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Methods of function approximation and modern computer-based technologies (Review)

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Abstract

The paper reviews publications over the last five years devoted to efficient solutions of some important practical problems using interlineation and interflation (blending function interpolation) and blending function approximation.

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References

  1. J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications, Acad. Press, New York (1967).

    MATH  Google Scholar 

  2. O. M. Anikeyenko, O. M. Lytvyn, V. L. Rvachov, and M. O. Safonov, “A formula for expansion in the neighborhood of an angle,” Dop. URSR, Ser. A, No. 2, 99–101 (1972).

  3. K. E. Babenko and O. M. Lytvyn, “The mathematical modeling in the computer tomography with using direct Fourier method and spline-interlineation of the functions,” in: Proc. Int. Conf. ACIT-Signal and Image Processing (ACIT-SIP 2005), Novosibirsk, 20–24 June, 2005.

  4. Yu. P. Vashchuk and O. M. Lytvyn, “Recovery of the structure of a 3D object based on tomograms in three systems of mutually perpendicular planes and blending approximation with Bernstein polynomials,” in: Proc. Sci. Conf. UkrObraz’204 [in Ukrainian], Kyiv (2004), pp. 210–213.

  5. L. I. Gulik and O. M. Lytvyn, “An approach to solving the Dirichlet boundary-value problem for the Poisson equation using interlineation,” Dop. NANU, No. 1, 13–18 (2004).

  6. L. I. Gulik and O. M. Lytvyn, “A new approach to solving the Dirichlet boundary-value problem for the Poisson equation using interlineation of functions,” Komp. Matem., No. 1 (2004).

  7. L. I. Gulik and O. M. Lytvyn, “Interlineation of functions of three variables on a system of orthogonal straight lines,” Dop. NANU, No. 11, 16–22 (2004).

    Google Scholar 

  8. L. I. Gulik, “Exact satisfaction of the boundary Dirichlet conditions on the boundary of a three-dimensional domain of complex form using interflation,” in: Abstracts of Interstate Sci.-Method. Conf., Dneprodzerzhinsk, 26-28 May 2004 [in Russian] (2004), pp. 14–15.

  9. L. I. Gulik and O. M. Lytvyn, “Exact satisfaction of boundary conditions for a three-dimensional domain of complex form using interflation,” in: Proc. Intern. Conf. on Optimization of Computations (POO-XXXII) [in Ukrainian], Inst. Cybern., Kyiv (2005).

    Google Scholar 

  10. L. I. Gulik and O. N. Litvin, “Interflation of functions of three variables on pyramids with one curvilinear face,” Cybern. Syst. Analysis, 41, No. 6, 818–831 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  11. E. I. Drobot, O. M. Lytvyn, and I. V. Sergienko, “Numerical implementation of the LIDE method for the nonstationary heat-conduction equation with three spatial variables,” Dop. NANU, No. 2, 67–73 (2000).

  12. O. F. Kalaida, “On two-sided approximations of functions by polynomials,” Vych. Prikl. Matem., Issue 67, 31–37 (1989).

    Google Scholar 

  13. V. V. Kamyshan, O. M. Lytvyn, and O. R. Maksymovych, “On numerical implementation of the interlineational method of finite elements for the Poisson equation in domains composed of rectangulars,” Dop. NAN Ukr., Ser. A, No. 11, 34–38 (1995).

  14. N. P. Korneichuk, Splines in the Theory of Approximation [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  15. S. I. Kulik and O. M. Lytvyn, “Generalized Haar operators based on a two-dimensional blended approximation by Haar wavelets,” in: Proc. Sci. Conf. UkrObraz’2004 [in Ukrainian], Kyiv (2004), pp. 225–228.

  16. S. I. Kulik and O. M. Lytvyn, “On numerical implementation of Haar wavelet transforms of one and two variables, constructed using experimental data,” in: Proc. Intern. Conf. on Optimization of Calculations (POO-XXXII) [in Ukrainian], Kyiv (2005).

  17. L. N. Kutsenko and O. N. Litvin, “Generalized interpolation on triangulated domains and its application in FEM,” Dep. VINITI, January 13 1976, Pt. 1, No. 1379; February 20 1977, Pt. 2, No. 2303.

  18. O. M. Lytvyn, Interlineation of Functions and Some Applications [in Ukrainian], Kyiv (2002).

  19. O. N. Litvin, “Polynomial Taylor’s interlineation of functions of two variables on several straight lines,” Izv. Vuzov., Mat., No. 12, 19–27 (1989).

  20. O. N. Litvin, “Polynomial interflation of functions of three variables on M (M ≥ 3 ) intersected planes without preserving the class C r(R 3),” Dokl. AN USSR, No. 11, 35–41 (1991).

  21. O. M. Lytvyn, “Interlineation and interflation functions of many variables (blending function interpolation) and economical algorithms in the approximation theory,” in: Proc. Int. Conf. Comput. Methods, ICCM-2004, Singapour, 15-17 Dec. (2004).

  22. O. M. Lytvyn, “Interlineation and interflation functions of many variables (the blending function interpolation) and multidimensional signal processing,” Int. Conf. ACIT-Signal and Image Processing (ACIT-SIP 2005), Novosibirsk, 20–24 June (2005).

  23. O. M. Lytvyn, Computational Methods. Additional Sections: A Textbook [in Ukrainian], Naukova Dumka, Kyiv (2005).

    Google Scholar 

  24. O. M. Lytvyn and O. P. Nechuiviter, “Computing Fourier coefficients using formulas of spline interlineation with splines of the second order,” in: Proc. Intern. Sci. Conf. UkrObraz’2004 [in Ukrainian], Kyiv (2004), pp. 214–217.

  25. O. M. Lytvyn and V. V. Mikhalkin, “Computing Fourier coefficients using Radon data based on the Filon formula,” in: Proc. Intern. Sci. Conf. UkrObraz’2004 [in Ukrainian], Kyiv (2004), pp. 205–209.

  26. O. M. Lytvyn and V. V. Mikhalkin, “Approximate solution of a problem of X-ray and computer-aided tomography by using piecewise-constant splines,” in: Proc. Intern. Conf on Optimization of Computations (POO-XXXII) [in Ukrainian], Kyiv (2005).

  27. O. N. Litvin and V. A. Pasechnik, “Optimizing a mathematical model of a 3D body surface,” Cybern. Syst. Analysis, 42, No. 1, 90–97 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  28. O. M. Lytvyn and Yu. I. Pershina, “Recovering 3-D objects from their traces on a system of intersecting planes using interflation of functions,” in: Proc. Intern. Sci. Conf. UkrObraz’2004 [in Ukrainian], Kyiv (2004), pp. 221–224.

  29. O. Lytvyn and Yu. Pershina, “Reconstruction of three-dimensional objects under their tomograms on system of three groups of the crossed planes with use interflation of functions,” in: Proc. Int. Conf. ACIT-Signal and Image Processing (ACIT-SIP 2005), Novosibirsk, 20–24 June (2005).

  30. O. M. Lytvyn and V. M. Udovichenko, “Operators of two-dimensional finite discrete-continuous Fourier transform that are based on piecewise-constant splines and exact on trigonometric polynomials of prescribed order,” in: Vestn. NTU “KhPI”, Special Issue: Power and Thermal Processes and Equipment, 9’2003, Vol. 2 [in Ukrainian], Kharkov (2003), pp. 108–117.

    Google Scholar 

  31. O. M. Lytvyn and V. M. Udovichenko, “Operators of two-dimensional finite discrete-continuous Fourier transform that are based on first-order splines and exact on trigonometric polynomials of prescribed order,” in: Vestn. NTU “KhPI”, Special Issue: Radiophysics and Ionosphere, 7’2003, Vol. 4 [in Ukrainian], Kharkov (2003), pp. 49–58.

    Google Scholar 

  32. O. M. Lytvyn and V. M. Udovichenko, “Operators of two-dimensional finite discrete-continuous Hartley transform that are based on piecewise-constant splines and exact on trigonometric polynomials of prescribed order, ” in: Vestn. NTU “KhPI”, Special Issue: Automation and Instrumentation, 7’2003, Vol. 3 [in Ukrainian], Kharkov (2003), pp. 105–114.

    Google Scholar 

  33. O. M. Lytvyn and V. M. Udovichenko, “Three-dimensional finite Fourier and Hartley transforms using interflation of functions,” in: Vestn. NTU “KhPI”, Special Issue: Automation and Instrumentation, 16’2005 [in Ukrainian], Kharkov (2005), pp. 117–157.

  34. O. M. Lytvyn and V. M. Udovichenko, “A generalization of the concept of trend for nonperiodic functions of two variables,” in: Proc. Sci. Conf. UkrObraz’2004 [in Ukrainian], Kyiv (2004), pp. 217–220.

  35. O. M. Lytvyn and V. M. Udovichenko, “Operators of finite three-dimensional discrete-continuous Hartley transform that are based on the Filon method and three-linear splines and exact on trigonometric polynomials of a prescribed order,” in: Radioelectronic and Computer Systems [in Ukrainian], KhAI, Kharkov, No. 2 (6), April–June (2004), pp. 123–127.

    Google Scholar 

  36. O. M. Lytvyn and V. M. Udovichenko, “Operators of finite three-dimensional discrete-continuous Hartley transform that are based on the Filon method and piecewise-constant splines and exact on trigonometric polynomials of a prescribed order,” in: Vestn. NTU “KhPI,” Special Issue: Dynamics and Strength of Machines, 19’2004 [in Ukrainian], Kharkov (2004), pp. 133–140.

  37. O. M. Lytvyn and V. M. Udovichenko, “Two-dimensional finite Fourier and Hartley transforms using interlineations of functions,” in: Vestn. NTU “KhPI,” Special Issue: Automation and Instrumentation, 7’2005 [in Ukrainian], Kharkov (2005), pp. 100–111.

  38. O. M. Lytvyn and V. M. Udovichenko, “Operators of finite three-dimensional Fourier transform,” Radioelektronika Informatika, No. 4 (29), 130–133 (2004).

  39. O. M. Lytvyn and V. M. Udovichenko, “Operators of finite three-dimensional discrete-continuous Fourier transform that are based on the Filon method and three-linear splines and are exact on trigonometric polynomials of prescribed order,” in: Inform.-Keruyuchi Systemy na Zalizn. Transporti, 1, 2(51, 52), 19–23 (2005).

    Google Scholar 

  40. O. M. Lytvyn and V. M. Udovichenko, “Three-dimensional Fourier and Hartley transforms using interflation of functions in: Vestn. NTU “KhPI,” Special Issue: Automation and Instrumentation, 38, No. 5 (2005).

  41. O. N. Litvin and V. V. Fed’ko, “Generalized piecewise-Hermitian interpolation,” Ukr. Mat. Zh., 28, No. 6, 812–819 (1976).

    MATH  MathSciNet  Google Scholar 

  42. O. N. Litvin and V. V. Fed’ko, “On a solution of the first boundary-value problem for the biharmonic equation using the Hermite generalized interpolation,” Dep. VINITI, Sept. 21, 1976, No. 4329.

  43. A. M. Pidgornyi and O. M. Lytvyn, “A method of solving boundary-value problems for partial derivative equations by reducing them to systems of ordinary integrodifferential equations,” Dop. AN URSR, Ser. A, No. 6, 522–525 (1974).

  44. V. L. Rvachev, Geometrical Application of Logic Algebra [in Russian], Tekhnika, Kyiv (1967).

    Google Scholar 

  45. I. V. Sergienko and O. M. Lytvyn, “Numerical implementation the LIDE method for a heat-conduction equation,” Dop. AN URSR, Ser. A, No. 10, 69–73 (1990).

  46. J. C. Cavendish, W. J. Gordon, and C. A. Hall, “Ritz-Galerkin approximation in blending function spaces,” Numer. Mathem., 26, No. 2, 155–178 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  47. S. A. Coons, “Surface for computer-aided design of space forms,” in: Project MAC report MAC-TR-41, Cambridge, June (1967), pp. 3–30.

  48. W. Haussmann, K. Getter, and B. Steinhaus, “Degree of best approximation by trigonometric blending functions,” Math. Ztschr., 189, No. 1, 143–150 (1985).

    Article  MATH  Google Scholar 

  49. W. Haussmann and K. Zeller, “Blending interpolation and best L 1-approximation,” Arch. Math. (Basel), 40, No. 6, 545–552 (1983).

    MATH  MathSciNet  Google Scholar 

  50. D. I. Mangeron, “Sopra un problema al contorno per un’equatione differentiable alle derivate parziali di quartordine conle caratteristice reali dopie,” Rend. Accad. Sci. Fis. Mat. Napoli., No. 2, 28–40 (1932).

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Authors and Affiliations

  1. Ukrainian Engineering and Pedagogical Academy, Kharkov, Ukraine

    O. N. Litvin

  2. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    I. V. Sergienko

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  2. I. V. Sergienko
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Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 64–81, January–February 2007.

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Litvin, O.N., Sergienko, I.V. Methods of function approximation and modern computer-based technologies (Review). Cybern Syst Anal 43, 50–65 (2007). https://doi.org/10.1007/s10559-007-0025-y

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