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Year 2021, , 20 - 33, 01.03.2021
https://doi.org/10.33205/cma.780906

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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.

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

Year 2021, , 20 - 33, 01.03.2021
https://doi.org/10.33205/cma.780906

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.

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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.
There are 6 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Sorın Gal 0000-0002-5743-3144

Ionut T. Iancu This is me 0000-0002-5743-3144

Project Number Not applicable
Publication Date March 1, 2021
Published in Issue Year 2021

Cite

APA Gal, S., & Iancu, I. T. (2021). Grüss and Grüss-Voronovskaya-type estimates for complex convolution polynomial operators. Constructive Mathematical Analysis, 4(1), 20-33. https://doi.org/10.33205/cma.780906
AMA Gal S, Iancu IT. Grüss and Grüss-Voronovskaya-type estimates for complex convolution polynomial operators. CMA. March 2021;4(1):20-33. doi:10.33205/cma.780906
Chicago Gal, Sorın, and Ionut T. Iancu. “Grüss and Grüss-Voronovskaya-Type Estimates for Complex Convolution Polynomial Operators”. Constructive Mathematical Analysis 4, no. 1 (March 2021): 20-33. https://doi.org/10.33205/cma.780906.
EndNote Gal S, Iancu IT (March 1, 2021) Grüss and Grüss-Voronovskaya-type estimates for complex convolution polynomial operators. Constructive Mathematical Analysis 4 1 20–33.
IEEE S. Gal and I. T. Iancu, “Grüss and Grüss-Voronovskaya-type estimates for complex convolution polynomial operators”, CMA, vol. 4, no. 1, pp. 20–33, 2021, doi: 10.33205/cma.780906.
ISNAD Gal, Sorın - Iancu, Ionut T. “Grüss and Grüss-Voronovskaya-Type Estimates for Complex Convolution Polynomial Operators”. Constructive Mathematical Analysis 4/1 (March 2021), 20-33. https://doi.org/10.33205/cma.780906.
JAMA Gal S, Iancu IT. Grüss and Grüss-Voronovskaya-type estimates for complex convolution polynomial operators. CMA. 2021;4:20–33.
MLA Gal, Sorın and Ionut T. Iancu. “Grüss and Grüss-Voronovskaya-Type Estimates for Complex Convolution Polynomial Operators”. Constructive Mathematical Analysis, vol. 4, no. 1, 2021, pp. 20-33, doi:10.33205/cma.780906.
Vancouver Gal S, Iancu IT. Grüss and Grüss-Voronovskaya-type estimates for complex convolution polynomial operators. CMA. 2021;4(1):20-33.