The analogue of Bernstein’s inverse theorem for the one
class of the space of sequences

V. K. Maslyuchenko; H. A. Voloshyn

doi:10.15330/ms.47.2.160-168

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.

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. – М.: 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. – Р. 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 А.Т. The course of Higher Algebra. – М.: 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 – М.: Science, 1989. – 624 p. (in Russian)

	The analogue of Bernstein’s inverse theorem for the one class of the space of sequences (in Ukrainian)
Author	V. K. Maslyuchenko, H. A. Voloshyn galja.vlshin@gmail.com; v.maslyuchenko@gmail.com 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. – М.: 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. – Р. 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 А.Т. The course of Higher Algebra. – М.: 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 – М.: Science, 1989. – 624 p. (in Russian)
Pages	160-168
Volume	47
Issue	2
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

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