Research Article
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Year 2020, , 64 - 74, 01.06.2020
https://doi.org/10.33205/cma.688661

Abstract

References

  • U. Abel: An identity for a general class of approximation operators. J. Approx. Theory 142 (2006), 20--35.
  • U. Abel, O. Agratini: Asymptotic behaviour of Jain operators. Numer. Algor. 71 (2016), 553--565.
  • U. Abel, O. Agratini: On the variation detracting property of operators of Balazs and Szabados. Acta Math. Hungar. 150 (2016), 383--395.
  • U. Abel, B. della Vecchia: Asymptotic approximation by the operators of K. Balazs and Szabados. Acta Sci. Math. (Szeged) 66 (1-2) (2000), 137--145.
  • U. Abel, W. Gawronski and T. Neuschel: Complete monotonicity and zeros of sums of squared Baskakov functions. Appl. Math. Comput. 258 (2015), 130--137.
  • T. Acar: Quantitative q-Voronovskaya and q-Gruss-Voronovskaya-type results for q-Szasz operators. Georgian Math. J. 23 (2016), 459--468.
  • A.M. Acu, H. Gonska and I. Raşa: Gruss-type and Ostrowski-type in approximation theory. Ukr. Math. J. 63 (2011), 843--864.
  • O. Agratini: On approximation properties of Balazs-Szabados operators and their Kantorovich extension. Korean J. Comput. & Appl. Math. 9 (2002), 361--372.
  • O. Agratini: Properties of discrete non-multiplicative operators. Anal. Math. Phys. 9 (2019), 131--141.
  • D. Andrica, C. Badea: Gruss inequality for positive linear functionals. Period. Math. Hungar. 19 (1988), 155--167.
  • C. Atakut: On the approximation of functions together with derivatives by certain linear positive operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 46 (1997), 57--65.
  • K. Balâzs: Approximation by Bernstein type rational functions. Acta Math. Acad. Sci. Hungar. 26 (1975), 123--134.
  • C. Balâzs, J. Szabados: Approximation by Bernstein type rational functions. II. Acta Math. Acad. Sci. Hungar. 40 (1982), 331--337.
  • E. Berdysheva: Studying Baskakov-Durrmeyer operators and quasi-interpolants via special functions. J. Approx. Theory 149 (2007), 131--150.
  • P. L. Chebyshev: Sur les expressions approximatives des integrales definies par les autres prises entre les meme limites. Proc. Math. Soc. Kharkov 2 (1882), 93--98.
  • E. Deniz: Quantitative estimates for Jain–Kantorovich operators. Commun. Fac. Sci. Univ. Ank. Sêr. A1 Math. Stat. 65 (2016), 121--132.
  • A. Farcaş: An asymptotic formula for Jain's operators. Stud. Univ. Babeş-Bolyai Math. 57 (2012), 511--517.
  • S. G. Gal, H. Gonska: Gruss and Gruss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables. Jaen J. Approx. 7 (2015), 97--122.
  • B. Gavrea,I. Gavrea: Ostrowski type inequalities from a linear functional point of view. J. Inequal. Pure Appl. Math. 1 (2000), Article 11.
  • H. Gonska, I. Raşa and M. D. Rusu: Cebysev-Gruss inequalities revisited. Math. Slov. 63 (2013), 1007--1024.
  • H. Gonska, I. Raşa and M. D. Rusu: Chebyshev-Gruss-type inequalities via discrete oscillations. Bul. Acad. Ştiinte Repub. Mold. Mat. 1 (74) (2014), 63--89.
  • H. Gonska, G. Tachev}: Gruss type inequality for positive linear operators with second order moduli. Mat. Vesn. 63 (2011), 247--252.
  • G. C. Greubel: A note on Jain basis functions. arXiv:1612.09385 [math.CA], (2016)
  • G. Grûss: Uber das Maximum des Absoluten Betrages von $\frac{1}{b-a}\int_a^b f(x)g(x)dx- \frac{1}{(b-a)^2}\int_a^b f(x)dx\ \int_a^b g(x)dx$. Math. Z. 39 (1935), 215--226.
  • A. Holhoş: Quantitative Estimates of Voronovskaya Type in Weighted Spaces. Results Math. 73 (2018), 53.
  • A. Holhoş: A Voronovskaya-Type Theorem for the First Derivatives of Positive Linear Operators. Results Math. 74 (2019), 76, https://doi.org/10.1007/s00025-019-0992-0
  • C. Impens, I. Gavrea: A Leibniz differentiation formula for positive operators. J. Math. Anal. Appl. 271 (2002), 175--181.
  • M. E. H. Ismail, C. P. May: On a Family of Approximation Operators. J. Math. Anal. Appl. 63 (1978), 446--462.
  • G. C. Jain: Approximation of functions by a new class of linear operators. J. Aust. Math. Soc. 13 (1972), 271--276.
  • A. Kajla, S. Deshwal and P. N. Agrawal: Quantitative Voronovskaya and Gruss-Voronovskaya type theorems for Jain-Durrmeyer operators of blending type. Anal. Math. Phys. 9 (2019), 1241--1263.
  • C. P. May: Saturation and inverse theorems for combinations of a class of exponential-type operators. Canad. J. Math 28 (1976), 1224--1250.
  • I. Raşa: Entropies and Heun functions associated with positive linear operators. Appl. Math. Comput. 268 (2015), 422--431.
  • M. D. Rusu: On Gruss-type inequalities for positive linear operators. Stud. Univ. Babes-Bolyai Math. 56 (2011), 551--565.
  • V. Totik: Saturation for Bernstein type rational functions. Acta Math. Hungar. 43 (1984), 219--250.
  • G. Ulusoy,T. Acar: q-Voronovskaya type theorems for q-Baskakov operators. Math. Methods Appl. Sci. 39 (2016), 3391--3401.
  • A. Wafi, S. Khatoon: Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators. Cent. Eur. J. Math. 6 (2008), 325--334.

The Product of Two Functions Using Positive Linear Operators

Year 2020, , 64 - 74, 01.06.2020
https://doi.org/10.33205/cma.688661

Abstract

In this paper we estimate the speed of convergence of the difference $L_n(fg)-(L_n f)\cdot (L_n g)$ towards 0, where $(L_n)$ are positive linear operators used in the approximation of continuous functions. We also study in what conditions the formula ${L'_n}(fg)-f {L'_n}g-g {L'_n}f \to 0$ holds true.

References

  • U. Abel: An identity for a general class of approximation operators. J. Approx. Theory 142 (2006), 20--35.
  • U. Abel, O. Agratini: Asymptotic behaviour of Jain operators. Numer. Algor. 71 (2016), 553--565.
  • U. Abel, O. Agratini: On the variation detracting property of operators of Balazs and Szabados. Acta Math. Hungar. 150 (2016), 383--395.
  • U. Abel, B. della Vecchia: Asymptotic approximation by the operators of K. Balazs and Szabados. Acta Sci. Math. (Szeged) 66 (1-2) (2000), 137--145.
  • U. Abel, W. Gawronski and T. Neuschel: Complete monotonicity and zeros of sums of squared Baskakov functions. Appl. Math. Comput. 258 (2015), 130--137.
  • T. Acar: Quantitative q-Voronovskaya and q-Gruss-Voronovskaya-type results for q-Szasz operators. Georgian Math. J. 23 (2016), 459--468.
  • A.M. Acu, H. Gonska and I. Raşa: Gruss-type and Ostrowski-type in approximation theory. Ukr. Math. J. 63 (2011), 843--864.
  • O. Agratini: On approximation properties of Balazs-Szabados operators and their Kantorovich extension. Korean J. Comput. & Appl. Math. 9 (2002), 361--372.
  • O. Agratini: Properties of discrete non-multiplicative operators. Anal. Math. Phys. 9 (2019), 131--141.
  • D. Andrica, C. Badea: Gruss inequality for positive linear functionals. Period. Math. Hungar. 19 (1988), 155--167.
  • C. Atakut: On the approximation of functions together with derivatives by certain linear positive operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 46 (1997), 57--65.
  • K. Balâzs: Approximation by Bernstein type rational functions. Acta Math. Acad. Sci. Hungar. 26 (1975), 123--134.
  • C. Balâzs, J. Szabados: Approximation by Bernstein type rational functions. II. Acta Math. Acad. Sci. Hungar. 40 (1982), 331--337.
  • E. Berdysheva: Studying Baskakov-Durrmeyer operators and quasi-interpolants via special functions. J. Approx. Theory 149 (2007), 131--150.
  • P. L. Chebyshev: Sur les expressions approximatives des integrales definies par les autres prises entre les meme limites. Proc. Math. Soc. Kharkov 2 (1882), 93--98.
  • E. Deniz: Quantitative estimates for Jain–Kantorovich operators. Commun. Fac. Sci. Univ. Ank. Sêr. A1 Math. Stat. 65 (2016), 121--132.
  • A. Farcaş: An asymptotic formula for Jain's operators. Stud. Univ. Babeş-Bolyai Math. 57 (2012), 511--517.
  • S. G. Gal, H. Gonska: Gruss and Gruss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables. Jaen J. Approx. 7 (2015), 97--122.
  • B. Gavrea,I. Gavrea: Ostrowski type inequalities from a linear functional point of view. J. Inequal. Pure Appl. Math. 1 (2000), Article 11.
  • H. Gonska, I. Raşa and M. D. Rusu: Cebysev-Gruss inequalities revisited. Math. Slov. 63 (2013), 1007--1024.
  • H. Gonska, I. Raşa and M. D. Rusu: Chebyshev-Gruss-type inequalities via discrete oscillations. Bul. Acad. Ştiinte Repub. Mold. Mat. 1 (74) (2014), 63--89.
  • H. Gonska, G. Tachev}: Gruss type inequality for positive linear operators with second order moduli. Mat. Vesn. 63 (2011), 247--252.
  • G. C. Greubel: A note on Jain basis functions. arXiv:1612.09385 [math.CA], (2016)
  • G. Grûss: Uber das Maximum des Absoluten Betrages von $\frac{1}{b-a}\int_a^b f(x)g(x)dx- \frac{1}{(b-a)^2}\int_a^b f(x)dx\ \int_a^b g(x)dx$. Math. Z. 39 (1935), 215--226.
  • A. Holhoş: Quantitative Estimates of Voronovskaya Type in Weighted Spaces. Results Math. 73 (2018), 53.
  • A. Holhoş: A Voronovskaya-Type Theorem for the First Derivatives of Positive Linear Operators. Results Math. 74 (2019), 76, https://doi.org/10.1007/s00025-019-0992-0
  • C. Impens, I. Gavrea: A Leibniz differentiation formula for positive operators. J. Math. Anal. Appl. 271 (2002), 175--181.
  • M. E. H. Ismail, C. P. May: On a Family of Approximation Operators. J. Math. Anal. Appl. 63 (1978), 446--462.
  • G. C. Jain: Approximation of functions by a new class of linear operators. J. Aust. Math. Soc. 13 (1972), 271--276.
  • A. Kajla, S. Deshwal and P. N. Agrawal: Quantitative Voronovskaya and Gruss-Voronovskaya type theorems for Jain-Durrmeyer operators of blending type. Anal. Math. Phys. 9 (2019), 1241--1263.
  • C. P. May: Saturation and inverse theorems for combinations of a class of exponential-type operators. Canad. J. Math 28 (1976), 1224--1250.
  • I. Raşa: Entropies and Heun functions associated with positive linear operators. Appl. Math. Comput. 268 (2015), 422--431.
  • M. D. Rusu: On Gruss-type inequalities for positive linear operators. Stud. Univ. Babes-Bolyai Math. 56 (2011), 551--565.
  • V. Totik: Saturation for Bernstein type rational functions. Acta Math. Hungar. 43 (1984), 219--250.
  • G. Ulusoy,T. Acar: q-Voronovskaya type theorems for q-Baskakov operators. Math. Methods Appl. Sci. 39 (2016), 3391--3401.
  • A. Wafi, S. Khatoon: Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators. Cent. Eur. J. Math. 6 (2008), 325--334.
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Adrian Holhoş

Publication Date June 1, 2020
Published in Issue Year 2020

Cite

APA Holhoş, A. (2020). The Product of Two Functions Using Positive Linear Operators. Constructive Mathematical Analysis, 3(2), 64-74. https://doi.org/10.33205/cma.688661
AMA Holhoş A. The Product of Two Functions Using Positive Linear Operators. CMA. June 2020;3(2):64-74. doi:10.33205/cma.688661
Chicago Holhoş, Adrian. “The Product of Two Functions Using Positive Linear Operators”. Constructive Mathematical Analysis 3, no. 2 (June 2020): 64-74. https://doi.org/10.33205/cma.688661.
EndNote Holhoş A (June 1, 2020) The Product of Two Functions Using Positive Linear Operators. Constructive Mathematical Analysis 3 2 64–74.
IEEE A. Holhoş, “The Product of Two Functions Using Positive Linear Operators”, CMA, vol. 3, no. 2, pp. 64–74, 2020, doi: 10.33205/cma.688661.
ISNAD Holhoş, Adrian. “The Product of Two Functions Using Positive Linear Operators”. Constructive Mathematical Analysis 3/2 (June 2020), 64-74. https://doi.org/10.33205/cma.688661.
JAMA Holhoş A. The Product of Two Functions Using Positive Linear Operators. CMA. 2020;3:64–74.
MLA Holhoş, Adrian. “The Product of Two Functions Using Positive Linear Operators”. Constructive Mathematical Analysis, vol. 3, no. 2, 2020, pp. 64-74, doi:10.33205/cma.688661.
Vancouver Holhoş A. The Product of Two Functions Using Positive Linear Operators. CMA. 2020;3(2):64-7.