Bivariate Generalized $\lambda$-Szász--Kantorovich Operators and Their Approximation Properties

Abstract

In this study, we construct a two-dimensional extension of Szász--Kantorovich operators involving the shape parameter $\lambda \in [-1, 1]$. The novelty of the proposed operators lies in the simultaneous incorporation of three tuning parameters $(\alpha, \beta, \lambda)$ within a bivariate setting, which substantially enhances the flexibility and accuracy of approximation compared with existing univariate and bivariate $\lambda$-Szász--Mirakjan--Kantorovich operators. The rate of approximation is established via the modulus of continuity. In addition, Peetre's $K$-functional and the Lipschitz-type maximal function are examined through the use of appropriate weight functions. The approximation behavior of the proposed operators within Bögel-type functional spaces is analyzed by employing the mixed modulus of smoothness. Graphical illustrations, numerical error estimates, and the convergence characteristics of the operators over various function classes are also presented. The main findings confirm that the proposed bivariate operators converge to the target function at a rate governed by $O(v_1^{-1/2} + v_2^{-1/2})$ in the weighted norm, and that the GBS-type counterparts outperform the classical tensor-product extensions in numerical experiments.

Keywords

• Szasz Operators
• Bogal space
• Lipschitz-maximal function

References

1. [1] S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Communications de la Société Mathématique de Kharkov, 13 (1912), 1–2.
2. [2] O. Szász, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Standards, 45(3) (1950), 239–245.
3. [3] S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Soc. Math. Kharkov, 13 (1912), 1–2.
4. [4] N. Rao, M. Heshamuddin and M. Shadab, Approximation properties of bivariate Szász–Durrmeyer operators via Dunkl analogue, Filomat, 35 (2021), 4515–4532.
5. [5] M. Ayman-Mursaleen, Approximation and convergence analysis of blending-type q-Baskakov operators using wavelet transformations, Filomat, 39(31) (2025).
6. [6] M. Ayman-Mursaleen, N. Rao, M. Rani, A. Kilicman, A. A. H. A. Al-Abied and P. Malik, A note on approximation of blending type Bernstein–Schurer–Kantorovich operators with shape parameter α, J. Math., 2023 (2023), 5245806.
7. [7] 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.
8. [8] S. A. Mohiuddine, N. Ahmad, F. Özger, A. Alotaibi and B. Hazarika, Approximation by the parametric generalization of Baskakov–Kantorovich operators linking with Stancu operators, Iran. J. Sci. Technol. Trans., 45 (2021), 593–605.

9. [9] M. Mursaleen, K. J. Ansari and A. Khan, Approximation properties and error estimation of q-Bernstein shifted operators, Numer. Algorithms, 84 (2020), 207–227.
10. [10] M. Mursaleen, A. Naaz and A. Khan, Improved approximation and error estimations by King type (p,q)-Szász–Mirakjan–Kantorovich operators, Appl. Math. Comput., 348 (2019), 2175–2185.
11. [11] A. Khan, M. Mansoori, K. Khan and M. Mursaleen, Phillips-type q-Bernstein operators on triangles, J. Funct. Spaces, 2021 (2021), 1–13.
12. [12] N. L. Braha, V. Loku, T. Mansour and M. Mursaleen, A new weighted statistical convergence and some associated approximation theorems, Math. Methods Appl. Sci., 45(10) (2022), 5682–5698.
13. [13] M. Nasiruzzaman, M. Mursaleen and N. Rao, On Szász–Jakimovski–Leviatan beta-type integral operators enhanced by Appell polynomials, Iranian Journal of Science, 48(4) (2024), 1013–1022.
14. [14] F. Özger, Weighted statistical approximation properties of univariate and bivariate λ-Kantorovich operators, Filomat, 33 (2019), 3473–3486.
15. [15] A. Alotaibi, On the approximation by bivariate Szász–Jakimovski–Leviatan-type operators of unbounded sequences of positive numbers, Mathematics, 11(4) (2023), 1009.
16. [16] Q. Qi, D. Guo and G. Yang, Approximation properties of λ-Szász–Mirakjan operators, Int. J. Eng. Res., 12 (2019), 662–669.
17. [17] R. Aslan, Some approximation results on λ-Szász–Mirakjan–Kantorovich operators, Fundam. J. Math. Appl., 4 (2021), 150–158.
18. [18] R. Aslan, Approximation by Szász–Mirakjan–Durrmeyer operators based on shape parameter λ, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(2) (2022), 407–421.
19. [19] H. Çiçek and A. İzgi, Approximation by modified bivariate Bernstein–Durrmeyer and GBS bivariate Bernstein–Durrmeyer operators on a triangular region, Fundam. J. Math. Appl., 5(2) (2022), 135–144.
20. [20] R. Aslan and L. Rathour, On approximation properties of Stancu variant λ-Szász–Mirakjan–Durrmeyer operators, Korean Journal of Mathematics, 30(3) (2022), 539–553.
21. [21] R. Aslan, On a Stancu form Szász–Mirakjan–Kantorovich operators based on shape parameter λ, Advanced Studies: Euro-Tbilisi Mathematical Journal, 15(1) (2022), 151–166.
22. [22] V. N. Mishra, M. Raiz and N. Rao, Dunkl analogue of Szász–Schurer Beta bivariate operators, Math. Found. Comput., 6(4) (2023), 651–669.
23. [23] A. D. Gadjiev and H. Hacısalihoglu, Convergence of the sequences of linear positive operators, Ank. Univ., 3(3.7) (1995), 3–8.
24. [24] K. Bögel, Mehrdimensionale differentiation von Funktionen mehrerer veränderlichen, J. Reine Angew. Math., 170 (1934), 197–217.
25. [25] K. Bögel, Über die mehrdimensionale differentiation, Jber. Deutsch. Math.-Verein., 65 (1962/63), 45–71.
26. [26] M. Ayman-Mursaleen, M. Nasiruzzaman, S. K. Sharma and Q.-B. Cai, Invariant means and lacunary sequence spaces of order (α,β), Demonstratio Mathematica, 57(1) (2024), 20240003.
27. [27] M. Nasiruzzaman, M. A. M. Tom, S. Serra-Capizzano, N. Rao and M. Ayman-Mursaleen, Approximation results for Beta Jakimovski–Leviatan type operators via q-analogue, Filomat, 37(24) (2023), 8389–8404.
28. [28] M. Ayman-Mursaleen, Quadratic function preserving wavelet type Baskakov operators for enhanced function approximation, Computational and Applied Mathematics, 44(8) (2025), 395.
29. [29] J. Aljedani, M. Nasiruzzaman and S. A. Mohiuddine, Bézier form of quantum λ Bernstein–Schurer operators with associated approximation properties, J. Math., 2026 (2026), 6615193.
30. [30] M. Ayman-Mursaleen, E. Alshaban and M. Nasiruzzaman, Shape preserving approximation properties to family of α-Bernstein shifted knot operators, J. Inequal. Appl., 2025 (2025), 107.
31. [31] M. Nasiruzzaman, Wavelets associated approximation enhanced by Szász-type operators initiated by q-properties, J. Inequal. Appl., 2026 (2026), 1.