Abstract and Applied Analysis

On Approximations by Trigonometric Polynomials of Classes of Functions Defined by Moduli of Smoothness

Nimete Sh. Berisha, Faton M. Berisha, Mikhail K. Potapov, and Marjan Dema

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we give a characterization of Nikol’skiĭ-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to such a class are given. In order to prove our results, we make use of certain recent reverse Copson-type and Leindler-type inequalities.

Article information

Source
Abstr. Appl. Anal., Volume 2017 (2017), Article ID 9323181, 11 pages.

Dates
Received: 24 November 2016
Accepted: 19 January 2017
First available in Project Euclid: 12 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1491962538

Digital Object Identifier
doi:10.1155/2017/9323181

Mathematical Reviews number (MathSciNet)
MR3630651

Zentralblatt MATH identifier
06929570

Citation

Berisha, Nimete Sh.; Berisha, Faton M.; Potapov, Mikhail K.; Dema, Marjan. On Approximations by Trigonometric Polynomials of Classes of Functions Defined by Moduli of Smoothness. Abstr. Appl. Anal. 2017 (2017), Article ID 9323181, 11 pages. doi:10.1155/2017/9323181. https://projecteuclid.org/euclid.aaa/1491962538


Export citation

References

  • B. Laković, “Ob odnom klasse funktsiĭ,” Matematički Vesnik, vol. 39, no. 4, pp. 405–415, 1987.
  • S. Tikhonov, “Characteristics of Besov-Nikol'skiĭ class of function,” Electronic Transactions on Numerical Analysis, vol. 19, pp. 94–104, 2005.
  • O. V. Besov, V. P. Il'in, and S. M. Nikol'skiĭ, Integral Representations of Functions and Imbedding Theorems, vol. 2 of Scripta Series in Mathematics, V. H. Winston & Sons, Washington, DC, USA; Halsted Press [John Wiley & Sons], New York, NY, USA, 1979, Edited by M. H. Taibleson.
  • M. K. Potapov, F. M. Berisha, N. S. Berisha, and R. Kadriu, “Some reverse lp-type inequalities involving certain quasi monotone sequences,” Mathematical Inequalities and Applications, vol. 18, no. 4, pp. 1245–1252, 2015.
  • S. Tikhonov, “Trigonometric series with general monotone coefficients,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 721–735, 2007.
  • E. Liflyand and S. Tikhonov, “A concept of general monotonicity and applications,” Mathematische Nachrichten, vol. 284, no. 8-9, pp. 1083–1098, 2011.
  • A. A. Konyushkov, “O klassakh lipshitsa,” Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, vol. 21, no. 3, pp. 423–448, 1957.
  • M. K. Potapov and M. Q. Berisha, “Moduli of smoothness and the Fourier coefficients of periodic functions of one variable,” Publications de l'Institut Mathématique (Beograd), vol. 26, no. 40, pp. 215–228, 1979.
  • G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 2nd edition, 1988.
  • E. T. Copson, “Note on series of positive terms,” Journal of the London Mathematical Society, vol. 1-3, no. 1, pp. 49–51, 1928.
  • L. Leindler, “Generalization of inequalities of Hardy and Littlewood,” Acta Scientiarum Mathematicarum (Szeged), vol. 31, pp. 279–285, 1970.
  • L. Leindler, “Power-monotone sequences and Fourier series with positive coefficients,” JIPAM Journal of Inequalities in Pure and Applied Mathematics, vol. 1, no. 1, article 1, 10 pages, 2000.
  • S. Tikhonov and M. Zeltser, “Weak monotonicity concept and its applications,” Trends in Mathematics, vol. 63, pp. 357–374, 2014.
  • A. Zygmund, Trigonometric Series, vol. 2, Cambridge University Press, Cambridge, UK, 1988, Russian translation, Gosudarstv. Izdat. Inostrannoĭ Literatury, Moscow, 1965.
  • A. F. Timan, Theory of Approximation of Functions of a Real Variable, vol. 34 of International Series of Monographs in Pure and Applied Mathematics, A Pergamon Press Book. The Macmillan Co., New York, NY, USA, 1963, Translated from the Russian by J. Berry, English translation edited and editorial preface by J. Cossar.
  • M. Q. Berisha and F. M. Berisha, “On monotone Fourier coefficients of a function belonging to Nikol'skiĭ-Besov classes,” Mathematica Montisnigri, vol. 10, pp. 5–20, 1999. \endinput