Discrete system of Wiener–Hopf type for coefficients of interpolation formulas
Keywords:
Hilbert space, extremal function, error functional, interpolation formulaAbstract
There are algebraic and variational approaches of construction in the spline theory. In algebraic approach splines are considered as some smooth piecewise polynomial functions. In the variational approach splines are elements of Hilbert or Banach spaces minimizing certain functionals. Then we study the problems of existence, uniqueness, and convergence of splines and algorithms for constructing them based on their own properties of splines. In this paper, we study the problem of constructing an optimal interpolation formula in a Hilbert space. Here, using the Sobolev method, the first part of the problem is solved, i.e. an explicit expression of the square of the norm of the error functional is found and a system of linear algebraic equations for the coefficients of the optimal interpolation formula is obtained.
References
Holladay J.C. Smoothest curve approximation. Math. Tables Aids Comput. – vol. 11. – 1957. – P. 223—243.
de Boor C. Best approximation properties of spline functions of odd degree. J. Math. Mech. – 1963. – vol.12. – P. 747–749.
Ahlberg J.H., Nilson E.N., Walsh J.L. The theory of splines and their applications. New York: Academic Press, – 1967. – 316 p.
Arcangeli R., Lopez de Silanes M.C., Torrens J.J. Multidimensional minimizing splines. Boston: Kluwer Academic publishers, – 2004. – 278 p.
Attea M. Hilbertian kernels and spline functions. North-Holland, – 1992. – 393 p.
de Boor C. A practical guide to splines. Springer, – 1978. – 342 p.
Игнатьев М.И., Певний А.Б. Натуральные сплайны многих переменных. М.: Наука, – 1991. – 154 с.
Лоран П.Ж. Аппроксимация и оптимизация. М.: Мир, – 1975. – 496 с.
Mastroianni G., Milovanovi’c G.V. Interpolation processes. Basic theory and applications. — Berlin: Springer, – 2008. –262 p.
Schoenberg I.J. On trigonometric spline interpolation. J. Math. Mech. – 1964. – vol. 13. – P. 795–825.
Schumaker L.L. Spline functions: basic theory. Cambridge Univ. Press, – 2007. – 600 p.
Стечкин С.Б., Субботин Ю.Н. Сплайны в вычислительной математике. М.: Наука, – 1976. – 248 с.
Василенко В. А. Сплайн-функции: теория, алгоритмы, программы. Н.: Наука, – 1983. – 215 с.
Болтаев А.К., Шоназаров С.К. Система для нахождения оптимальных коэффициентов интерполяционных формул. Проблемы вычислительной и прикладной математики. Ташкент, – 2022. – № 5/1(44) – C. 54–63.
Соболев С.Л. Введение в теорию кубатурных формул. М. Наука, – 1974. – 808 с.
Sobolev S.L. The coefficients of optimal quadrature formulas, in: Selected works of S.L.Sobolev. Springer US, – 2006. – P. 561—566.
Соболев С.Л., Васкевич В.Л. Теория кубатурных формул. СО АН России. Новосибирск, – 1996. – 484 с.
Shadimetov Kh.M., Boltaev A.K. Optimal interpolation formulas on classes of differentiable functions. Uzbek Mathematical Journal. – 2021. – Vol.65. – Issue 4. – P. 166–74.
Владимиров В.С. Обобщенные функции в математической физике. М.: Наука, – 1979. – 320 с.
