In the spaces L ψ (T m) of periodic functions with metric
where ψ is a function of the type of modulus of continuity, we study the direct Jackson theorem in the case of approximation by trigonometric polynomials. It is proved that the direct Jackson theorem is true if and only if the lower dilation index of the function ψ is not equal to zero.
Similar content being viewed by others
References
S. A. Pichugov, “On the Jackson theorem for periodic functions in spaces with integral metric,” Ukr. Mat. Zh., 52, No. 1, 122–133 (2000); English translation: Ukr. Math. J., 52, No. 1, 133–147 (2000).
É. A. Storozhenko, P. Oswald, and V. G. Krotov, “Direct and inverse theorems of the Jackson type in the spaces L p , 0 < p < 1,” Mat. Sb., 98, No. 3, 395–415 (1975).
S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University, Princeton (1971).
V. V. Goncharov, Theory of Interpolation and Approximation of Functions [in Russian], Gostekhizdat, Moscow (1954).
K. Runovski, “On Jackson’s type inequalities in Orlicz classes,” Revista Mat. Comp., 14, No. 2, 394–404 (2001).
É. A. Storozhenko and P. Oswald, “Jackson theorems in the spaces L p (Tn), 0 < p < 1,” Sib. Mat. Zh., 19, No. 4, 888–901 (1978).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 11, pp. 1524–1533, November, 2011.
Rights and permissions
About this article
Cite this article
Pichugov, S.A. On the Jackson theorem for periodic functions in metric spaces with integral metric. II. Ukr Math J 63, 1733–1744 (2012). https://doi.org/10.1007/s11253-012-0609-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-012-0609-1