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

Yıl 2019, , 15 - 21, 01.03.2019

Sorin G. Gal , Sorin Trıfa

https://doi.org/10.33205/cma.481186

Cited By: 2

Öz

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.

Anahtar Kelimeler

{Monotone and submodular set function, Choquet integral, Bernstein-Kantorovich-Choquet polynomial, $L^{p}$ quantitative estimates, $K$-functional, Distorted Lebesgue measure

Kaynakça

• [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.