Hahns pairs and zero inverse problem (in Ukrainian)

Author
V. K. Maslyuchenko, V. S. Melnyk, H. A. Voloshyn
Chernivtsi National University, Chernivtsi, Ukraine
Abstract
We prove that for a function $\alpha_0\colon [0,1] \rightarrow \mathbb{R}$ there exists a separately continuous function $f\colon [0,1]^{2} \rightarrow \mathbb{R}$ such that $E_{0}(f^{x})=\alpha_0(x)$ on [0,1] if and only if $\alpha_0$ is the nonnegative lower semicontinuous function, where $f^{x}(y)=f(x,y)$ for any $x, y, \in [0,1]$ and $E_{0}(g)$ is the best approximation of a function $g$ by a constant.
Keywords
inverse Bernsteins theorem; Hahns pair; separately continuous function
DOI
doi:10.15330/ms.48.1.74-81
Reference
1. Bernstein S.N. Sur probleme inverse de la theorie de la meilleure approximation des fonctions continues// Comp. Rend. - 1938. - P. 1520-1523.

2. Bernstein S.N. On an inverse problem of approximation theory// Collected works in 4 volumes. - M.:Publ. house of the Academy of Sciences of the USSR. 1954. - V.2. - P. 292-294. (in Russian)

3. Vasiliev A.I. An inverse problem of the theory of the best approximation in F-spaces// Reports of the Academy of Sciences. - 1999. - P. 583-585. (in Russian)

4. Voloshyn H.A., Maslyuchenko V.K. The generalization of one Bernsteinfs theorem// Mat. Visn. NTSh. - 2009. - V.6. - P. 62-72. (in Ukrainian)

5. Vlasyuk H., Maslyuchenko V.K. Bernsteins polinomials and separately continuous functions// Nauk. Visn. Cherniv. Univ., Mat. - V.336-337. - 2007. - P. 52-59. (in Ukrainian)

6. Voloshyn H.A., Maslyuchenko V.K. The functional generalization of one Bernsteins theorem// Mat. Stud. - 2010. - V.33, 2. - P. 220-224. (in Ukrainian)

7. Engelking R. General Topology. - Moscow: Mir, 1986. - 752 p. (in Russian)

8. Hahn H. Uber halbstetige und unstetige Functionen// Sitzungsberichte Acad. Wiss. Wien. Math. - naturwiss. Kl. Abt. IIa. - 1917. - V.126. - S. 91-110.

9. Maslyuchenko V.K., Mykhayuk V.V., Sobchuk O.V. Construction of the separately continuous function of n variables with a given diagonal// Mat. Stud. - 1999. - V.12, 1. - P. 101-107. (in Ukrainian)

Pages
74-81
Volume
48
Issue
1
Year
2017
Journal
Matematychni Studii
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