Hahn’s pairs and zero inverse problem

V. K. Maslyuchenko; V. S. Mel’nyk; H. A. Voloshyn

doi:10.15330/ms.48.1.74-81

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.

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 BernsteinЃfs 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)

	Hahn’s pairs and zero inverse problem (in Ukrainian)
Author	V. K. Maslyuchenko, V. S. Mel’nyk, H. A. Voloshyn galja.vlshin@gmail.com, v.maslyuchenko@gmail.com, windchange7@gmail.com 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 Bernstein’s theorem; Hahn’s 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 BernsteinЃfs 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
Full text of paper	pdf
Table of content of issue	html

Hahn’s pairs and zero inverse problem (in Ukrainian)