On the connection of Bernstein and Kantorovich polynomials for a symmetric module function

I. V. Okorochkov   ; I. V. Tikhonov   ; V. B. Sherstyukov

Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 1, pp. 87-99

Résumé

The paper is based on the report which made by authors at the XVI International Scientific Conference “Order analysis and related problems of mathematical modeling. Operator theory and differential equations” (Vladikavkaz, September 2021). A brief review of our recent results is presented. We study the connection of Bernstein and Kantorovich polynomials for an important example with the symmetric module function. It is well known that such nonsmooth functions play a special role in approximation theory. By means of the obtained relations, the investigation of Kantorovich polynomials can be reduced to the using of the Bernstein polynomials properties. In particular, the deviation of Kantorovich polynomials from the symmetric module function is considered. In addition to accurate two-sided estimates on the interval $[0,1]$, a simple asymptotic formula for deviation is noted. The character of the convergence of Kantorovich polynomials differs from that of Bernstein polynomials give on the interval $[0,1]$. We also present new results on the convergence of Kantorovich polynomials in the complex plane. The convergence set is the same as for Bernstein polynomials. This is so-called Kantorovich compact, which limited by the lemniscate $|4z(1-z)|=1$. Everywhere here the rate of convergence of Kantorovich polynomials is established. In view of the limited size of the article, we present only the schemes of proofs. The proofs in details is planned to be given separately.

Bibliographie

[1] Lorentz G. G., Bernstein Polynomials, University of Toronto Press, Toronto, 1953, x+130 pp. | MR | Zbl

[2] DeVore R. A., Lorentz G. G., Constructive Approximation, Springer-Verlag, Berlin–Heidelberg–N. Y., 1993, x+450 pp. | MR | Zbl

[3] Videnskij V. S., Bernstein Polynomials, Textbook for the Special Course, LSPI n. a. A. I. Herzen, L., 1990, 64 pp. (in Russian)

[4] Bustamante J., Bernstein Operators and Their Properties, Birkhauser, Basel, 2017, xii+420 pp. | MR | Zbl

[5] Tikhonov I. V., Sherstyukov V. B., Petrosova M. A., “Bernstein Polynomials: the Old and the New”, Studies on Mathematical Analysis, Math. Forum. Results of Science. South of Russia, 8, no. 1, SMI VSC RAS and RNO-A, Vladikavkaz, 2014, 126–175 (in Russian)

[6] Tikhonov I. V., Sherstyukov V. B., “Approximation of the Module Function with Bernstein Polynomials”, Bulletin of Chelyabinsk State University. Matematics. Mechanics. Informatics, 15:26 (2012), 6–40 (in Russian)

[7] Tikhonov I. V., Sherstyukov V. B., “Approximation of the Module Function with Bernstein Polynomials: New Advances and Possible Generalizations”, Modern problems of the theory of functions and their applications, materials of the 20th international Saratov winter school, Nauchnaya kniga, Saratov, 2020, 409–414 (in Russian)

[8] Tikhonov I. V., Sherstyukov V. B., Tsvetkovich D. G., “Generalized Popoviciu Expansions for Bernstein Polynomials of a Rational Module”, Proceedings of the Voronezh Winter Mathematical School “Modern Methods of Function Theory and Related Problems” (January 28 – February 2, 2019), Progress in Science and Technology. Contemporary Mathematics and Its Applications. Thematic Surveys, 170, 2019, 71–117 (in Russian) | DOI

[9] Tikhonov I. V., Sherstyukov V. B., Tsvetkovich D. G., “Comparative Analysis of Two-Sided Estimates of the Central Binomial Coefficient”, Chelyabinsk Physical and Mathematical Journal, 5:1 (2020), 70–95 (in Russian) | DOI | MR | Zbl

[10] Popov A. Yu., “Two-Sided Estimates of the Central Binomial Coefficient”, Chelyabinsk Physical and Mathematical Journal, 5:1 (2020), 56–69 (in Russian) | DOI | MR | Zbl

[11] Popov A. Yu., “The Upper Bound of the Remainder of Power Series with Positive Coefficients of a Special Class”, Chelyabinsk Physical and Mathematical Journal, 2:2 (2017), 192–197 (in Russian)

[12] Graham R. L., Knuth D. E., Patashnik O., Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley Longman Publ. Co, Inc., Reading, Massachusetts, 1994, ix+658 pp. | MR | MR | Zbl

[13] Telyakovskii S. A., “On the Approximation of Differentiable Functions by Bernstein Polynomials and Kantorovich Polynomials”, Proc. Steklov Inst. Math., 260 (2008), 289–296 | Zbl

[14] Kantoroviĉ L. V., “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 nat, 1931, no. 8, 1103–1115

[15] Gal S. G., Approximation by Complex Bernstein and Convolution Type Operators, World Scientific, New Jersey–London–Singapore, 2009, xii+338 pp. | MR | Zbl

[16] Tikhonov I. V., Tsvetkovich D. G., Sherstyukov V. B., “Computer Analysis of the Attractors of Zeros for Classical Bernstein Polynomials”, Journal of Mathematical Sciences, 245:2 (2020), 217–233 | DOI | MR | Zbl

[17] 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 Russian) | Zbl

[18] Tikhonov I. V., Sherstyukov V. B., Tsvetkovich D. G., “On Some Method for Finding the Convergence Domain of Bernstein Polynomials in the Complex Plane”, Some Actual Problems of Modern Mathematics and Mathematical Education. Herzen Readings – 2018, Publishing House of RSPU n. a. A. I. Herzen, Saint Petersburg, 2018, 145–153 (in Russian)

[19] Tsvetkovich D. G., “Detailed Atlas of Attractors of Zeros for the Classical Bernstein Polynomials”, Chelyabinsk Physical and Mathematical Journal, 3:1 (2018), 58–89 (in Russian) | MR | Zbl