Geodesic
Algebraic representation for Bernstein polynomials on the symmetric interval and combinatorial relations
Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 3, pp. 68-86 
M. A. Petrosova; I. V. Tikhonov; V. B. Sherstyukov. Algebraic representation for Bernstein polynomials on the symmetric interval and combinatorial relations. Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 3, pp. 68-86. http://geodesic.mathdoc.fr/item/VMJ_2019_21_3_a5/
@article{VMJ_2019_21_3_a5,
     author = {M. A. Petrosova and I. V. Tikhonov and V. B. Sherstyukov},
     title = {Algebraic representation for {Bernstein} polynomials on the symmetric interval and combinatorial relations},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {68--86},
     year = {2019},
     volume = {21},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2019_21_3_a5/}
}
TY  - JOUR
AU  - M. A. Petrosova
AU  - I. V. Tikhonov
AU  - V. B. Sherstyukov
TI  - Algebraic representation for Bernstein polynomials on the symmetric interval and combinatorial relations
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2019
SP  - 68
EP  - 86
VL  - 21
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VMJ_2019_21_3_a5/
LA  - ru
ID  - VMJ_2019_21_3_a5
ER  - 
%0 Journal Article
%A M. A. Petrosova
%A I. V. Tikhonov
%A V. B. Sherstyukov
%T Algebraic representation for Bernstein polynomials on the symmetric interval and combinatorial relations
%J Vladikavkazskij matematičeskij žurnal
%D 2019
%P 68-86
%V 21
%N 3
%U http://geodesic.mathdoc.fr/item/VMJ_2019_21_3_a5/
%G ru
%F VMJ_2019_21_3_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

We pose the question of explicit algebraic representation for Bernstein polynomials. The general statement of the problem on an arbitrary interval $[a,b]$ is briefly discussed. For completeness, we recall Wigert formulas for the polynomials coefficients on the standard interval $[0,1]$. However, the focus of the paper is the case of the symmetric interval $[-1,1]$, which is of fundamental interest for approximation theory. The exact expressions for the coefficients of Bernstein polynomials on $[-1,1]$ are found. For the interpretation of the results we introduce a number of new numerical objects named Pascal trapeziums. They are constructed by analogy with a classical triangle, but with their own “initial” and “boundary” conditions. The elements of Pascal trapeziums satisfy various relations which remind customary combinatorial identities. A systematic research on such properties is fulfilled, and summaries of formulas are given. The obtained results are applicable for the study of the behavior of the coefficients in Bernstein polynomials on $[-1,1]$. For example, it appears that there exists a universal connection between two coefficients $a_{2m,m}(f)$ and $a_{m,m}(f)$, and this is true for all $m\in\mathbb N$ and for all functions $f\in C[-1,1]$. Thus, it is set up that the case of symmetric interval $[-1,1]$ is essentially different from the standard case of $[0,1]$. Perspective topics for future research are proposed. A number of this topics is already being studied.

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

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

[3] Natanson I. P., Constructive Theory of Functions, GITTL, M.–L., 1949, 688 pp. (in Russian) | MR

[4] Korovkin P. P., Linear Operators and the Theory of Approximation, Fizmatgiz, M., 1959, 212 pp. (in Russian)

[5] Davis P. J., Interpolation, Approximation, Dover, N.Y., 1975, xvi+394 pp. | MR | Zbl

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

[7] G. M. Phillips, Interpolation and Approximation by Polynomials, Springer, N.Y.–Berlin–Heidelberg, 2003, xiv+312 pp. | MR | Zbl

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

[9] Tikhonov I. V., Sherstyukov V. B., Petrosova M. A., “Gluing rule for Bernstein polynomials on the symmetric interval”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 15:3 (2015), 288–300 (in Russian) | DOI | Zbl

[10] S. Wigert, “Réflexions sur le polynome d'approximation $\sum_{\nu=0}^n\binom\nu n \varphi(\frac\nu n)x^\nu(1-x)^{n-\nu}$”, Arkiv för Matematik, Astronomi och Fysik, 20:2 (1927), 1–15

[11] Dzyadyk V. K., Introduction to the Theory of Uniform Approximation of Functions by Polynomials, Nauka, M., 1977, 512 pp. (in Russian)

[12] Graham R. L., Knuth D. E., Patashnik O., Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley Longman Publ. Co., Inc., Boston, 1994, 672 pp. | MR | MR | Zbl

[13] Tikhonov I. V., Sherstyukov V. B., “On the Behavior of the Coefficients of Bernstein Polynomials as Algebraic Notation on a Standard Interval”, Some Current Issues Modern Mathematics and Math. Education, Publishing House of RSPU n.a. A. I. Herzen, Saint Petersburg, 2015, 115–121 (in Russian)

[14] Tikhonov I. V., Sherstyukov V. B., Petrosova M. A., “Explicit Expressions for Coefficients Bernstein Polynomials in Algebraic Notation on a Symmetric Interval”, Some Current Issues Modern Mathematics and Math. Education, Publishing House of RSPU n.a. A. I. Herzen, Saint Petersburg, 2015, 121–124 (in Russian)

[15] Petrosova M. A., Tikhonov I. V., Sherstyukov V. B., “The Case of a Symmetric Interval in the Theory of Classical Bernstein Polynomials”, Computer Mathematics Systems and Their Applications, 15, Smolensk State Univ., Smolensk, 2014, 184–186 (in Russian)

[16] Petrosova M. A., Tikhonov I. V., Sherstyukov V. B., “Combinatorial Relations Related with Bernstein Polynomials on a Symmetric Interval”, Computer Mathematics Systems and Their Applications, 17, Smolensk State Univ., Smolensk, 2016, 177–182 (in Russian)

[17] J. D. Stafney, “A permissible restriction on the coefficients in uniform polynomial approximation to $C[0, 1]$”, Duke Math. J., 34:3 (1967), 393–396 | MR | Zbl

[18] J. A. Roulier, “Permissible bounds on the coefficients of approximating polynomials”, J. Approx. Theory, 3:2 (1970), 117–122 | MR | Zbl

[19] V. I. Gurarii, M. A. Meletidi, “A Functional Analysis and Its Applications”, Funct. Anal. Appl., 5:1 (1971), 60–62 | DOI | MR

[20] N. E. Nörlund, Vorlesungenuber Differenzenrechnung, Springer Verlag, Berlin, 1924, ix+551 pp.

[21] Tikhonov I. V., Sherstyukov V. B., Petrosova M. A., “New Research, Associated with the Algebraic Representation of Bernstein Polynomials on a Symmetric Interva”, Computer Mathematics Systems and Their Applications, 19, Smolensk State Univ., Smolensk, 2018, 336–347 (in Russia)

[22] P. Feinsilver, J. Kocik, “Krawtchouk polynomials and Krawtchouk matrices”, Recent Advances in Applied Probability, eds. Baeza-Yates R., Glaz J., Gzyl H., Hüsler J., Palacios J. L., Springer, Boston, MA, 2005, 115–141 | MR | Zbl

[23] Ivchenko G. I., Medvedev Yu. I., Mironova V. A., “Analysis of the Spectrum of Random Symmetric Boolean Functions”, Mat. Vopr. Kriptogr., 4:1 (2013), 59–76 (in Russian) | DOI

[24] Ivchenko G. I., Medvedev Yu. I., Mironova V. A., “Krawtchouk Polynomials and their Applications in Cryptography and Coding Theory”, Mat. Vopr. Kriptogr., 6:1 (2015), 33–56 (in Russian) | DOI | MR