Geodesic
Generalized Popoviciu expansions for Bernstein polynomials of a rational module
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 1, Tome 170 (2019), pp. 71-117

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

We prove that Bernstein polynomials for simple nonsmooth functions such as a rational module can be represented as special sums of a regular structure called “generalized Popoviciu decompositions.” To write generalized expansions, a certain formalism based on combinatorial calculations is developed. Based on the formulas obtained, we obtain a complete description of the convergence set of Bernstein polynomials of a rational module. The connection between the Popoviciu expansions and the distribution of zeros of Bernstein polynomials on the complex plane is discussed. In conclusion, a number of additional, new relations for Bernstein polynomials of a rational module are presented.
Keywords: Bernstein polynomial, piecewise linear function, generalized Popoviciu expansion, Kantorovich lemniscate
Mots-clés : rational module, domain of convergence, distribution of zeros of a polynomial. 
@article{INTO_2019_170_a6,
     author = {I. V. Tikhonov and V. B. Sherstyukov and D. G. Tsvetkovich},
     title = {Generalized {Popoviciu} expansions for {Bernstein} polynomials of a rational module},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {71--117},
     publisher = {mathdoc},
     volume = {170},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_170_a6/}
}
TY  - JOUR
AU  - I. V. Tikhonov
AU  - V. B. Sherstyukov
AU  - D. G. Tsvetkovich
TI  - Generalized Popoviciu expansions for Bernstein polynomials of a rational module
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2019
SP  - 71
EP  - 117
VL  - 170
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2019_170_a6/
LA  - ru
ID  - INTO_2019_170_a6
ER  - 
%0 Journal Article
%A I. V. Tikhonov
%A V. B. Sherstyukov
%A D. G. Tsvetkovich
%T Generalized Popoviciu expansions for Bernstein polynomials of a rational module
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2019
%P 71-117
%V 170
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2019_170_a6/
%G ru
%F INTO_2019_170_a6
I. V. Tikhonov; V. B. Sherstyukov; D. G. Tsvetkovich. Generalized Popoviciu expansions for Bernstein polynomials of a rational module. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 1, Tome 170 (2019), pp. 71-117. http://geodesic.mathdoc.fr/item/INTO_2019_170_a6/