Geodesic
The module function approximation by Bernstein polynomials
Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika, no. 15 (2012), pp. 6-40 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We obtain all the basic formulas connected with the approximation of $f(x)=|2x-1|$ by Bernstein polynomials. The estimates of the maximal coefficients in the polynomials are given. Degree of approximation for different points $x\in[0,1]$ is investigated. It is shown that near the point $x=1/2$ acceptable degree of approximation achieved on polynomials which are not computable from a practical point of view.
Keywords: the module function approximation, Bernstein polynomials. 
@article{VCHGU_2012_15_a1,
     author = {I. V. Tikhonov and V. B. Sherstyukov},
     title = {The module function approximation by {Bernstein} polynomials},
     journal = {Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika},
     pages = {6--40},
     year = {2012},
     number = {15},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VCHGU_2012_15_a1/}
}
TY  - JOUR
AU  - I. V. Tikhonov
AU  - V. B. Sherstyukov
TI  - The module function approximation by Bernstein polynomials
JO  - Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika
PY  - 2012
SP  - 6
EP  - 40
IS  - 15
UR  - http://geodesic.mathdoc.fr/item/VCHGU_2012_15_a1/
LA  - ru
ID  - VCHGU_2012_15_a1
ER  - 
%0 Journal Article
%A I. V. Tikhonov
%A V. B. Sherstyukov
%T The module function approximation by Bernstein polynomials
%J Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika
%D 2012
%P 6-40
%N 15
%U http://geodesic.mathdoc.fr/item/VCHGU_2012_15_a1/
%G ru
%F VCHGU_2012_15_a1
I. V. Tikhonov; V. B. Sherstyukov. The module function approximation by Bernstein polynomials. Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika, no. 15 (2012), pp. 6-40. http://geodesic.mathdoc.fr/item/VCHGU_2012_15_a1/

[1] S. Bernstein, “Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités”, Soobscheniya Kharkov. mat. o-va. Vtoraya seriya, 13:1 (1912), 1–2 | MR | Zbl

[2] I. N. Khlodovskii, “O nekotorykh svoistvakh polinomov S. N. Bernshteina”, Trudy pervogo Vsesoyuznogo s'ezda matematikov (Kharkov 1930 g.), ONTI NKTP SSSR, M.–L., 1936, 22

[3] E. V. Voronovskaya, “Opredelenie asimptoticheskogo vida priblizheniya funktsii polinomami S. N. Bernshteina”, Dokl. Akad. nauk CCCR, 1932, no. 4, 79–85 | Zbl

[4] Tiberiu Popoviciu, “Sur l'approximation des fonctions convexes d'ordre supérieur”, Mathematica, 10 (1935), 49–54 | Zbl

[5] G. G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953 | MR | Zbl

[6] P. C. Sikkema, “Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein–Polynomen”, Numerische Mathematik, 3:1 (1961), 107–116 | DOI | MR | Zbl

[7] I. P. Natanson, Konstruktivnaya teoriya funktsii, GITTL, M.–L., 1949 | MR

[8] V. L. Goncharov, Teoriya interpolirovaniya i priblizheniya funktsii, GITTL, M., 1954 | MR

[9] P. P. Korovkin, Lineinye operatory i teoriya priblizhenii, GIFML, M., 1959

[10] T. J. Rivlin, An Introduction to the Approximation of Functions, Blaisdell Publ., Waltham–Toronto–London, 1969 | MR | Zbl

[11] P. J. Davis, Interpolation and Approximation, Dover, N.Y., 1975 | Zbl

[12] V. K. Dzyadyk, Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Nauka, M., 1977

[13] V. S. Videnskii, Mnogochleny Bernshteina, ucheb. posobie k spetskursu, LGPI im. A. I. Gertsena, L., 1990

[14] B. M. Makarov, M. G. Goluzina, A. A. Lodkin, A. N. Podkorytov, Izbrannye zadachi po veschestvennomu analizu, Nauka, M., 1992

[15] R. A. DeVore, G. G. Lorentz, Constructive Approximation, Springer–Verlag, Berlin–Heidelberg–N. Y., 1993 | Zbl

[16] G. M. Phillips, Interpolation and Approximation by Polynomials, Springer, N. Y.–Berlin–Heidelberg, 2003 | Zbl

[17] M. Kac, “Une remarque sur les polynomes de M. S. Bernstein”, Studia Mathematica, 7 (1938), 49–51

[18] O. Aramă, “Rroprientăţi privind monotonia şirului polinoamelor de interpolare ale lui S. N. Bernstein şi aplicarea lor la studiul aproximării funcţiilor”, Studii şi cercetări de Matematică (Cluj), 8:3–4 (1957), 195–210

[19] I. J. Schoenberg, “On Variation Diminishing Approximation Methods”, On Numerical Approximation : proceedings of a Symposium conducted by the Math. Research Center US Army (University of Wisconsin, Madison, April 21–23, 1958), ed. R. E. Langer, University of Wisconsin Press, Madison, 1959, 249–274

[20] L. I. Strukov, A. F. Timan, “Utochnenie i obobschenie nekotorykh teorem o priblizhenii polinomami S. N. Bernshteina”, Teoriya priblizheniya funktsii, Tr. mezhdunar. konf. po teorii priblizhenii (Kaluga, 24–28 iyulya 1975 g.), Nauka, M., 1977, 338–341

[21] V. S. Videnskii, “O priblizhenii funktsii $|x-c|$ mnogochlenami Bernshteina i interpolyatsionnymi lomanymi”, Priblizhenie funktsii spetsialnymi klassami operatorov, Mezhvuz. sb. nauch. tr., Vologod. gos. ped. in-t, Vologda, 1987, 25–28

[22] S. A. Telyakovskii, “O priblizhenii differentsiruemykh funktsii mnogochlenami Bernshteina i mnogochlenami Kantorovicha”, Tr. MIAN, 260, 2008, 289–296 | Zbl

[23] S. A. Telyakovskii, “O skorosti priblizheniya funktsii mnogochlenami Bernshteina”, Tr. In-ta matematiki i mekhaniki UrO RAN, 14, no. 3, 2008, 162–169

[24] S. A. Telyakovskii, “O priblizhenii mnogochlenami Bernshteina v tochkakh razryva proizvodnykh”, Matem. zametki, 85:4 (2009), 622–629 | DOI | Zbl

[25] V. O. Tonkov, “O priblizhenii mnogochlenami Bernshteina v tochkakh razryva proizvodnykh”, Tr. MIAN, 269, 2010, 265–270

[26] D. O. Shklyarskii, N. N. Chentsov, I. M. Yaglom, Izbrannye zadachi i teoremy elementarnoi matematiki, Biblioteka matematicheskogo kruzhka, 1, Nauka, M., 1965