Relative Widths of Some Sets of l^{m}_{p}

ABSTRACT

n this paper, the relative widths of some sets in are studied. Relative widths is the further development of Kolmogorov widths and it is a new problem in approximation theory which aroused some mathematics workers great interest recently. We present some basic propositions of relative widths and investigate relative widths of some sets (ball or ellipsoid) of

n this paper, the relative widths of some sets in are studied. Relative widths is the further development of Kolmogorov widths and it is a new problem in approximation theory which aroused some mathematics workers great interest recently. We present some basic propositions of relative widths and investigate relative widths of some sets (ball or ellipsoid) of

Cite this paper

nullW. Xiao and W. Luan, "Relative Widths of Some Sets of l^{m}_{p}," *Advances in Pure Mathematics*, Vol. 1 No. 2, 2011, pp. 30-32. doi: 10.4236/apm.2011.12008.

nullW. Xiao and W. Luan, "Relative Widths of Some Sets of l

References

[1] V. N. Konovalov, “Estimates of Diameters of Kolmogorov Type for Classes of Differentiable Periodic Functions,” Mate-maticheskie Zametki, Vol. 35, No. 3, 1984, pp. 369-380.

[2] V. F. Babenko, “On the Relative Widths of Classes of Functions with Bounded Mixed Derivative,” East Journal on Approximations, Vol. 2, No. 3, 1996, pp. 319-330.

[3] V. F. Babenko, “Approximation in the Mean under Restriction on the Derivatives of the Approximating Functions, Questions in Analysis and Approximations,” Insti-tute of the Mathematics of the Ukrainean Academy of Science, Kiev, 1989, pp. 9-18. (in Russian)

[4] V. F. Babenko, “On Best Uniform Approximations by Splines in the Presence of Restrictions on Their Derivatives,” Mathematical Notes, Vol. 50, No. 6, 1991, pp. 1227-1232. Translated from Mate-maticheskie Zametki, Vol. 50, No. 6, 1991, pp. 24-30.

[5] V. N. Konovalov, “Approximation of Sobolev Classes by Their Sections of Finite Dimensional,” Ukrainian Mathematical Journal, Vol. 54, No. 5, 2002, pp. 795-805. doi:10.1023/A:1021635530578

[6] V. N. Konovalov, “Ap-proximation of Sobolev Classes by Their Finite-Dimensional Sections,” Matematicheskie Za- metki, Vol. 72, No. 3, 2002, pp. 370-382.

[7] Y. N. Subbotin and S. A. Telyakovskii, “Exact Values of Relative Widths of Classes of Differentiable Func-tions,” Mathematical Notes, Vol. 65, No. 6, 1999, pp. 731-738. Translated from Matematicheskie Zametki, Vol. 65, No. 6, 1999, pp. 871-879.

[8] Y. N. Subbotin and S. A. Telyakovskii, “Relative Widths of Classes of Differentiable Functions in the Metric,” Russian Mathematical Surveys, Vol. 56, No. 4, 2001, pp. 159-160.

[9] Y. N. Subbotin and S. A. Telyakovskii, “On Relative Widths of Classes of Differentiable Functions,” Proceedings of the Steklov Institute of Mathematics, 2005, pp. 243-254.

[10] V. T. Shevaldin, “Approximation by Local Trigonometric Splines,” Mathematical Notes, Vol. 77, No. 3, 2005, pp. 326-334.

[11] V. M. Tikhomirov, “Some Remarks on Relative Diameters,” Banach Center Publications, Vol. 22, 1989, pp. 471-474.

[12] W. W. Xiao, “Relative Widths of Classes of Functions Defined by a Self-Conjugate Linear Dif-ferential Operator in (in Chinese),” Chinese annals of Mathematics, Vol. 29A, No. 5, 2008, pp. 679-688.

[13] W. W. Xiao, “Relative Infinite-Dimensional Width of Sobolev Classes,” Journal of Mathematical Analysis and Applications, Vol. 369, No. 2, 2010, pp. 575-582. doi:10.1016/j.jmaa.2010.03.050

[14] Y. P. Liu and W. W. Xiao, “Relative Average Widths of Sobolev Spaces in ,” Analysis Mathematica, Vol. 34, No. 1, 2008, pp. 71-82. doi:10.1007/s10476-008-0107-8

[15] Y. P. Liu and L. H. Yang, “Relative Width of Smooth Classes of Multivariate Periodic Functions with Restrictions on Interated Laplace Derivatives in The -Met- Ric,” Acta Mathematica Scientia, Vol. 26B, No. 4, 2006, pp. 720-728. doi:10.1016/S0252-9602(06)60098-2

[16] Y. P. Liu and L. H. Yang, “Relative Widths of Smooth Factions Determined by Fractional Order Derivatives,” Journal of complexity, Vol. 24, No. 2, 2008, pp. 259-282.

[17] L. H. Yang and Y. P. Liu, “Relative Widths of Smooth Functions Determined by Linear Differential Operator,” Journal of Mathematical Analysis and Applications, Vol. 351, No. 2, 2009, pp. 734-746. doi:10.1016/j.jmaa.2008.11.009

[18] G. G. Lorentz, M. V. Golitschek, and Y. Makovoz, “Constructive Approximation,” Springer, Berlin, 1996.

[19] A. Pinkus, “N-Widths in Ap-proximation Theory,” Springer, Berlin, 1985.

[1] V. N. Konovalov, “Estimates of Diameters of Kolmogorov Type for Classes of Differentiable Periodic Functions,” Mate-maticheskie Zametki, Vol. 35, No. 3, 1984, pp. 369-380.

[2] V. F. Babenko, “On the Relative Widths of Classes of Functions with Bounded Mixed Derivative,” East Journal on Approximations, Vol. 2, No. 3, 1996, pp. 319-330.

[3] V. F. Babenko, “Approximation in the Mean under Restriction on the Derivatives of the Approximating Functions, Questions in Analysis and Approximations,” Insti-tute of the Mathematics of the Ukrainean Academy of Science, Kiev, 1989, pp. 9-18. (in Russian)

[4] V. F. Babenko, “On Best Uniform Approximations by Splines in the Presence of Restrictions on Their Derivatives,” Mathematical Notes, Vol. 50, No. 6, 1991, pp. 1227-1232. Translated from Mate-maticheskie Zametki, Vol. 50, No. 6, 1991, pp. 24-30.

[5] V. N. Konovalov, “Approximation of Sobolev Classes by Their Sections of Finite Dimensional,” Ukrainian Mathematical Journal, Vol. 54, No. 5, 2002, pp. 795-805. doi:10.1023/A:1021635530578

[6] V. N. Konovalov, “Ap-proximation of Sobolev Classes by Their Finite-Dimensional Sections,” Matematicheskie Za- metki, Vol. 72, No. 3, 2002, pp. 370-382.

[7] Y. N. Subbotin and S. A. Telyakovskii, “Exact Values of Relative Widths of Classes of Differentiable Func-tions,” Mathematical Notes, Vol. 65, No. 6, 1999, pp. 731-738. Translated from Matematicheskie Zametki, Vol. 65, No. 6, 1999, pp. 871-879.

[8] Y. N. Subbotin and S. A. Telyakovskii, “Relative Widths of Classes of Differentiable Functions in the Metric,” Russian Mathematical Surveys, Vol. 56, No. 4, 2001, pp. 159-160.

[9] Y. N. Subbotin and S. A. Telyakovskii, “On Relative Widths of Classes of Differentiable Functions,” Proceedings of the Steklov Institute of Mathematics, 2005, pp. 243-254.

[10] V. T. Shevaldin, “Approximation by Local Trigonometric Splines,” Mathematical Notes, Vol. 77, No. 3, 2005, pp. 326-334.

[11] V. M. Tikhomirov, “Some Remarks on Relative Diameters,” Banach Center Publications, Vol. 22, 1989, pp. 471-474.

[12] W. W. Xiao, “Relative Widths of Classes of Functions Defined by a Self-Conjugate Linear Dif-ferential Operator in (in Chinese),” Chinese annals of Mathematics, Vol. 29A, No. 5, 2008, pp. 679-688.

[13] W. W. Xiao, “Relative Infinite-Dimensional Width of Sobolev Classes,” Journal of Mathematical Analysis and Applications, Vol. 369, No. 2, 2010, pp. 575-582. doi:10.1016/j.jmaa.2010.03.050

[14] Y. P. Liu and W. W. Xiao, “Relative Average Widths of Sobolev Spaces in ,” Analysis Mathematica, Vol. 34, No. 1, 2008, pp. 71-82. doi:10.1007/s10476-008-0107-8

[15] Y. P. Liu and L. H. Yang, “Relative Width of Smooth Classes of Multivariate Periodic Functions with Restrictions on Interated Laplace Derivatives in The -Met- Ric,” Acta Mathematica Scientia, Vol. 26B, No. 4, 2006, pp. 720-728. doi:10.1016/S0252-9602(06)60098-2

[16] Y. P. Liu and L. H. Yang, “Relative Widths of Smooth Factions Determined by Fractional Order Derivatives,” Journal of complexity, Vol. 24, No. 2, 2008, pp. 259-282.

[17] L. H. Yang and Y. P. Liu, “Relative Widths of Smooth Functions Determined by Linear Differential Operator,” Journal of Mathematical Analysis and Applications, Vol. 351, No. 2, 2009, pp. 734-746. doi:10.1016/j.jmaa.2008.11.009

[18] G. G. Lorentz, M. V. Golitschek, and Y. Makovoz, “Constructive Approximation,” Springer, Berlin, 1996.

[19] A. Pinkus, “N-Widths in Ap-proximation Theory,” Springer, Berlin, 1985.