Abstract
The approximation properties of the spaces S pϕ introduced by Stepanets’ were studied in a series of works of Stepanets’ and his disciples. In these works, problems related to the determination of exact values of n-term approximations of q-ellipsoids in these spaces were reduced to some extremal problems for series with terms that are products of elements of two nonnegative sequences one of which is fixed and the other varies on a certain set.
Since solutions of these extremal problems may be of independent interest, in the present work we develop a new method for finding these solutions that enables one to obtain the required result in a substantially shorter and more transparent way.
Similar content being viewed by others
References
A. I. Stepanets, “Approximation characteristics of spaces S pϕ ,” Ukr. Mat. Zh., 53, No. 3, 392–416 (2001).
A. I. Stepanets, “Approximation characteristics of spaces S pϕ in different metrics,” Ukr. Mat. Zh., 53, No.8, 1121–1146 (2001).
V. R. Voitsekhivs’kyi, “Jackson-type inequalities in the approximation of functions from the space S p by Zygmund sums,” in: Theory of Approximation of Functions and Related Problems [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2002), pp. 33–46.
A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 2, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2002).
A. I. Stepanets, “Approximation characteristics of spaces S p,” in: Proceedings of the Ukrainian Mathematical Congress-2001 [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2002), pp. 208–226.
A. I. Stepanets and A. S. Serdyuk, “Direct and inverse theorems in the theory of approximation of functions in the space S p,” Ukr. Mat. Zh., 54, No. 1, 106–124 (2002).
S. B. Vakarchuk, “On some extremal problems of approximation theory in spaces S p (1 ≤ p < ∞),” in: Proceedings of the Voronezh Winter Mathematical School “Modern Methods in the Theory of Functions and Related Problems” [in Russian], (Voronezh, January 26–February 2, 2003), Voronezh University, Voronezh (2003), pp. 47–48.
V. R. Voitsekhivs’kyi, “Widths of some classes from the space S p,” in: Extremal Problems in the Theory of Functions [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2003), pp. 17–26.
V. I. Rukasov, “Best n-term approximations in spaces with nonsymmetric metric,” Ukr. Mat. Zh., 55, No. 6, 806–816 (2003).
A. S. Serdyuk, “Widths of classes of functions defined by the moduli of continuity of their ψ-derivatives in the space S p,” in: Extremal Problems in the Theory of Functions [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2003), pp. 229–248.
A. I. Stepanets, “Extremal problems of approximation theory in linear spaces,” Ukr. Mat. Zh., 55, No. 10, 1392–1423 (2003).
A. I. Stepanets and V. I. Rukasov, “Spaces S p with nonsymmetric metric,” Ukr. Mat. Zh., 55, No. 2, 264–277 (2003).
A. I. Stepanets and V. I. Rukasov, “Best “continuous” n-term approximations in the spaces S pϕ ,” Ukr. Mat. Zh., 55, No. 5, 663–670 (2003).
O. I. Stepanets’ and A. L. Shydlich, “Best n-term approximations by Λ-methods in the spaces S pϕ ,” Ukr. Mat. Zh., 55, No. 8, 1107–1126 (2003).
A. L. Shydlich, “Best n-term approximations by Λ-methods in the spaces S pϕ , ” in: Extremal Problems in the Theory of Functions [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (2003), pp. 283–306.
S. B. Vakarchuk, “Jackson-type inequalities and exact values of widths of classes of functions in the spaces S p, 1 ≤ p < ∞,” Ukr. Mat. Zh., 56, No. 5, 595–605 (2004).
A. I. Stepanets, “Best approximations of q-ellipsoids in spaces S pϕ ,” Ukr. Mat. Zh., 56, No. 10, 1378–1383 (2004).
A. L. Shydlich, “On saturation of linear summation methods for Fourier series in the spaces S pϕ ,” Ukr. Mat. Zh., 56, No. 1, 133–138 (2004).
A. I. Stepanets, “Best n-term approximations with restrictions,” Ukr. Mat. Zh., 57, No. 4, 533–553 (2005).
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge (1934).
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer, Dordrecht (1993).
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1677–1683, December, 2005.
Rights and permissions
About this article
Cite this article
Stepanets’, O.I., Shydlich, A.L. On one extremal problem for positive series. Ukr Math J 57, 1968–1976 (2005). https://doi.org/10.1007/s11253-006-0042-4
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-006-0042-4