[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
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.
[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.
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. Mart 2019;2(1):15-21. doi:10.33205/cma.481186
Chicago
Gal, Sorin G., ve Sorin Trıfa. “Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials With Respect to Distorted Lebesgue Measures”. Constructive Mathematical Analysis 2, sy. 1 (Mart 2019): 15-21. https://doi.org/10.33205/cma.481186.
EndNote
Gal SG, Trıfa S (01 Mart 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 ve S. Trıfa, “Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials with Respect to Distorted Lebesgue Measures”, CMA, c. 2, sy. 1, ss. 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 (Mart 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. ve Sorin Trıfa. “Quantitative Estimates for $L^p$-Approximation by Bernstein-Kantorovich-Choquet Polynomials With Respect to Distorted Lebesgue Measures”. Constructive Mathematical Analysis, c. 2, sy. 1, 2019, ss. 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.