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.К., Mykhaylyk V.V., Sobchuk О.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.К., Maslyuchenko O.V., Voloshyn H.А. 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.К., Voloshyn H.А. 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.К., Nesterenko V.V. The weak Darboux property and transitional in topological vector spaces// Carp. Mаth. Publ. – 2013. – V.5, №1. – P. 79–87. (in Ukrainian)

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

9. Kelly J., General Topology. – М.: 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
Reference	1. Lebesgue H. Sur l’approximation des fonctions// Bull. Sci. Math. – 1898. – V.22. – P. 278–287. 2. Maslyuchenko V.К., Mykhaylyk V.V., Sobchuk О.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.К., Maslyuchenko O.V., Voloshyn H.А. 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.К., Voloshyn H.А. 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.К., Nesterenko V.V. The weak Darboux property and transitional in topological vector spaces// Carp. Mаth. Publ. – 2013. – V.5, №1. – P. 79–87. (in Ukrainian) 8. Maslyuchenko V.К., First types of topological vector spaces. – Chernivtsi: Ruta, 2002. – 72 p. (in Ukrainian) 9. Kelly J., General Topology. – М.: 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)