Recovery of functional regularities based on Gegenbauer polynomials
Abstract
The choice of a base approximating function in the recovery model of functional dependencies in additive and multiplicative forms as Gegenbauer polynomials is justified. A comparative analysis of the applications of the approximating functions with the results of approximation with the help of the Chebyshev and Legendre polynomials, who are special cases of Gegenbauer polynomials is performed. It is shown that the Gegenbauer polynomials are more versatile and comfortable, allowing for a constant computational complexity to achieve a high accuracy of approximation for a wide range of restored dependencies.References
Geste T., Tibshirani R., Fridman D. Elementy statisticheskogo obucheniya // Intellektual’nyy analiz dannykh, vyvoda i prognozirovaniye posledovatel’nostey Springera v statistike. Springer. — 2008. — 764 s.
ZHuchko O.V., Pyt’yev YU.P. Vosstanovleniye funktsional’noy zavisimosti teoretiko-vozmozhnostnymi metodami // ZH. vychisl. matem. i matem. Fiz. — 2003. — 43, № 5. — S. 767–783.
Mihaila B., Mihaila I. Numerical approximations using Chebyshev polynomial expansions: El-gendi’s method revisited // J. Phys. A: Math. Gen. — 2002. — 35(43), № 5. — P. 731–746.
Kolmogorov А.N. O predstavlenii nepreryvnykh funktsiy neskol’kikh peremennykh v vide superpozitsii nepreryvnykh funktsiy odnogo peremennogo i slozheniya // DАN SSSR. — 1957. — 114, № 5. — S. 953–956.
Pankratova N.D. Formirovaniye tselevykh funktsiy v sistemnoy zadache kontseptual’noy neopredelennosti // Dop. NAN Ukrayiny. — 2000. — # 9. — S. 67–73.
Sayyed K.A.M., Metwally M.S., Batahan R.S. Gegenbauer matrix polynomials and second order matrix differential equations, Department of Mathematics // Divulgaciones Matem´aticas. — 2004. — 12, № 2. —P. 101–115.
David Gottlieb, Chi-Wang Shu. Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piece wise analytic function // Math. Comp. — 1995. — 64. — P. 1081–1095.
Guo Ben-yu. Gegenbauer approximation and its applications to differential equations on the whole line // Journal of Mathematical Analysis and Applications. — 1998. — 226, № 1 — P. 180–206.
Boyd John P. Trouble with Gegenbauer reconstruction for defeating Gibbs phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations // Journal of Computational Physics. — 2005. — 204. — P. 53–264.