On linear interpolation of vector valued functions and
its applications

H. A. Voloshyn; V. K. Maslyuchenko

We prove a theorem on the uniform and point-wise approximation of a mapping $\varphi\colon [a,b]\rightarrow Z$ taking values in a topological vector space by its linear interpolations. Using this theorem we prove that if $Y$ is a compact space then every separately continuous function $f\colon [a,b]\times Y\rightarrow \mathbb{R}$ with at most countable projection of the discontinuity point set $D(f)$ to $[a,b]$ belongs to the sequential closure of the subspace of all continuous functions $g\colon [a,b]\times Y\rightarrow \mathbb{R}$ in the space $S([a,b]\times Y)$ of all separately continuous functions $f\colon [a,b]\times Y\rightarrow \mathbb{R}$ with the topology of the layer-wise uniform convergence.

1. Lebesgue H. Sur l'approximation des fonctions// Bull. Sci. Math. - 1898. - V.22. - P. 278-287.

2. Maslyuchenko V.K., Mykhaylyk V.V., Sobchuk O.V. Investigation on separately continuous functions// Mathematical Proceedings of the International Conference dedicated to the memory of Hans Hahn. - Chernivtsi: Ruta, 1995. - P. 192-246. (in Ukrainian)

3. Tsuji M. On Bair's Theorem concerning a function f(x; y) which is continuous with respect to each variable x and y// J. Math. Soc. Japan. - 1951. - V.2, №3-4. - P. 210-212.

4. Baire R. Sur les fonctions de variables reelles// Ann. Mat. Pura Appl., ser.3. - 1899. - V.3. - P. 1-123.

5. Maslyuchenko V.K., Maslyuchenko O.V., Voloshyn H.A. On layer-wise uniform approximation of separately continuous functioins by polynomials// Math. Bull. of SSS. - 2013. - V.10. - P. 135-158. (in Ukrainian)

6. Maslyuchenko V.K., Voloshyn H.A. The linear interpolation of vector valued functions and its applications// IV international Hahn conference dedicated to the 135 anniversary of Hans Hahn, 2014, Chernivtsi. Abstract. 2014. - P. 20-23. (in Ukrainian)

7. Maslyuchenko V.K., Nesterenko V.V. The weak Darboux property and transitional in topological vector spaces// Carp. Math. Publ. - 2013. - V.5, №1. - P. 79-87. (in Ukrainian)

8. Maslyuchenko V.K., First types of topological vector spaces. - Chernivtsi: Ruta, 2002. - 72 p. (in Ukrainian)

9. Kelly J., General Topology. - M.: Nauka, 1981. - 432 p. (in Russian)

	On linear interpolation of vector valued functions and its applications (in Ukrainian)
Author	H. A. Voloshyn, V. K. Maslyuchenko vmaslyuchenko@gmail.com, galja.vlshin@gmail.com Yuriy Fedkovych Chernivtsi National University; Bukovyna State University of Finance and Economics
Abstract	We prove a theorem on the uniform and point-wise approximation of a mapping $\varphi\colon [a,b]\rightarrow Z$ taking values in a topological vector space by its linear interpolations. Using this theorem we prove that if $Y$ is a compact space then every separately continuous function $f\colon [a,b]\times Y\rightarrow \mathbb{R}$ with at most countable projection of the discontinuity point set $D(f)$ to $[a,b]$ belongs to the sequential closure of the subspace of all continuous functions $g\colon [a,b]\times Y\rightarrow \mathbb{R}$ in the space $S([a,b]\times Y)$ of all separately continuous functions $f\colon [a,b]\times Y\rightarrow \mathbb{R}$ with the topology of the layer-wise uniform convergence.
Keywords	uniform and point-wise approximation; linear interpolations; separately continuous functions; vector valued functions; sequential closure
DOI	doi:10.30970/ms.42.2.129-133
Reference	1. Lebesgue H. Sur l'approximation des fonctions// Bull. Sci. Math. - 1898. - V.22. - P. 278-287. 2. Maslyuchenko V.K., Mykhaylyk V.V., Sobchuk O.V. Investigation on separately continuous functions// Mathematical Proceedings of the International Conference dedicated to the memory of Hans Hahn. - Chernivtsi: Ruta, 1995. - P. 192-246. (in Ukrainian) 3. Tsuji M. On Bair's Theorem concerning a function f(x; y) which is continuous with respect to each variable x and y// J. Math. Soc. Japan. - 1951. - V.2, №3-4. - P. 210-212. 4. Baire R. Sur les fonctions de variables reelles// Ann. Mat. Pura Appl., ser.3. - 1899. - V.3. - P. 1-123. 5. Maslyuchenko V.K., Maslyuchenko O.V., Voloshyn H.A. On layer-wise uniform approximation of separately continuous functioins by polynomials// Math. Bull. of SSS. - 2013. - V.10. - P. 135-158. (in Ukrainian) 6. Maslyuchenko V.K., Voloshyn H.A. The linear interpolation of vector valued functions and its applications// IV international Hahn conference dedicated to the 135 anniversary of Hans Hahn, 2014, Chernivtsi. Abstract. 2014. - P. 20-23. (in Ukrainian) 7. Maslyuchenko V.K., Nesterenko V.V. The weak Darboux property and transitional in topological vector spaces// Carp. Math. Publ. - 2013. - V.5, №1. - P. 79-87. (in Ukrainian) 8. Maslyuchenko V.K., First types of topological vector spaces. - Chernivtsi: Ruta, 2002. - 72 p. (in Ukrainian) 9. Kelly J., General Topology. - M.: Nauka, 1981. - 432 p. (in Russian)
Pages	129-133
Volume	42
Issue	2
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On linear interpolation of vector valued functions and its applications (in Ukrainian)