Skip to main navigation menu Skip to main content Skip to site footer

Vol. 46 No. 3 (2020)
Serdica Mathematical Journal

Articles

Strong converse inequalities for the weighted simultaneous approximation by the Szász-Mirakjan operator

Published 18-06-2021

  • Borislav R. Draganov

Borislav R. Draganov
St. Kl. Ohridski University of Sofia and Institute of Mathematics and Informatics–BAS

Abstract

We establish two-term strong converse estimates of the rate of weighted simultaneous approximation by the Szász-Mirakjan operator for smooth functions in the supremum norm on the non-negative semi-axis. We consider Jacobi-type weights. The estimates are stated in terms of appropriate moduli of smoothness or \(K\)-functionals.

2020 Mathematics Subject Classification:

41A17, 41A25, 41A27, 41A28, 41A35, 41A40, 41A81

Keywords

Szász-Mirakjan operator, strong converse inequality, converse estimate, simultaneous approximation, modulus of smoothness, K-functional

Full text

Author Details

Borislav R. Draganov

Department of Mathematics and Informatics
Sofia University “St. Kliment Ohridski”
5, James Bourchier Blvd
1164 Sofia, Bulgaria
and
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., Bl. 8
1113 Sofia, Bulgaria
e-mail: bdraganov@fmi.uni-sofia.bg

References

  1. T. Acar, A. Aral, I. Raşa. Approximation by k-th order modifications of Szász-Mirakyan operators. Studia Sci. Math. Hungar. 53, 3 (2016), 379–398.
  2. A. Aral, G. Tachev. Quantitative Voronovskaya type theorems for a general sequence of linear positive operators. Filomat 33, 8 (2019), 2507–2518.
  3. R. A. DeVore, G. G. Lorentz. Constructive Approximation. Fundamental Principles of Mathematical Sciences, vol. 303. Berlin, Springer-Verlag, 1993.
  4. Z. Ditzian, K. G. Ivanov. Strong converse inequalities. J. Anal. Math. 61 (1993), 61–111.
  5. Z. Ditzian, V. Totik. Moduli of Smoothness. Springer Series in Computational Mathematics, vol. 9. New York, Springer-Verlag, 1987.
  6. Z. Ditzian, V. Totik. K-functionals and weighted moduli of smoothness. J. Approx. Theory 63, 1 (1990), 3–29.
  7. B. R. Draganov. Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums. J. Approx. Theory 200 (2015), 92–135.
  8. B. R. Draganov. Direct estimates of the weighted simultaneous approximation by the Szász-Mirakjan operator. Period. Math. Hungar. (2020), https://doi.org/10.1007/s10998-020-00370-x.
  9. B. R. Draganov, K. G. Ivanov. A characterization of weighted approximations by the Post-Widder and the Gamma operators (II). J. Approx. Theory 162, 10 (2010), 1805–1851.
  10. B. R. Draganov, I. Gadjev. Approximation of functions by the Szász–Mirakjan–Kantorovich Operator. Numer. Funct. Anal. Optim. 40, 7 (2019), 803–824.
  11. R. Grassl, O. Levin. More Discrete Mathematics: via Graph Theory, 2018, http://discrete.openmathbooks.org/more/mdm/frontmatter.html.
  12. V. Gupta, G. Tachev. Approximation with Positive Linear Operators and Linear Combinations. Developments in Mathematics, vol. 50. Cham, Springer, 2017.
  13. R. Martini. On the approximation of functions together with their derivatives by certain linear positive operators. Indag. Math. 31 (1969), 473–481.
  14. V. Totik. Uniform approximation by Szász–Mirakjan type operators. Acta Math. Hungar. 41, 3–4 (1983), 291–307.
  15. V. Totik. Uniform approximation by positive operators on infinite intervals. Anal. Math. 10, 2 (1984), 163–182.