## Journal of Applied Mathematics

### Bernstein Widths of Some Classes of Functions Defined by a Self-Adjoint Operator

Guo Feng

#### Abstract

We consider the classes of periodic functions with formal self-adjoint linear differential operators ${W}_{p}({\scr L}_{r})$, which include the classical Sobolev class as its special case. Using the iterative method of Buslaev, with the help of the spectrum of linear differential equations, we determine the exact values of Bernstein width of the classes ${W}_{p}({\scr L}_{r})$ in the space ${L}_{q}$ for $1.

#### Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 495054, 9 pages.

Dates
First available in Project Euclid: 15 March 2012

https://projecteuclid.org/euclid.jam/1331817618

Digital Object Identifier
doi:10.1155/2012/495054

Mathematical Reviews number (MathSciNet)
MR2844126

Zentralblatt MATH identifier
1230.41003

#### Citation

Feng, Guo. Bernstein Widths of Some Classes of Functions Defined by a Self-Adjoint Operator. J. Appl. Math. 2012 (2012), Article ID 495054, 9 pages. doi:10.1155/2012/495054. https://projecteuclid.org/euclid.jam/1331817618

#### References

• V. M. Tikhomirov, Some Questions in Approximation Theory, Izdat. Moskov. Univ., Moscow, Russia, 1976.
• A. Pinkus, n-Widths in Approximation Theory, Springer, New York, NY, USA, 1985.
• G. G. Magaril-Il'yaev, “Mean dimension, widths, and optimal recovery of Sobolev classes of functions on the line,” Mathematics of the USSR–-Sbornik, vol. 74, no. 2, pp. 381–403, 1993.
• A. P. Buslaev, G. G. Magaril-Il'yaev, and Nguen T'en Nam, “Exact values of Bernstein widths for Sobolev classes of periodic functions,” Matematicheskie Zametki, vol. 58, no. 1, pp. 139–143, 1995 (Russian).
• A. P. Buslaev and V. M. Tikhomirov, “Spectra of nonlinear differential equations and widths of Sobolev classes,” Mathematics of the USSR–-Sbornik, vol. 71, no. 2, pp. 427–446, 1992.
• S. I. Novikov, “Exact values of widths for some classes of periodic functions,” The East Journal on Approximations, vol. 4, no. 1, pp. 35–54, 1998.
• Nguen Thi Thien Hoa, Optimal quadrature formulae and methods for recovery on function classds defined by variation diminishing convolutions, Candidate's Dissertation, Moscow State University, Moscow, Russia, 1985.
• V. A. Jakubovitch and V. I. Starzhinski, Linear Differential Equations with Periodic Coeflicients and Its Applications, Nauka, Moscow, Russia, 1972.
• A. Pinkus, “n-widths of Sobolev spaces in ${L}^{p}$,” Constructive Approximation, vol. 1, no. 1, pp. 15–62, 1985.
• K. Borsuk, “Drei Sätze über die n-dimensionale euklidische Sphäre,” Fundamenta Mathematicae, vol. 20, pp. 177–190, 1933.
• A. P. Buslaev and V. M. Tikhomirov, “Some problems of nonlinear analysis and approximation theory,” Soviet Mathematics–-Doklady, vol. 283, no. 1, pp. 13–18, 1985.