Research Article
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Year 2019, , 15 - 21, 01.03.2019
https://doi.org/10.33205/cma.481186

Abstract

References

  • [1] F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, deGruyter Studies in Mathematics, vol. 17. Walter de Gruyter, New York, 1994.
  • [2] E. E. Berdysheva and B.-Z. Li, On $L^{p}$-convergence of Bernstein-Durrmeyer operators with respect to arbitrary measure, Publ. Inst. Math. (Beograd) (N.S.). 96(110) (2014), 23-29.
  • [3] M. Campiti and G. Metafune, $L^{p}$-convergence of Bernstein-Kantorovich-type operators, Ann. Polon. Math., LXIII (1996), 273-280.
  • [4] J. Cerdà, J., Martín and P., Silvestre, Capacitary function spaces, Collect. Math., 62 (2011), 95-118.
  • [5] G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble), 5 (1954), 131-295.
  • [6] D. Denneberg, Non-Additive Measure and Integral, Kluwer Academic Publisher, Dordrecht, 1994.
  • [7] S. G. Gal and B. D. Opris, Uniform and pointwise convergence of Bernstein-Durrmeyer operators with respect to monotone and submodular set functions, J. Math. Anal. Appl. 424 (2015), 1374-1379.
  • [8] S. G. Gal, Approximation by Choquet integral operators, Ann. Mat. Pura Appl., 195 (2016), No. 3, 881-896.
  • [9] S. G. Gal and S. Trifa, Quantitative estimates in uniform and pointwise approximation by Bernstein-Durrmeyer-Choquet operators, Carpath. J. Math., 33 (2017), 49-58.
  • [10] S. G. Gal and S. Trifa, Quantitative estimates in $L^{p}$-approximation by Bernstein-Durrmeyer-Choquet operators with respect to distorted Borel measures, Results Math., 72 (2017), no. 3, 1405-1415.
  • [11] S. G. Gal, Uniform and pointwise quantitative approximation by Kantorovich-Choquet type integral operators with respect to monotone and submodular set functions, Mediterr. J. Math., 14 (2017), no. 5, Art. 205, 12 pp.
  • [12] S. G. Gal, The Choquet integral in capacity, Real Analysis Exchange, 43 (2) (2018), 263-280.
  • [13] S. G. Gal, Shape preserving properties and monotonicity properties of the sequences of Choquet type integral operators, J. Numer. Anal. Approx. Theory, under press.
  • [14] S. G. Gal, Quantitative approximation by Stancu-Durrmeyer-Choquet-Šipoš operators, Math. Slovaca, under press.
  • [15] S. G. Gal, Fredholm-Choquet integral equations, J. Integral Equations Applications, https://projecteuclid.org/ euclid. jiea/1542358961, under press.
  • [16] S. G. Gal, Volterra-Choquet integral equations, J. Integral Equations Applications, https://projecteuclid.org/ euclid. jiea/1541668067. under press.
  • [17] L. V. Kantorovich, Sur certains développements suivant les polynômes de la forme de S. Bernstein, I, II, C.R. Acad. Sci. URSS (1930) 563-568, 595-600.
  • [18] B.-Z. Li, Approximation by multivariate Bernstein-Durrmeyer operators and learning rates of least-square regularized regression with multivariate polynomial kernel, J. Approx. Theory, 173 (2013), 33-55.
  • [19] M. Sugeno, Theory of Fuzzy Integrals and its Applications, Ph.D. dissertation, Tokyo Institute of Technology, Tokyo (1974).
  • [20] Wang, R. S., Some inequalities and convergence theorems for Choquet integrals, J. Appl. Math. Comput., 35 (2011), 305-321.
  • [21] Z. Wang and G. J. Klir, Generalized Measure Theory, Springer, New York, 2009.

Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials with Respect to Distorted Lebesgue Measures

Year 2019, , 15 - 21, 01.03.2019
https://doi.org/10.33205/cma.481186

Abstract

For the univariate Bernstein-Kantorovich-Choquet polynomials written in terms of the Choquet integral with respect to a distorted probability Lebesgue measure, we obtain quantitative approximation estimates for the $L^{p}$-norm, $1\le p<+\infty$, in terms of a $K$-functional.

References

  • [1] F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, deGruyter Studies in Mathematics, vol. 17. Walter de Gruyter, New York, 1994.
  • [2] E. E. Berdysheva and B.-Z. Li, On $L^{p}$-convergence of Bernstein-Durrmeyer operators with respect to arbitrary measure, Publ. Inst. Math. (Beograd) (N.S.). 96(110) (2014), 23-29.
  • [3] M. Campiti and G. Metafune, $L^{p}$-convergence of Bernstein-Kantorovich-type operators, Ann. Polon. Math., LXIII (1996), 273-280.
  • [4] J. Cerdà, J., Martín and P., Silvestre, Capacitary function spaces, Collect. Math., 62 (2011), 95-118.
  • [5] G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble), 5 (1954), 131-295.
  • [6] D. Denneberg, Non-Additive Measure and Integral, Kluwer Academic Publisher, Dordrecht, 1994.
  • [7] S. G. Gal and B. D. Opris, Uniform and pointwise convergence of Bernstein-Durrmeyer operators with respect to monotone and submodular set functions, J. Math. Anal. Appl. 424 (2015), 1374-1379.
  • [8] S. G. Gal, Approximation by Choquet integral operators, Ann. Mat. Pura Appl., 195 (2016), No. 3, 881-896.
  • [9] S. G. Gal and S. Trifa, Quantitative estimates in uniform and pointwise approximation by Bernstein-Durrmeyer-Choquet operators, Carpath. J. Math., 33 (2017), 49-58.
  • [10] S. G. Gal and S. Trifa, Quantitative estimates in $L^{p}$-approximation by Bernstein-Durrmeyer-Choquet operators with respect to distorted Borel measures, Results Math., 72 (2017), no. 3, 1405-1415.
  • [11] S. G. Gal, Uniform and pointwise quantitative approximation by Kantorovich-Choquet type integral operators with respect to monotone and submodular set functions, Mediterr. J. Math., 14 (2017), no. 5, Art. 205, 12 pp.
  • [12] S. G. Gal, The Choquet integral in capacity, Real Analysis Exchange, 43 (2) (2018), 263-280.
  • [13] S. G. Gal, Shape preserving properties and monotonicity properties of the sequences of Choquet type integral operators, J. Numer. Anal. Approx. Theory, under press.
  • [14] S. G. Gal, Quantitative approximation by Stancu-Durrmeyer-Choquet-Šipoš operators, Math. Slovaca, under press.
  • [15] S. G. Gal, Fredholm-Choquet integral equations, J. Integral Equations Applications, https://projecteuclid.org/ euclid. jiea/1542358961, under press.
  • [16] S. G. Gal, Volterra-Choquet integral equations, J. Integral Equations Applications, https://projecteuclid.org/ euclid. jiea/1541668067. under press.
  • [17] L. V. Kantorovich, Sur certains développements suivant les polynômes de la forme de S. Bernstein, I, II, C.R. Acad. Sci. URSS (1930) 563-568, 595-600.
  • [18] B.-Z. Li, Approximation by multivariate Bernstein-Durrmeyer operators and learning rates of least-square regularized regression with multivariate polynomial kernel, J. Approx. Theory, 173 (2013), 33-55.
  • [19] M. Sugeno, Theory of Fuzzy Integrals and its Applications, Ph.D. dissertation, Tokyo Institute of Technology, Tokyo (1974).
  • [20] Wang, R. S., Some inequalities and convergence theorems for Choquet integrals, J. Appl. Math. Comput., 35 (2011), 305-321.
  • [21] Z. Wang and G. J. Klir, Generalized Measure Theory, Springer, New York, 2009.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Sorin G. Gal

Sorin Trıfa This is me

Publication Date March 1, 2019
Published in Issue Year 2019

Cite

APA Gal, S. G., & Trıfa, S. (2019). Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials with Respect to Distorted Lebesgue Measures. Constructive Mathematical Analysis, 2(1), 15-21. https://doi.org/10.33205/cma.481186
AMA Gal SG, Trıfa S. Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials with Respect to Distorted Lebesgue Measures. CMA. March 2019;2(1):15-21. doi:10.33205/cma.481186
Chicago Gal, Sorin G., and Sorin Trıfa. “Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials With Respect to Distorted Lebesgue Measures”. Constructive Mathematical Analysis 2, no. 1 (March 2019): 15-21. https://doi.org/10.33205/cma.481186.
EndNote Gal SG, Trıfa S (March 1, 2019) Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials with Respect to Distorted Lebesgue Measures. Constructive Mathematical Analysis 2 1 15–21.
IEEE S. G. Gal and S. Trıfa, “Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials with Respect to Distorted Lebesgue Measures”, CMA, vol. 2, no. 1, pp. 15–21, 2019, doi: 10.33205/cma.481186.
ISNAD Gal, Sorin G. - Trıfa, Sorin. “Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials With Respect to Distorted Lebesgue Measures”. Constructive Mathematical Analysis 2/1 (March 2019), 15-21. https://doi.org/10.33205/cma.481186.
JAMA Gal SG, Trıfa S. Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials with Respect to Distorted Lebesgue Measures. CMA. 2019;2:15–21.
MLA Gal, Sorin G. and Sorin Trıfa. “Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials With Respect to Distorted Lebesgue Measures”. Constructive Mathematical Analysis, vol. 2, no. 1, 2019, pp. 15-21, doi:10.33205/cma.481186.
Vancouver Gal SG, Trıfa S. Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials with Respect to Distorted Lebesgue Measures. CMA. 2019;2(1):15-21.