APM  Vol.2 No.3 , May 2012
The Best m-Term One-Sided Approximation of Besov Classes by the Trigonometric Polynomials
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 metricLp(Td(1≤p≤∞ are given.

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.
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.

 
 
Top