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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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How Do Attractors of Zeros for Classical Bernstein Polynomials Look Like

Author(s):

Ivan Vladimirovich Tikhonov

Doctor of Physical and Mathematical Sciences,
Professor of Department of Mathematical Physics,
Faculty of Computational Mathematics and Cybernetics,
Lomonosov Moscow State University

ivtikh@mail.ru

Vladimir Borisovich Sherstyukov

Doctor of Physical and Mathematical Sciences,
Professor of Department of High Mathematics NRNU «MEPhI».
Kashirskoe shosse, 31,
Moscow, Russia, 115409.

shervb73@gmail.com

Diana Goranovna Tsvetkovich

Postgraduate Student of Mathematical Analysis Department,
Moscow State University of Education.
Mathematical Analysis Department,
Moscow State University of Education,
Krasnoprudnaya Str., 14,
Moscow, Russia, 107140.

Abstract:

The paper contains material for the plenary report at the scientific conference «Herzen Readings - 2017». The problem of the distribution of zeros for the classical Bernstein polynomials is discussed. The set of zeros is considered as a discrete dynamical system on the complex plane.For a fixed choice of the generating function we introduce a special mapping such that to the polynomial number corresponds the set of the polynomial zeros on the plane. The study is carried out by computer tools. The attention is paid to the existence of the attractor to which almost all zeros converge with increasing the number of the Bernstein polynomials. A list of six rules allows us to construct the attractor of zeros for a piecewise linear generating function. Examples and illustrations are given. Some unsolved problems are formulated. Prospective themes for further research are pointed out. Our exposition is supplemented with additional information.

Keywords

References:

  1. Novikov I. Ya. Asimptotika kornei polinomov Bernsteina, ispol'zuemyh v postroenii modificirovannyh vspleskov Dobeshi. Matem. zametki. 2002. T. 71. Vyp. 2. S. 239-253
  2. Tikhonov I. V., Sherstyukov V. B., Tsvetkovich D. G. Zadacha o nulyah polinomov Bernsteina na model'nom primere simmetrichnogo modulya. Sovremennye problemy teorii funkcii i ih prilozheniya: Materialy 18th mezhdunar. Sarat. zimnei schkoly. Saratov: OOO Izdatel'stvo “Nauchnaya kniga”, 2016. S. 271-275
  3. Tikhonov I. V., Sherstyukov V. B., Tsvetkovich D. G. Special'nye zadachi dlya polinomov Bernsteina v kompleksnoi oblasti. Nekotorye aktual'nye problemy sovremennoi matematiki i matematicheskogo obrazovaniya. Hercenovskie chteniya - 2016. Materialy nauchnoi konferencii, 11-15 aprelya 2016 g. SPb: Izd. RGPU im. A. I. Hercena, 2016. S. 139-145
  4. Tikhonov I. V., Sherstyukov V. B., Tsvetkovich D. G. Komp'yuternaya klassifikaciya attraktorov dlya nulei polinomov Bernsteina. Sistemy komp'yuternoi matematiki i ih prilozheniya: materialy XVII mezhdunarodnoi nauchnoi konferencii. Vypusk 17. Smolensk: SmolGU, 2016. S. 224-227
  5. Duren P., Herbig A. -K., Khavinson D. Robert Jentzsch, mathematician and poet. The Mathematical Intelligencer. 2008. Vol. 30. Issue 3. P. 18-24
  6. Jentzsch R. Untersuchungen zur Theorie der Folgen analytischer Funktionen. Acta Math. 1918. Vol. 41. P. 219-251
  7. Montel' P. Normal'nye semeistva analiticheskih funkcii. M. -L. : ONTI NKTP SSSR, 1936. 240 s
  8. Titchmarsh E. Teoriya funkcii. M. : Nauka, 1980. 463 s
  9. Rahman Q. I., Schmeisser G. Analytic Theory of Polynomials. Oxford: Clarendon Press, 2002. xiv+742 p
  10. Izumi S. On the Distribution of the Zero Points of Sections of a Power Series. Japanese Journal of Math. 1927. Vol. 4. P. 29-32
  11. Korobeinik Yu. F. O nekotoryh svoistvah funkcional'nyh ryadov. Uspekhi matem. nauk. 1960. T. 15. Vyp. 4. S. 149-156
  12. Lebedeva E. A. Ob odnom obobshchenii teoremy R. Jencha. Matem. zametki. 2010. T. 88. Vyp. 5. S. 753-758
  13. Bernstein S. N. Dé monstration du thé orè me de Weierstrass fondé e sur la calcul des probabilities. Soobshcheniya Har'kovskago Matematicheskago Obshchestva. 1912. T. 13. № 1. S. 1-2
  14. Lorentz G. G. Bernstein Polynomials. Toronto: Univ. of Toronto Press, 1953. x+130 p
  15. Videnskij V. S. Mnogochleny Bernsteina. Uchebnoe posobie k speckursu. L. : LGPI im. A. I. Hercena, 1990. 64 s
  16. Bustamante J. Bernstein Operators and Their Properties. Birkhä user, 2017. xii+420 p
  17. Kantorovich L. V. O skhodimosti posledovatel'nosti polinomov S. N. Bernsteina za predelami osnovnogo interval. Izvestiya AN SSSR. Ser. VII. Otdelenie matem. i estestv. nauk. 1931. № 8. S. 1103-1115
  18. Bernstein S. N. O skhodimosti mnogochlenov v kompleksnoi oblasti. Izvestiya AN SSSR. Ser. matem. 1943. T. 7. № 2. S. 49-88
  19. Videnskii V. S. Raboty L. V. Kantorovicha o polinomah S. N. Bernsteina. Vestnik SPbGU. Ser. 1. 2013. Vyp. 2. S. 50-53
  20. Gal S. G. Approximation by Complex Bernstein and Convolution Type Operators. (Series on Concrete and Applicable Mathematics. Vol. 8. ) New Jersey - London - Singapore: World Scientific, 2009. xii+338 p
  21. Tikhonov I. V., Sherstyukov V. B., Petrosova M. A. Polinomy Bernsteina: staroe i novoe. Matematicheskii forum. T. 8, ch. 1. Issledovaniya po matematicheskomu analizu. - Vladikavkaz: YUMI VNC RAN i RSO-A, 2014. S. 126-175

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