A., Marichev O. I. Fractional Integrals and Derivatives. Theory and Applications. New York-Philadephia-London-Paris-Mountreux-Tokyo-Melbourne, Gordon and Breach Science Publishers, 1993. 1006 р.
3. Titmarsh E. Introduction in the theory of Fourier integrals. Moscow, Leningrad, State Publishing House of Technical and Theretical Literature, 1948. 418 р. (in Russian.).
4. Krylov V. I., Skoblya N. S. A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Moscow, Mir Publ., 1977. 273 р.
5. Bateman G., Erdelyi A. Tables of integral transforms. Vol. 1: Transformations of Fourier, Laplace, Mellin. Moscow, Nauka Publ., 1969. 344 р. (in Russian.).
6. Yanovich L. A., Ignatenko M. V. About interrelation of interpolation of operators and functions. Doklady Natsional'noy akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 1998, vol. 42, no 3, pp. 9–16 (in Russian.).
7. Makarov V. L., Khlobystov V. V., Yanovich L. A. Methods of Operator Interpolation. Pratsi institutu matematiki NAN Ukraїni [Institute of Mathematics of the National Academy of Sciences of Ukraine]. Kiev, 2010, vol. 83, pp. 1–517.
8. Yanovich L. A., Ignatenko M. V. Bases of the theory of interpolation of functions of matrix variables. Minsk, Belaruskaya Navuka Publ., 2016. 281 р. (in Russian.).
Review
Views: 875
Email this article 