# The analogue of Bernstein’s inverse theorem for the one class of the space of sequences (in Ukrainian)

Author
Chernivtsi National University, Chernivtsi, Ukraine
Abstract
We introduce the space of numerical sequences $l_{\mathbf{p}}=\{{x=(\xi_{k})_{k=1}^{\infty}\colon} |x|=\sum\nolimits_{k=1}^{\infty}|{\xi_{k}}|^{p_k}\le+\infty\}$ with a quasi-norm $|\cdot|$ for an every sequence $\mathbf{p}=(p_k)_{k=1}^{\infty}$ of numbers $p_k$ from the interval $(0,1]$ and we prove the analogue of the inverse of Bernstein's theorem for this space.
Keywords
inverse Bernstein’s theorem; quasi-norm; quasi-normed space; bounded balls; sequence of finite dimensional linear subspaces
DOI
doi:10.15330/ms.47.2.160-168
Reference
1. Bernstein S.N. Sur le probleme inverse de la theorie de la meilleure approximation des fonctions continues// Comp. Rend. – 1938. – V.206. – P. 1520–1523.

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

3. Nikolsky S.M. Approximation by polynomials of functions of a real variable// Mathematics in the USSR for 30 years. –M.-L.: GITTL, 1948. – P. 288–318. (in Russian)

4. Nesterenko O.N. The inverse problem of paproximation and evaluation of norms of entire functions of exponential type and polynomials: dis ... candidate of physical and mathematical sciences. Sciences: 01.01.01 - Mathematical analysis. – Kiev, 2006. – 148 p. (in Ukrainian)

5. Voloshyn H.A., Maslyuchenko V.K. The generalization of one Bernstein’s theorem// Mat. Visn. NTSh. – 2009. – V.6. – P. 62–72. (in Ukrainian)

6. Voloshyn H.A. Separately continuous functions and the theory of approximations: dis ... candidate of physical and mathematical sciences. Sciences: 01.01.01 – Mathematical analysis. – Chernivtsi, 2012. – 138 p. (in Ukrainian)

7. Orlicz W. Uber konjugierte Exponentenfolgen// Stud. Math. – 1931. – V.3. – Ð. 200–211.

8. Maligranda L. Hidegoro Nakano (1909-1974) – on the centenary of his birth// Proceedings of the International Symposium on Banach and Function Spaces IV (Kitakyushu, Japan, 2009). – 2011. – P. 99–171.

9. Maslyuchenko V.K. The first types of topological vector spaces. – Chernivtsi: Ruta, 2002. – 72 p. (in Ukrainian)

10. Maslyuchenko V.K. Linear Continuous Operators. – Chernivtsi: Ruta, 2002. – 72 p. (in Ukrainian)

11. Maslyuchenko V.K. Lectures on functional analysis. Ch.1.Metric and normalized spaces. – Chernivtsi: ChNU Ruta, 2010. – 184p. (in Ukrainian)

12. Kurosh À.Ò. The course of Higher Algebra. – Ì.: Science, 1971. – 432p. (in Russian)

13. Robertson A.P., Robertson W.G. Topological vector spaces. – M.: Peace, 1967. – 258 p. (in Russian)

14. Shefer H. Topological vector spaces. – M.: Peace, 1971. – 360 p. (in Russian)

15. Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis – Ì.: Science, 1989. – 624 p. (in Russian)

Pages
160-168
Volume
47
Issue
2
Year
2017
Journal
Matematychni Studii
Full text of paper
Table of content of issue