Abstract
We discuss the relations among the best approximation E n (f) and the Fourier coefficients of a function, \( \hat f(n) \in C,n = 0, \pm 1, \pm 2, \ldots , \) under the conditions that \( \{ \hat f(n)\} _{n = 0}^\infty \in MVBVS^* \) and \( \{ \hat f(n) + f( - n)\} _{n = 0}^\infty \in MVBVS^* \), where MVBVS* is the class of the so-called Strong Mean Value Bounded Variation Sequences.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belov A S. On sequential estimate of best approximation and moduli of continuity by sums of trigonometric series with quasimonotone coefficients (in Russian). Matem Zemetki, 51: 132–134 (1992)
Xiao W, Xie T F, Zhou S P. The best approximation rate of certain trigonometric series (in Chinese). Chinese Ann Math Ser A, 21: 81–88 (2000)
Le R J, Zhou S P. A new condition for the uniform convergence of certain trigonometric series. Acta Math Hungar, 108: 161–169 (2005)
Leindler L. On the uniform convergence and boundedness of a certain class of sine series. Anal Math, 27: 279–285 (2001)
Yu D S, Zhou S P. A generalization of monotone condition and applications. Acta Math Hungar, 115: 247–267 (2007)
Zhou S P. What strong monotonicity condition on Fourier coefficients can make the ratio |f-S n (f)|/E n (f) bounded? Proc Amer Math Soc, 121: 779–785 (1994)
Zhou S P, Le R J. Some remarks on the best approximation rate of certain trigonometric series (in Chinese). Acta Math Sinica Chin Ser, 49: 509–518 (2006)
Zhou S P, Zhou P, Yu D S. The ultimate condition to generalize monotonicity for uniform convergence of trigonometric series. arXiv: math.CA/0611805 v1, November 27, 2006, preprint
Zygmund A. Trigonometric Series, Vol. I. 2nd ed. Cambridge: Cambridge University Press, 1959
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by NSERC RCD grant and AARMS of Canada (Yu D S), by NSERC of Canada (Zhou P), and by the National Natural Science Foundation of China (Grant No. 10471130) and the Open Fund (Grant No. PLN0613) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University) (Zhou S P)
Rights and permissions
About this article
Cite this article
Yu, D., Zhou, P. & Zhou, S. On the relations among best approximation and Fourier coefficients. Sci. China Ser. A-Math. 51, 1883–1894 (2008). https://doi.org/10.1007/s11425-007-0162-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-007-0162-9