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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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On boundedness inequalities in the variation of certain Schurer-type operators; pp. 1–9

(Full article in PDF format) https://doi.org/10.3176/proc.2017.1.01


Authors

Andi Kivinukk, Tarmo Metsmägi

Abstract

This paper is concerned with boundedness inequalities in the variation for the higher order derivatives of general Schurertype operators. In particular, the boundedness inequalities in the variation for the higher order derivatives of the Bernstein–Schurer, Kantorovich–Schurer, and Durrmeyer–Schurer operators are derived.

Keywords

approximation theory, Schurer-type operators, boundedness inequalities, variation detracting property, absol continuous functions.

References

1. Bărbosu , D. Durrmeyer–Schurer type operators. Facta Univ. Ser. Math. Inform. , 2004 , 19 , 65–72.

2. Bardaro , C. , Butzer , P. L. , Stens , R. L. , and Vinti , G. Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals. Analysis (Munich) , 2003 , 23(4) , 299–340.
https://doi.org/10.1524/anly.2003.23.4.299

3. Butzer , P. L. and Nessel , R. J. Fourier Analysis and Approximation. Birkhäuser Verlag , Basel , and Academic Press , New York , 1971.
https://doi.org/10.1007/978-3-0348-7448-9

4. Căbulea , L. Some properties of the Schurer type operators. Acta Univ. Apulensis Math. Inform. , 2008 , 15 , 255–261.

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https://doi.org/10.1016/0021-9045(81)90101-5

6. DeVore , R. A. and Lorentz , G. G. Constructive Approximation. Springer , Berlin , 1993. A. Kivinukk and T. Metsm¨agi: Boundedness inequalities in the variation of Schurer-type operators 9

7. Durrmeyer , J. L. Une formule d’inversion , de la transformée de Laplace: Applications à la théorie des moments , Thèse de 3e cycle. Faculté des Sciences de l’Universitée de Paris , 1967.

8. Fichtenholz , G. M. Differential- und Integralrechnung. II. VEB Deutscher Verlag der Wissenschaften , Berlin , 1966.

9. Goodman , T. N. T. Variation diminishing properties of Bernstein polynomials on triangles. J. Approx. Theory , 1987 , 50 , 111–126.
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10. Lorentz , G. G. Zur Theorie der Polynome von S. Bernstein. Mat. Sb. , 1937 , 2 , 543–556.

11. Schurer , F. Linear Positive Operators in Approximation Theory. Math. Inst. Techn. Univ. Delft: report , 1962.

12. Schurer , F. On the order of approximation with generalized Bernstein polynomials. Indag. Math. , 1962 , 24 , 484–488.
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13. Timan , A. F. Theory of Approximation of Functions of a Real Variable. Dover Publications , Inc. , New York , 1994.

14. Zhuk , V. V. Lectures on the Theory of Approximation. Saint Petersburg , 2008 (in Russian).

 
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Current Issue: Vol. 68, Issue 3, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December