The development of approximation theory and some proposed applications

  • Authors

    • Malik Saad Al-Muhja School of Quantitative Sciences, College of Arts and Sciences, University Utara Malaysia, Sintok 06010
    • Masnita Misiran School of Quantitative Sciences, College of Arts and Sciences, University Utara Malaysia, Sintok 06010, Malaysia.
    • Zurni Omar School of Quantitative Sciences, College of Arts and Sciences, University Utara Malaysia, Sintok 06010, Malaysia.
    2019-07-31
    https://doi.org/10.14419/ijet.v7i4.24816
  • Approximation Theory, Chebyshev’s Polynomial, Weierstrass’ Theorem, 2010 MSC, 42A10, 42A15, 41A52, 41Axx.

  • Abstract

    In this survey article, we review the early history of approximation theorem that was introduced by Weierstrass in the late 18th century, together with its extension works. We also propose some applicable scenarios that best fit this theory. Such applications include the kinetics conditions related to manual and automatic vehicle transmission by using convex function, and the theory of calculating deviations of eye’s layers (normal vision, hyperopia and myopia) in some patients by using monotone function.

     

     

     

     


  • References

    1. [1] Knuth, D., The art of computer programming: fundamental algorithms, AN: Addison Wesley Longman, (1969).

      [2] Nikol’skii, S., "Approximation of periodic functions by trigonometrical polynomials", Travaux Inst. Math. Stekloff, Vol. 15, (1945), pp. 3–76.

      [3] Kolmogorov, A., Selected works: Mathematics and Mechanics, Moscow Leningrad, ML: Izdatel'stvo Akad. Nauk SSR, (1985).

      [4] Steffens, K., The history of approximation theory: From Euler to Bernstein, New York, NY: Birkhauser Boston, (2006).

      [5] Akhiezer, N., General theory of Chebyshev polynomials, In: Scientific heritage of P.L. Chebyshev, Moscow Leningrad, ML: Izdatel'stvo Akad. Nauk SSR, (1945).

      [6] Goncharov, V., Theory of best approximation of functions, In: Scientific heritage of P.L. Chebyshev, Moscow Leningrad, ML: Izdatel'stvo Akad. Nauk SSR, (1945).

      [7] Kolmogorov, A., & Tikhomirov, V., "∈-entropy and ∈-capacity of sets in function space", Usp. Mat. Nauk, Vol. 14, No. 2, (1959), pp. 3–86.

      [8] Tikhomirov, V. M., Convex analysis, in: Gamkrelidze, R. V., (Ed.). Analysis II: Convex analysis and approximation theory, New York, NY: Encyclopaedia of Mathematical Sciences, (1990). https://doi.org/10.1007/978-3-642-61267-1_1.

      [9] Osborne, M., Locally convex spaces, New York, NY: Springer International Publishing Switzerland, (2014). https://doi.org/10.1007/978-3-319-02045-7_3.

      [10] Ng, K-F. "On the Stone-Weierstrass theorem", Journal of the Australian Mathematical Society, Vol. 21, (1976), pp. 337–340. https://doi.org/10.1017/S1446788700018632.

      [11] Kopotun, K., Leviatan, D., & Shevchuk, I., "Convex polynomial approximation in the uniform norm: conclusion", Canadian Journal of Mathematics, Vol. 57, No. 6, (2005), pp. 1224–1248. https://doi.org/10.4153/CJM-2005-049-6.

      [12] Kopotun, K., Leviatan, D., & Shevchuk, I., "Coconvex approximation in the uniform norm: the final frontier", Acta Mathematica Hungarica, Vol. 110, (2006), pp. 117–151. https://doi.org/10.1007/s10474-006-0010-3.

      [13] Leviatan, D., & Shevchuk, I., "Coconvex approximation", Journal of Approximation Theory, Vol. 118, (2002), pp. 20–65. https://doi.org/10.1006/jath.2002.3695.

      [14] Kopotun, K., & Listopad, V., "On monotone and convex approximation by algebraic polynomials", Ukrainian Mathematical Journal, Vol. 9, No. 46, (1994), pp. 1393–1398. https://doi.org/10.1007/BF01059430.

      [15] K. A. Kopotun and D. a. S. I. A. Leviatan, "Interpolatory pointwise estimates for monotone polynomial approximation", J. Math. Anal. Appl., Vol. 2, No. 459, (2018), pp. 1260–1295. https://doi.org/10.1016/j.jmaa.2017.11.038.

      [16] Kopotun, K., Leviatan, D., & Shevchuk, I., "Are the degree of best (co)convex and unconstrained polynomial approximation the same?", Acta Mathematica Hungarica, Vol. 3, No. 123, (2009), pp. 273–290. https://doi.org/10.1007/s10474-009-8111-4.

      [17] Kopotun, K., Leviatan, D., & Shevchuk, I., "Are the degree of the best (co)convex and unconstrained polynomial approximation the same? II", Ukrainian Mathematical Journal, Vol. 3, No. 62, (2010), pp. 420–440. https://doi.org/10.1007/s11253-010-0362-2.

      [18] Leviatan, D., "Monotone and comonotone polynomial approximation revisited, Journal of Approximation Theory", No. 53, (1988), pp. 1–16. https://doi.org/10.1016/0021-9045(88)90071-8.

      [19] DeVore, R., Hu, Y., & Leviatan, D., "Convex Polynomial and Spline Approximation in , ", Constr. Approx., No. 12, (1996), pp. 409–422. https://doi.org/10.1007/BF02433051.

      [20] Leviatan, D., & Petrova, I., "Interpolatory estimates in monotone piecewise polynomial approximation", Journal of Approximation Theory, No. 223, (2017), pp. 1–8. https://doi.org/10.1016/j.jat.2017.07.006.

      [21] Gustavsson, A., Development and analysis of synchronization process control algorithms in a dual clutch transmission, Ph.D. thesis, (2009).

      [22] Zemmrich, T., "New automated five-speed manual transmission–easytronic 3.0", ATZ worldwide, Vol. 6, No. 117, (2015), pp. 34–37. https://doi.org/10.1007/s38311-015-0021-1.

      [23] Rosander, P., Bednarek, G., Seetharaman, S., & Kahraman, A., "Development of an efficiency model for manual transmissions", ATZ worldwide, Vol. 4, No. 110, (2008), pp. 36–43. https://doi.org/10.1007/BF03225001.

      [24] A. A. 2. YouTube, "The new S-tronic 7 Speed transmission". [Film].

      [25] T. website, "https://www.allaboutvision.com/resources/anatomy.htm," All About Vision. [Online].

      [26] Ao M., Feng, Y., Xiao, G., Xu, Y., & Hong, J., "Clinical outcome of Descemet stripping automated endothelial keratoplasty in 18 cases with iridocorneal endothelial syndrome", Eye, Vol. 32, No. 4, (2017), pp. 1–8. https://doi.org/10.1038/eye.2017.282.

      [27] Eghrari, A., Riazuddin, S., & Gottsch, J., "Overview of the cornea: structure, function, and development", Prog. Mol. Biol. Transl. Sci., No. 134, (2015), pp. 7–23. https://doi.org/10.1016/bs.pmbts.2015.04.001.

      [28] Priyadarsini, S., Rowsey, T., Ma, J-X., & Karamichos, D., "Unravelling the stromal-nerve interactions in the human diabetic cornea", Experimental Eye Research, No. 164, (2017), pp. 22–30. https://doi.org/10.1016/j.exer.2017.08.003.

  • Downloads

  • How to Cite

    Saad Al-Muhja, M., Misiran, M., & Omar, Z. (2019). The development of approximation theory and some proposed applications. International Journal of Engineering & Technology, 8(2), 90-94. https://doi.org/10.14419/ijet.v7i4.24816

    Received date: 2018-12-26

    Accepted date: 2019-06-08

    Published date: 2019-07-31