Approximations on Stancu variant of Szász-Mirakjan-Kantorovich type operators

Abstract

This paper discusses the properties of a modified version of the Stancu variant Szász-Mirakjan Kantorovich type operators. We determine the order of approximation in terms of the modulus of continuity and second-order of smoothness, and we obtain the rate of convergence using the Lipschitz space. We also establish a relation using the Peetre-$K$ functional for the proposed operators. An asymptotic formula for these operators is also established to understand the asymptotic behavior. Along with these discussions, we give some convergence properties in weighted spaces using the weight function. Moving ahead in weighted spaces, the Quantitative Voronovskaya-type and Grüss Voronovskaya-type theorems are proved using the weighted modulus of continuity. Furthermore, we determine the rate of convergence of the proposed operators in terms of functions with derivatives of bounded variations. Finally, we provide graphical representations and numerical analysis for our theoretical findings.

Figures (5)

• Figure 1: Convergence of operators $\mathcal{RL}_{n}^{[\alpha]}(g;x)$ to the function $g(x)$.
• Figure 2: Comparison of convergence the operators $\mathcal{RL}_{n}^{[\alpha]}(g;x)$ with operators $\mathcal{L}_{n}^{[\alpha]}(g;x)$ to the function $g(x)$.
• Figure 3: Comparison of operators $\mathcal{RL}_{n}^{[\alpha]}(g;x)$ with operators $\mathcal{L}_{n}^{[\alpha]}(g;x)$.
• Figure 4: Comparison of operators $\mathcal{RL}_{n}^{[\alpha]}(g;x)$ with operators $\mathcal{L}_{n}^{[\alpha]}(g;x)$ in the approximation of function $g(x)$.
• Figure 5: Comparison of $\mathcal{RL}_{n}^{[\alpha]}(g;x)$ to the function $g(x)(blue)$.

Theorems & Definitions (41)

• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Lemma 1.3
• Theorem 2.1
• proof