Detailed atlas of attractors of zeros for the classical Bernstein polynomials

D. G. Tsvetkovich

Čelâbinskij fiziko-matematičeskij žurnal, Tome 3 (2018) no. 1, pp. 58-89.

Résumé

The distribution of zeros for the classical Bernstein polynomials taken from piecewise linear generating functions is discussed. The classification of attractors to which subtend zeros at increasing of number of the Bernstein polynomials is given. We have established that the attractor structure subjects to its regular rules. By means of systems of computer mathematics large series of examples for each of the formulated rules are considered and illustrations to all typical situations are given.

Détail

Keywords: Bernstein polynomials, zeros of polynomials, attractor of zeros.

Bibliographie

[1] G. G. Lorentz, Bernstein Polynomials, Univ. of Toronto Press, Toronto, 1953, 130 pp. | MR

[2] V.S. Videnskii, Bernstein polynomials, A.I. Gertsen Leningrad State Pedagogical University, Leningrad, 1990, 64 pp. (In Russ.)

[3] J. Bustamante, Bernstein Operators and Their Properties, Birkhäuser, Basel, 2017, 420 pp. | MR

[4] I.Ya. Novikov, “Asymptotics of the Roots of Bernstein Polynomials Used in the Construction of Modified Daubechies Wavelets”, Mathematical Notes, 71:1–2 (2002), 217–229 | DOI | DOI | MR | MR

[5] Tikhonov I.V., Sherstyukov V.B., Tsvetkovich D.G., “The problem on zeros of Bernstein polynomials for the model example with the symmetric module function”, Contemporary problems of functions theory and their applications, Proceedings of 18th International Saratov Winter School, Nauchnaya Kniga Publ., Saratov, 2016, 271–275 (In Russ.)

[6] Tikhonov I.V., Sherstyukov V.B., Tsvetkovich D.G., “Special problems for the Bernstein polynomials in a complex domain”, Some actual problems of contemporary mathematica and mathematical education. Gertsen's Readings — 2016, Proceedings of Scientific Conference, A.I. Gertsen Russian State Pedagogical University, Saint Petersburg, 2016, 139–145 (In Russ.)

[7] Tikhonov I.V., Sherstyukov V.B., Tsvetkovich D.G., “Computer classification of attractors for zeros of the Bernstein polynomials”, Systems of computer mathematics and their applications, Proceedings of XVII International Scientific Conference, Smolensk State University, Smolensk, 2016, 224–227 (In Russ.)

[8] Tikhonov I.V., Tsvetkovich D.G., Sherstyukov V.B., “Computer analysis of the attractors of zeros for classical Bernstein polynomials”, Fundamental and Applied Mathematics, 21:4 (2016), 198–214 (In Russ.)

[9] Tikhonov I.V., Sherstyukov V.B., Tsvetkovich D.G., “How do attractors of zeros for classical Bernstein polynomials look like”, Differential Equations and Control Processes, 2017, no. 2, 59–73 (In Russ.)

[10] Tikhonov I.V., Sherstyukov V.B., Petrosova M.A., “Bernstein polynomials: the old and the new”, Mathematical forum. (The results of Science in the South of Russia), 8:1 (2014), 126–175 (In Russ.)

[11] L. Kantorovič, “Sur la convergence de la suite des polynômes de S. Bernstein en dehors de l'intervalle fondamental”, Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et naturelles, 1931, no. 8, 1103–1115 (In French.)

[12] S. G. Gal., Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, 8, World Scientific, New Jersey–London–Singapore, 2009, 338 pp. | DOI | MR

[13] Tikhonov I.V., Sherstyukov V.B., “The module function approximation by Bernstein polynomials”, Bulletin of Chelyabinsk State University, 2012, no. 26, 6–40 (In Russ.)

[14] Wolfram Mathematica, (accessed 02.02.2018) https://www.wolfram.com/mathematica