The Best m-Term One-Sided Approximation of Besov Classes by the Trigonometric Polynomials

Affiliation(s)

School of Information and Technology, Shandong Agricultural University, Tai’an, China.

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems,Ministry of Education, Beijing, China.

School of Information and Technology, Shandong Agricultural University, Tai’an, China.

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems,Ministry of Education, Beijing, China.

ABSTRACT

In this paper, we continue studying the so called best m-term one-sided approximation and Greedy-liked one-sided ap- proximation by the trigonometric polynomials. The asymptotic estimations of the best m-terms one-sided approximation by the trigonometric polynomials on some classes of Besov spaces in the metricL_{p}(T^{d}(1≤p≤∞ are given.

In this paper, we continue studying the so called best m-term one-sided approximation and Greedy-liked one-sided ap- proximation by the trigonometric polynomials. The asymptotic estimations of the best m-terms one-sided approximation by the trigonometric polynomials on some classes of Besov spaces in the metricL

Cite this paper

R. Li and Y. Liu, "The Best m-Term One-Sided Approximation of Besov Classes by the Trigonometric Polynomials,"*Advances in Pure Mathematics*, Vol. 2 No. 3, 2012, pp. 183-189. doi: 10.4236/apm.2012.23025.

R. Li and Y. Liu, "The Best m-Term One-Sided Approximation of Besov Classes by the Trigonometric Polynomials,"

References

[1] R. A. Devore and V. N. Temlyakov, “Nonlinear Approximation by Trigonometric Sums,” Journal of Fourier Analysis Application, Vol. 2, No. 1, 1995, pp. 29-48. doi:10.1007/s00041-001-4021-8

[2] V. N. Temlyakov, “Greedy Algorithm and m-Term Trigonometric Approximation,” Constructive Approximation, Vol. 14, No. 4, 1998, pp. 569-587. doi:10.1007/s003659900090

[3] R. S. Li and Y. P. Liu, “The Asymptotic Estimations of Best m-Term One-Sided Approximation of Function Classes Determined by Fourier Coefficients,” Advance in Mathematics (China), Vol. 37, No. 2, 2008, pp. 211-221.

[4] R. Li and Y. Liu, “Best m-Term One-Sided Trigonometric Approximation of Some Function Classes Defined by a Kind of Multipliers,” Acta Mathematica Sinica, English Series, Vol. 26, No. 5, 2010, pp. 975-984. doi:10.1007/s10114-009-6478-3

[5] T. Ganelius, “On One-Sided Approximation by Trigonometrical Polynomials,” Mathematica Scandinavica, Vol. 4, 1956, pp. 247-258.

[6] E. Schmidt, “Zur Theorie der Linearen und Nichtlinearen Integralgleichungen,” Annals of Mathematics, Vol. 63, 1907, pp. 433-476. doi:10.1007/BF01449770

[7] A. S. Romanyuk, “Best m-Term Trigonometric Approximations of Besov Classes of Periodic Functions of Several Variables,” Izvestiya: Mathematics, Vol. 67, No. 2, 2003, pp. 265-302. doi:10.1070/IM2003v067n02ABEH000427

[8] S. V. Konyagin and V. N. Temlyakov, “Convergence of Greedy Approximation II. The Trigonometric Systerm,” Studia Mathematica, Vol. 159, No. 2, 2003, pp. 161-184. doi:10.4064/sm159-2-1

[9] V. N. Temlyakov, “The Best m-Term Approximation and Greedy Algorithms,” Advances in Computational Mathematics, Vol. 8, No. 3, 1998, pp. 249-265. doi:10.1023/A:1018900431309

[10] R. Li and Y. Liu, “The Asymptotic Estimations of Best m-Term Approximation and Greedy Algorithm for Multiplier Function Classes Defined by Fourier Series,” Chinese Journal of Engineering Mathematics, Vol. 25, No. 1, 2008, pp. 89-96. doi:10.3901/JME.2008.10.089

[11] V. A. Popov, “Onesided Approximation of Periodic Functions of Serveral Variables,” Comptes Rendus de Academie Bulgare Sciences, Vol. 35, No. 12, 1982, pp. 1639-1642.

[12] V. A. Popov, “On the One-Sided Approximation of Multivariate Functions,” In: C. K. Chui, L. L. Schumaker and J. D. Ward, Eds., Approximation Theory IV, Academic Press, New York, 1983.

[13] A. Zygmund, “Trigonometric Series II,” Cambridge University Press, New York, 1959.

[14] R. A. Devore and G. G. Lorentz, “Constructive Approximation,” Spring-Verlag, New York, Berlin, Heidelberg, 1993.

[15] R. A. Devore and V. Popov, “Interpolation of Besov Spaces,” American Mathematical Society, Vol. 305, No. 1, 1988, pp. 397-414.

[1] R. A. Devore and V. N. Temlyakov, “Nonlinear Approximation by Trigonometric Sums,” Journal of Fourier Analysis Application, Vol. 2, No. 1, 1995, pp. 29-48. doi:10.1007/s00041-001-4021-8

[2] V. N. Temlyakov, “Greedy Algorithm and m-Term Trigonometric Approximation,” Constructive Approximation, Vol. 14, No. 4, 1998, pp. 569-587. doi:10.1007/s003659900090

[3] R. S. Li and Y. P. Liu, “The Asymptotic Estimations of Best m-Term One-Sided Approximation of Function Classes Determined by Fourier Coefficients,” Advance in Mathematics (China), Vol. 37, No. 2, 2008, pp. 211-221.

[4] R. Li and Y. Liu, “Best m-Term One-Sided Trigonometric Approximation of Some Function Classes Defined by a Kind of Multipliers,” Acta Mathematica Sinica, English Series, Vol. 26, No. 5, 2010, pp. 975-984. doi:10.1007/s10114-009-6478-3

[5] T. Ganelius, “On One-Sided Approximation by Trigonometrical Polynomials,” Mathematica Scandinavica, Vol. 4, 1956, pp. 247-258.

[6] E. Schmidt, “Zur Theorie der Linearen und Nichtlinearen Integralgleichungen,” Annals of Mathematics, Vol. 63, 1907, pp. 433-476. doi:10.1007/BF01449770

[7] A. S. Romanyuk, “Best m-Term Trigonometric Approximations of Besov Classes of Periodic Functions of Several Variables,” Izvestiya: Mathematics, Vol. 67, No. 2, 2003, pp. 265-302. doi:10.1070/IM2003v067n02ABEH000427

[8] S. V. Konyagin and V. N. Temlyakov, “Convergence of Greedy Approximation II. The Trigonometric Systerm,” Studia Mathematica, Vol. 159, No. 2, 2003, pp. 161-184. doi:10.4064/sm159-2-1

[9] V. N. Temlyakov, “The Best m-Term Approximation and Greedy Algorithms,” Advances in Computational Mathematics, Vol. 8, No. 3, 1998, pp. 249-265. doi:10.1023/A:1018900431309

[10] R. Li and Y. Liu, “The Asymptotic Estimations of Best m-Term Approximation and Greedy Algorithm for Multiplier Function Classes Defined by Fourier Series,” Chinese Journal of Engineering Mathematics, Vol. 25, No. 1, 2008, pp. 89-96. doi:10.3901/JME.2008.10.089

[11] V. A. Popov, “Onesided Approximation of Periodic Functions of Serveral Variables,” Comptes Rendus de Academie Bulgare Sciences, Vol. 35, No. 12, 1982, pp. 1639-1642.

[12] V. A. Popov, “On the One-Sided Approximation of Multivariate Functions,” In: C. K. Chui, L. L. Schumaker and J. D. Ward, Eds., Approximation Theory IV, Academic Press, New York, 1983.

[13] A. Zygmund, “Trigonometric Series II,” Cambridge University Press, New York, 1959.

[14] R. A. Devore and G. G. Lorentz, “Constructive Approximation,” Spring-Verlag, New York, Berlin, Heidelberg, 1993.

[15] R. A. Devore and V. Popov, “Interpolation of Besov Spaces,” American Mathematical Society, Vol. 305, No. 1, 1988, pp. 397-414.