Some extremal problems in the Orlicz spaces

A. L. Shidlich; S. O. Chaichenko

We obtained the exact values of analogues of the best $n$-term approximations and the basis widths in the Orlicz spaces $L_M(\mathbb{A};d\mu)$ of classes of non-negative functions, which can be represented as the product of a fixed function and functions from the unit balls of the spaces $L_p(\mathbb{A};d\mu)$ and $L_M(\mathbb{A};d\mu)$, respectively.

1. Stechkin S.B. On absolute convergence of orthogonal series// Dokl. Akad. Nauk SSSR. – 1955. – V.102, №1. – P. 37–40. (in Russian)

2. Stepanets A.I., Shidlich A.L. Best approximations of integrals by integrals of finite rank// J. Approx. Theory. – 2010. – V.162. – P. 323–348.

3. Stepanets A.I., Shidlich A.L. Extremal problems for integrals of nonnegative functions// Izv. Ross. Akad. Nauk Ser. Mat. – 2010. – V.74, №3. – P. 169–224. (in Russian)

4. De Vore R. Nonlinear approximation// Acta Numer. – 1998. – V.7. – P. 51–150.

5. Romanyuk A.S. Best M-term trigonometric approximations of Besov classes of periodic functions of several variables// Izv. Ross. Akad. Nauk Ser. Mat. – 2003. – V.67, №2. – P. 61–100. (in Russian)

6. Stepanets A.I. Extremal problems of approximation theory in linear spaces// Ukrain. Mat. Zh. – 2003. – V.55, №10. – P. 1378–1410. (in Russian)

7. Shidlich A.L., Chaichenko S.O. Approximative properties of diagonal operators in Orlicz spaces// arXiv preprint. – 2014. – arXiv:1405.1542. – 11 p.

8. Shidlich A.L., Chaichenko S.O. On some inequalities of Chebyshev type, in press.

9. Shidlich A.L., Chaichenko S.O. On some inequalities of Chebyshev type// arXiv preprint. – 2014. – arXiv:1405.1256. – 6 p.

	Some extremal problems in the Orlicz spaces(in Ukrainian)
Author	A. L. Shidlich, S. O. Chaichenko andy709@list.ru, shidlich@imath.kiev.ua, stolch@mail.ru Institute of Mathematics of Ukrainian National Academy of Sciences, Donbas State Pedagogical University
Abstract	We obtained the exact values of analogues of the best $n$-term approximations and the basis widths in the Orlicz spaces $L_M(\mathbb{A};d\mu)$ of classes of non-negative functions, which can be represented as the product of a fixed function and functions from the unit balls of the spaces $L_p(\mathbb{A};d\mu)$ and $L_M(\mathbb{A};d\mu)$, respectively.
Keywords	Orlicz spaces; best n-term approximations; basis widths
Reference	1. Stechkin S.B. On absolute convergence of orthogonal series// Dokl. Akad. Nauk SSSR. – 1955. – V.102, №1. – P. 37–40. (in Russian) 2. Stepanets A.I., Shidlich A.L. Best approximations of integrals by integrals of finite rank// J. Approx. Theory. – 2010. – V.162. – P. 323–348. 3. Stepanets A.I., Shidlich A.L. Extremal problems for integrals of nonnegative functions// Izv. Ross. Akad. Nauk Ser. Mat. – 2010. – V.74, №3. – P. 169–224. (in Russian) 4. De Vore R. Nonlinear approximation// Acta Numer. – 1998. – V.7. – P. 51–150. 5. Romanyuk A.S. Best M-term trigonometric approximations of Besov classes of periodic functions of several variables// Izv. Ross. Akad. Nauk Ser. Mat. – 2003. – V.67, №2. – P. 61–100. (in Russian) 6. Stepanets A.I. Extremal problems of approximation theory in linear spaces// Ukrain. Mat. Zh. – 2003. – V.55, №10. – P. 1378–1410. (in Russian) 7. Shidlich A.L., Chaichenko S.O. Approximative properties of diagonal operators in Orlicz spaces// arXiv preprint. – 2014. – arXiv:1405.1542. – 11 p. 8. Shidlich A.L., Chaichenko S.O. On some inequalities of Chebyshev type, in press. 9. Shidlich A.L., Chaichenko S.O. On some inequalities of Chebyshev type// arXiv preprint. – 2014. – arXiv:1405.1256. – 6 p.
Pages	21-32
Volume	42
Issue	1
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Some extremal problems in the Orlicz spaces(in Ukrainian)