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# Quantitative Voronovskaya type theorems and GBS operators of Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution

• *Corresponding author: Neha Bhardwaj
• The motivation behind the current paper is to elucidate the approximation properties of a Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution. We construct quantitative-Voronovskaya and Grüss-Voronovskaya type theorems and determine the convergence estimates of the above operators. We also contrive the statistical convergence and talk about the approximation degree of a bivariate extension of these operators by exhibiting the convergence rate in terms of the complete and partial moduli of continuity. We build GBS (Generalized Boolean Sum) operators allied with the bivariate operators and estimate their convergence rate using mixed modulus of smoothness and Lipschitz class of B$\ddot{o}$gel continuous functions. We also evaluate the order of approximation of the GBS operators in the spaces of B-continuous (Bögel continuous) and B-differentiable (Bögel differentiable) functions. In addition, we depict the comparison between the rate of convergence of the proposed bivariate operators and the corresponding GBS operators for some functions by graphical illustrations using MATLAB software.

Mathematics Subject Classification: 41A25, 41A35.

 Citation:

• Figure 1.  Graph showing convergence of $T_{10,10,\frac{1}{10},\frac{1}{10}}^{\left ( \alpha ,\beta \right )}$ and $S_{10,10\frac{1}{10},\frac{1}{10}}^{\left ( \alpha ,\beta \right )}$ to the function $f\left ( x,r \right ) = x ^{4}+x ^{3}r+x r^{3}+r^{4}$

Figure 2.  Graph showing convergence of $T_{50,50,\frac{1}{50},\frac{1}{50}}^{\left ( \alpha ,\beta \right )}$ and $S_{50,50\frac{1}{50},\frac{1}{50}}^{\left ( \alpha ,\beta \right )}$ to the function $f\left ( x,r \right ) = x ^{4}+x ^{3}r+x r^{3}+r^{4}$

Figure 3.  Graph showing convergence of $T_{20,20,\frac{1}{20},\frac{1}{20}}^{\left ( \alpha ,\beta \right )}$ and $S_{20,20\frac{1}{20},\frac{1}{20}}^{\left ( \alpha ,\beta \right )}$ to the function $f\left ( x,r \right ) = \cos \left ( \frac{5x ^{3}}{x ^{2}+r^{2}} \right )$

Figure 4.  Graph showing convergence of $T_{50,50,\frac{1}{50},\frac{1}{50}}^{\left ( \alpha ,\beta \right )}$ and $S_{50,50\frac{1}{50},\frac{1}{50}}^{\left ( \alpha ,\beta \right )}$ to the function $f\left ( x,r \right ) = \cos \left ( \frac{5x ^{3}}{x ^{2}+r^{2}} \right )$

Table 1.  Absolute error of operators $T_{n ,m,\frac{1}{n },\frac{1}{m}}^{\left ( \alpha ,\beta \right )}$ and $S_{n ,m,\frac{1}{n },\frac{1}{m}}^{\left ( \alpha ,\beta \right )}$ with function $f\left ( x,r \right ) = \cos \left ( \frac{5x ^{3}}{x ^{2}+r^{2}} \right )$ for $m = n = 50$ and $\alpha_{1} = \alpha_{2} = \beta_{1} = \beta_{2} = 5$

 $(x,r)$ $\left| T_{n ,m,\frac{1}{n },\frac{1}{m}}^{\left ( \alpha ,\beta \right )}-f\left ( x,r \right )\right|$ $\left| S_{n ,m,\frac{1}{n },\frac{1}{m}}^{\left ( \alpha ,\beta \right )}-f\left ( x,r \right )\right|$ (0.13, 0.20) .4362 .4406 (0.15, 0.33) .5185 .5210 (0.16, 0.39) .5481 .5505 (0.18, 0.29) .5274 .5330 (0.33, 0.63) .6050 .6104 (0.37, 0.81) .6365 .6404 (0.44, 0.37) .1642 .1714 (0.83, 0.55) .1761 .1827 (0.90, 1) .6696 .6732 (1, 0.83) .6696 .6732
•  [1] T. Acar, Quantitative $q$-Voronovskaya and $q$-Grüss–Voronovskaya-type results for $q$-Szász operators, Georgian Math. J., 23 (2016), 459-468.  doi: 10.1515/gmj-2016-0007. [2] A. M. Acu, H. Gonska and I. Raşa, Grüss-type and Ostrowski-type inequalities in approximation theory, Ukranian Math. J., 63 (2011), 843-864.  doi: 10.1007/s11253-011-0548-2. [3] P. N. Agrawal, B. Baxhaku and R. Chauhan, Quantitative Voronovskaya- and Grüss-Voronovskaya-type theorems by the blending variant of Szász operators including Brenke-type polynomials, Turkish J. Math., 42 (2018), 1610-1629.  doi: 10.3906/mat-1708-1. [4] P. N. Agrawal, N. Bhardwaj and J. K. Singh, Approximation degree of bivariate Kantorovich Stancu operators, J. Nonlinear Sci. Appl., 14 (2021), 423-439.  doi: 10.22436/jnsa.014.06.05. [5] P. N. Agrawal and P. Gupta, $q$-Lupas Kantorovich operators based on Pólya distribution, Ann. Univ. Ferrara Sez. Ⅶ Sci. Mat., 64 (2018), 1-23.  doi: 10.1007/s11565-017-0291-1. [6] P. N. Agrawal, N. İspir and A. Kajla, GBS operators of Lupaş–Durrmeyer type based on Pólya distribution, Results Math., 69 (2016), 397-418.  doi: 10.1007/s00025-015-0507-6. [7] G. A. Anastassiou and S. G. Gal, Approximation theory: Moduli of Continuity and Global Smoothness Preservation, Springer Science & Business Media, Birkhäuser, Boston, 2012. [8] C. Badea, I. Badea and H. H. Gonska, A test function theorem and apporoximation by pseudopolynomials, Bull. Austral. Math. Soc., 34 (1986), 53-64.  doi: 10.1017/S0004972700004494. [9] C. Badea and C. Cottin,, Korovkin-type theorems for generalized Boolean sum operators, in Approximation Theory (Kecskemét, 1990), Colloq. Math. Soc. János Bolyai, 58, North-Holland, Amsterdam, 1991, 51–68. [10] I. Badea, The modulus of continuity in the Bögel sense and some applications in approximation by a Bernšteĭn operator, Studia Univ. Babeş-Bolyai Ser. Math.-Mech., 18 (1973), 69-78. [11] D. Bărbosu, GBSoperators of Schurer-Stancu type, An. Univ. Craiova Ser. Mat. Inform., 30 (2003), 34-39. [12] D. Bărbosu, A.-M. Acu and C. V. Muraru, On certain GBS-Durrmeyer operators based on $q$-integers, Turkish J. Math., 41 (2017), 368-380. [13] S. Bernšteın, Démonstration du théoreme de Weierstrass fondée sur le calcul des probabilities, Comm. Soc. Math. Kharkov., 13 (1912), 1-2. [14] K. Bögel, Mehrdimensionale Differentiation von Funktionen mehrerer reeller Veränderlichen, J. Reine Angew. Math., 170 (1934), 197-217.  doi: 10.1515/crll.1934.170.197. [15] K. Bögel, Über mehrdimensionale Differentiation, Integration und beschränkte Variation, J. Reine Angew. Math., 173 (1935), 5-30.  doi: 10.1515/crll.1935.173.5. [16] Q.-B. Cai and G. Zhou,, Blending type approximation by $GBS$ operators of bivariate tensor product of $\lambda$-Bernstein–Kantorovich type, J. Inequal. Appl., 2018 (2018), 11pp. doi: 10.1186/s13660-018-1862-0. [17] J. S. Connor, The statistical and strong $p$-Cesàro convergence of sequences, Analysis, 8 (1988), 47-63.  doi: 10.1524/anly.1988.8.12.47. [18] E. Deniz, Quantitative estimates for Jain-Kantorovich operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 65 (2016), 121-132.  doi: 10.1501/Commua1_0000000764. [19] Z. Ditzian and V. Totik,, Moduli of Smoothness, Springer Series in Computational Mathematics, 9, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4778-4. [20] E. Dobrescu and I. Matei,, The approximation by Bernšteĭn type polynomials of bidimensionally continuous functions., An. Univ. Timişoara Ser. Şti. Mat.-Fiz., 4 (1966), 85–90. [21] P. Erdős and G. Tenenbaum, Sur les densités de certaines suites d'entiers, Proc. London Math. Soc.(3), 59 (1989), 417-438.  doi: 10.1112/plms/s3-59.3.417. [22] M. D. Farcaş, About approximation of B-continuous and B-differentiable functions of three variables by GBS operators of Bernstein type, Creat. Math. Inform., 17 (2008), 20-27. [23] M. D. Farcaş, About approximation of B-continuous functions of three variables by GBS operators of Bernstein type on a tetrahedron, Acta Univ. Apulensis Math. Inform., 16 (2008), 93-102. [24] Z. Finta,, Remark on Voronovskaja theorem for $q$-Bernstein operators, Stud. Univ. Babeş-Bolyai Math., 56 (2011), 335–339. [25] J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313.  doi: 10.1524/anly.1985.5.4.301. [26] J. A. Fridy and M. K. Khan, Tauberian theorems via statistical convergence, J. Math. Anal. Appl., 228 (1998), 73-95.  doi: 10.1006/jmaa.1998.6118. [27] J. A. Fridy and C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl., 173 (1993), 497-504.  doi: 10.1006/jmaa.1993.1082. [28] A. D. Gadjiev and A. M. Ghorbanalizadeh, Approximation properties of a new type Bernstein–Stancu polynomials of one and two variables, Appl. Math. Comput., 216 (2010), 890-901.  doi: 10.1016/j.amc.2010.01.099. [29] A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky. Mountain J. Math., 32 (2002), 129-138.  doi: 10.1216/rmjm/1030539612. [30] S. G. Gal and H. Gonska, Grüss and Grüss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables., Jaen J. Approx., 7 (2015), 97-122. [31] G. Grüss, Über das Maximum des absoluten Betrages von $\frac{1}{{b - a}}\int\limits_a^b {f\left(x \right)} g\left(x \right)dx - \frac{1}{{\left({b - a} \right)^2 }}\int\limits_a^b {f\left(x \right)dx} \int\limits_a^b g \left(x \right)dx$, Math. Z., 39 (1935), 215-226.  doi: 10.1007/BF01201355. [32] P. Gupta and P. N. Agrawal, Quantitative Voronovskaja and Grüss Voronovskaja-type theorems for operators of Kantorovich type involving multiple Appell polynomials, Iran J. Sci. Technol. Trans. A Sci., 43 (2019), 1679-1687.  doi: 10.1007/s40995-018-0613-x. [33] M. Heilmann, $L_p$-saturation of some modified Bernstein operators, J. Approx. Theory., 54 (1988), 260-273.  doi: 10.1016/0021-9045(88)90003-2. [34] G. İçöz, A Kantorovich variant of a new type Bernstein–Stancu polynomials, Appl. Math. Comput., 218 (2012), 8552-8560.  doi: 10.1016/j.amc.2012.02.017. [35] A. Kajla, S. Deshwal and P. N. Agrawal, Quantitative Voronovskaya and Grüss-Voronovskaya type theorems for Jain–Durrmeyer operators of blending type, Anal. Math. Phys., 9 (2019), 1241-1263.  doi: 10.1007/s13324-018-0229-5. [36] A. Kajla and D. Miclăuş, Blending type approximation by GBS operators of generalized Bernstein-Durrmeyer type, Results Math., 73 (2018), 21pp. doi: 10.1007/s00025-018-0773-1. [37] L. Kantorovich,, Sur certains développements suivant les polynômes de la forme de S., Bernstein, I, II, CR Acad. URSS., 563 (1930). [38] E. Kolk,, The statistical convergence in Banach spaces., Tartu Ül. Toimetised, 928 (1991), 41–52. [39] D. Miclăuş, On the GBS Bernstein-Stancu's type operators, Creat. Math. Inform, 22 (2013), 73-80.  doi: 10.37193/CMI.2013.01.09. [40] H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347 (1995), 1811-1819.  doi: 10.1090/S0002-9947-1995-1260176-6. [41] S. A. Mohiuddine, T. Acar and M. A. Alghamdi,, Genuine modified Bernstein–Durrmeyer operators, J. Inequal. Appl., 2018 (2018), 13pp. doi: 10.1186/s13660-018-1693-z. [42] S. A. Mohiuddine, T. Acar and A. Alotaibi, Construction of a new family of Bernstein-Kantorovich operators, Math. Methods Appl. Sci., 40 (2017), 7749-7759.  doi: 10.1002/mma.4559. [43] T. Neer and P. N. Agrawal,, Quantitative-Voronovskaya and Grüss-Voronovskaya type theorems for Szász-Durrmeyer type operators blended with multiple Appell polynomials, J. Inequal. Appl., 2017 (2017), 20pp. doi: 10.1186/s13660-017-1520-y. [44] S. Pehlivan and M. A. Mamedov, Statistical cluster points and turnpike, Optimization, 48 (2000), 93-106.  doi: 10.1080/02331930008844495. [45] O. T. Pop, Approximation of $B$-continuous and $B$-differentiable functions by GBS operators defined by finite sum, Facta Univ. Ser. Math. Inform., 22 (2007), 33-41. [46] O. T. Pop,, Approximation of $B$-continuous and $B$-differentiable functions by GBS operators defined by infinite sum, JIPAM. J. Inequal. Pure Appl. Math., 10 (2009), 8pp. [47] O. T. Pop and D. Bărbosu, GBS operators of Durrmeyer-Stancu type, Miskolc Math. Notes, 9 (2008), 53-60.  doi: 10.18514/MMN.2008.133. [48] O. T. Pop and M. Farcaş,, Approximation of $B$-continuous and {$B$-differentiable} functions by GBS operators of Bernstein bivariate polynomials, JIPAM. J. Inequal. Pure Appl. Math, 7 (2006), 9pp. [49] S. Rahman, M. Mursaleen and A. Khan,, A Kantorovich variant of Lupaş–Stancu operators based on Pólya distribution with error estimation, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), 26pp. doi: 10.1007/s13398-020-00804-8. [50] R. Ruchi, B. Baxhaku and P. N. Agrawal, GBS operators of bivariate Bernstein-Durrmeyer–type on a triangle, Math. Methods Appl. Sci., 41 (2018), 2673-2683.  doi: 10.1002/mma.4771. [51] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980), 139-150. [52] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math Monthly, 66 (1959), 361-775.  doi: 10.1080/00029890.1959.11989303. [53] D. D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl, 13 (1968), 1173-1194. [54] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math, 2 (1951), 73-74. [55] J. Tariboon and S. K. Ntouyas,, Quantum integral inequalities on finite intervals, J. Inequal. Appl., 2014 (2014), 13pp. doi: 10.1186/1029-242X-2014-121. [56] V. Totik, Problems and solutions concerning Kantorovich operators, J. Approx. Theory, 37 (1983), 51-68.  doi: 10.1016/0021-9045(83)90116-8. [57] G. Ulusoy and T. Acar, $q$-Voronovskaya type theorems for $q$-Baskakov operators, Math. Methods Appl. Sci., 39 (2016), 3391-3401.  doi: 10.1002/mma.3784. [58] A. Zygmund,  Trigonometric Series. Vols. Ⅰ, Ⅱ, 3$^{rd}$ edition, Cambridge University Press, Cambridge, 2002.

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