The exact-order estimates of linear widths are established for the classes *B*
^{Ω}_{
p,θ
}
of periodic functions of many variables in the space *L*
_{
q
} for certain relations between on the parameters *p* and *q.*

### Similar content being viewed by others

## References

N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two conjugated functions,”

*Tr. Mosk. Mat. Obshch.*,**5**, 483–522 (1956).Guiqiao Xu, “The

*n*-widths for a generalized periodic Besov classes,”*Acta Math. Sci.*,**25**, No. 4, 663–671 (2005).O. V. Besov, “On a family of functional spaces. Embedding theorems and their extensions,”

*Dokl. Akad. Nauk SSSR*,**126**, No. 6, 1163–1165 (1959).S. M. Nikol’skii, “Inequalities for the entire functions of finite power and their application to the theory of differentiable functions of many variables,”

*Tr. Mat. Inst. Akad. Nauk SSSR*,**38**, 244–278 (1951).V. M. Tikhomirov, “Widths of the sets in functional spaces and the theory of best approximations,”

*Usp. Mat. Nauk*,**15**, No. 3, 81–120 (1960).É. M. Galeev, “On the linear widths of the classes of periodic functions of many variables,”

*Vestn. Mosk. Univ., Ser. Mat., Mekh.*,**4**, 13–16 (1987).É. M. Galeev, “Linear widths of the H¨older–Nikol’skii classes of periodic functions of many variables,”

*Mat. Zametki*,**59**, No. 2, 189–199 (1996).A. S. Romanyuk, “Linear widths of the Besov classes of periodic functions of many variables. I,”

*Ukr. Mat. Zh.*,**53**, No. 5, 647–661 (2001);*English translation:**Ukr. Math. J.*,**53**, No. 5, 744–761 (2001).A. S. Romanyuk, “Linear widths of the Besov classes of periodic functions of many variables. II,”

*Ukr. Mat. Zh.*,**53**, No. 6, 820–829 (2001);*English translation:**Ukr. Math. J.*,**53**, No. 6, 965–977 (2001).A. S. Romanyuk, “Widths and the best approximation for the classes

*B*^{r}_{ p,θ }of periodic functions of many variables,”*Anal. Math.*,**37**, No. 3, 181–213 (2011).A. Kolmogoroff, “”Uber die beste Anneherung von Funktionen einer gegebenen Funktionenklasse,”

*Ann. Math.*,**37**, No. 1, 107–110 (1963).E. D. Gluskin, “Norms of random matrices and widths of finite-dimensional sets,”

*Mat. Sb.*,**120**, No. 2, 180–189 (1983).B. S. Kashin, “Some properties of the matrices of bounded operators from the space

*l*^{n}_{2}into*l*^{m}_{2}*,*”*Izv. Akad. Nauk Arm. SSR, Ser. Mat.*,**15**, No. 5, 379–394 (1980).A. Zygmund,

*Trigonometric Series*[Russian translation], Vol. 2, Mir, Moscow (1965).É. M. Galeev, “Kolmogorov widths of the classes of periodic functions of many variables \( {\tilde{W}}_p^{\overline{\alpha}} \) and \( {\tilde{H}}_p^{\overline{\alpha}} \) in the space \( {\tilde{L}}_q \)

*,*”*Izv. Akad. Nauk SSSR, Ser. Mat.*,**49**, No. 5, 916–934 (1985).P. I. Lizorkin, “Generalized Hölder spaces

*B*^{(r)}_{ p,θ }and their relationship with the Sobolev spaces*L*^{(r)}_{ p }*,*”*Sib. Mat. Zh.*,**9**, No. 5, 1127–1152 (1968).V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,”

*Tr. Mat. Inst. Akad. Nauk SSSR*,**178**, 3–113 (1986).D. Jackson, “Certain problems of closest approximation,”

*Bull. Amer. Math. Soc.*,**39**, No. 3, 889–906 (1993).N. V. Derev’yanko, “Trigonometric widths of classes of periodic functions of many variables,”

*Ukr. Mat. Zh.*,**64**, No. 8, 1041–1052 (2012);*English translation:**Ukr. Math. J.*,**64**, No. 8, 1185–1198 (2013).K. V. Solich, “Kolmogorov widths of the classes

*B*^{Ω}_{ p,θ }of periodic functions of many variables in the space*L*_{ q }*,*”*Ukr. Mat. Zh.*,**64**, No. 10, 1416–1425 (2012);*English translation:**Ukr. Math. J.*,**64**, No. 10, 1610–1620 (2013).O. V. Fedunyk, “Linear widths of the classes

*B*^{Ω}_{ p,θ }of periodic functions of many variables,” in:*Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv*[in Ukrainian],**1**, No. 1 (2004), pp. 375–388.A. F. Konohrai, “Linear widths of the classes

*B*^{Ω}_{ p,θ }of periodic functions of one and many variables,” in:*Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv*[in Ukrainian],**7**, No. 1 (2010), pp. 94–112.V. M. Tikhomirov, “Approximation theory,” in:

*VINITI Series in Contemporary Problems of Mathematics. Fundamental Directions*[in Russian],**14**, VINITI, Moscow (1987), pp. 103–260.B. S. Kashin, “Widths of some finite-dimensional sets and classes of smooth functions,”

*Izv. Akad. Nauk SSSR, Ser. Mat.*,**41**, No. 2, 334–351 (1977).S. A. Stasyuk, “Approximation of the classes

*B*^{ω}_{ p,θ }of periodic functions of many variables with spectra in cubic domains,”*Mat. Stud.*,**35**, No. 1, 66–73 (2011).O. V. Fedunyk, “Linear widths of the classes

*B*^{Ω}_{ p,θ }of periodic functions of many variables in the space*L*_{ q }*,*”*Ukr. Mat. Zh.*,**58**, No. 1, 93–104 (2006);*English translation:**Ukr. Math. J.*,**58**, No. 1, 103–117 (2006).

## Author information

### Authors and Affiliations

## Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 7, pp. 909–921, July, 2014.

## Rights and permissions

## About this article

### Cite this article

Derev’yanko, N.V. Estimations of Linear Widths of the Classes *B*
^{Ω}_{
p,θ
}
of Periodic Functions of Many Variables in the Space *L*
_{
q
}
.
*Ukr Math J* **66**, 1013–1027 (2014). https://doi.org/10.1007/s11253-014-0991-y

Received:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s11253-014-0991-y