Grüss and Grüss-Voronovskaya-type estimates for complex convolution polynomial operators

Year 2021, Volume: 4 Issue: 1, 20 - 33, 01.03.2021

Sorın Gal , Ionut T. Iancu

https://doi.org/10.33205/cma.780906

Cited By: 2

Abstract

A classical well-known result in approximation theory is the Grüss inequality for positive linear functionals, which gives an upper bound for the Chebyshev-type functional.

Starting also from a problem posed by Gavrea, this inequality was also investigated in terms of the least concave majorants of the moduli of continuity and for positive linear operators by Acu, Gonska, Rasa and Rusu, where the cases of classical Hermite-Fejer and Fejer-Korovkin convolution operators were considered.

Refined versions of the Grüss-type inequality in the spirit of Voronovskaya's theorem were obtained by Gal and Gonska
for Bernstein and Paltanea operators of real variable and for complex Bernstein, genuine Bernstein-Durrmeyer and Bernstein-Faber operators attached to analytic functions of complex variable.

After the appearance of these papers, several papers by other authors have developed these directions of research.

The goal of this paper is to continue the above mentioned directions of research, obtaining Grüss and Grüss-Voronovskaya exact estimates (with respect to the degree of polynomials) for the de la Vallee-Poussin complex polynomials in Section 2, for Zygmund-Riesz complex polynomials in Section 3 and for Jackson complex polynomials in Section 4.

Keywords

Complex convolution polynomials, Gruss-type estimate, Gruss-Voronovskaya-type estimate

Supporting Institution

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Project Number

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Thanks

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References

• A. M. Acu, H. Gonska and I. Ra¸sa: Grüss-type and Ostrovski-type inequalities in approximation theory. Ukrainian Mathematical Journal 63 (2011), No. 6, 843-864.
• S. G. Gal: Approximation by Complex Bernstein and Convolution Type Operators. World Scientific Publ. Co., New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2009.
• I. Gavrea, B. Gavrea: Ostrowski type inequalities from a linear functional point of view. JIPAM. J. Inequal. Pure Appl. Math. 1 (2000), Article 11.
• S. G. Gal, H. Gonska: Grüss and Grüss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables. Jaen J. Approx. 7 (1)(2015), 97-122.
• H. Gonska, I. Ra¸sa and M. Rusu: Cebysev-Grüss-type inequalities revisited. Mathematica Slovaca 63 (2013), No. 5, 1007-1024.
• W. Rudin: Principles of Mathematical Analysis. McGraw-Hill, Inc., New York, 1976.