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Riemann-Liouville type fractional a new generalization of Bernstein-Kantorovich operators

Reşat Aslan

TL;DR

The paper introduces a novel Riemann-Liouville type fractional Bernstein-Kantorovich operator $\Re_{m,\eta,\gamma}^{(\alpha,s)}$ that generalizes classical Bernstein-Kantorovich operators by incorporating a RL fractional integral with shape-parameterized Bernstein bases. It establishes the foundational moments, proves uniform convergence on $C[0,1]$ via a Bohman-Korovkin framework, and derives rate bounds through modulus of continuity and Peetre's $K$-functional, along with Lipschitz-type estimates. A comprehensive bivariate extension is developed, with corresponding moments, Korovkin-type convergence on $I^2$, and two-variable modulus bounds that quantify the degree of approximation. Numerical and graphical evaluations corroborate the theoretical results and indicate superior approximation performance of the RL-based operator, particularly in the bivariate setting, compared to existing operators.

Abstract

Approximation theory is a substantial field of mathematical analysis that emerged in the 19th century and has been developed by mathematicians across the globe ever since. Its importance has increased over time, as it provides solutions to numerous scientific challenges not only in mathematics but also in fields like as physics and engineering etc. In the present work, we construct Riemann-Liouville type fractional a new generalization of Bernstein-Kantorovich type operators. First, we obtain the moment and central moments from some basic calculations. Also, we study several direct and local approximation outcomes of the constructed operators. Next, we serve up certain graphical and numerical results to demonstrate the convergence, accuracy and significance of constructed operators. Further, we provide bivariate version of the newly constructed operators and establish degree of approximation through of partial and complete modulus of continuity. Lastly, we present some graphical representations and maximum error of approximation tables to verify the convergence behavior of bivariate form of related operators based on various parameters.

Riemann-Liouville type fractional a new generalization of Bernstein-Kantorovich operators

TL;DR

The paper introduces a novel Riemann-Liouville type fractional Bernstein-Kantorovich operator that generalizes classical Bernstein-Kantorovich operators by incorporating a RL fractional integral with shape-parameterized Bernstein bases. It establishes the foundational moments, proves uniform convergence on via a Bohman-Korovkin framework, and derives rate bounds through modulus of continuity and Peetre's -functional, along with Lipschitz-type estimates. A comprehensive bivariate extension is developed, with corresponding moments, Korovkin-type convergence on , and two-variable modulus bounds that quantify the degree of approximation. Numerical and graphical evaluations corroborate the theoretical results and indicate superior approximation performance of the RL-based operator, particularly in the bivariate setting, compared to existing operators.

Abstract

Approximation theory is a substantial field of mathematical analysis that emerged in the 19th century and has been developed by mathematicians across the globe ever since. Its importance has increased over time, as it provides solutions to numerous scientific challenges not only in mathematics but also in fields like as physics and engineering etc. In the present work, we construct Riemann-Liouville type fractional a new generalization of Bernstein-Kantorovich type operators. First, we obtain the moment and central moments from some basic calculations. Also, we study several direct and local approximation outcomes of the constructed operators. Next, we serve up certain graphical and numerical results to demonstrate the convergence, accuracy and significance of constructed operators. Further, we provide bivariate version of the newly constructed operators and establish degree of approximation through of partial and complete modulus of continuity. Lastly, we present some graphical representations and maximum error of approximation tables to verify the convergence behavior of bivariate form of related operators based on various parameters.
Paper Structure (6 sections, 13 theorems, 61 equations, 6 figures, 7 tables)

This paper contains 6 sections, 13 theorems, 61 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Direct and Local estimates of $\Re_{m,\eta,\gamma}^{(\alpha,s)}$
  4. Graphical and Numerical Evaluations
  5. Bivariate extension of operators of $\Re_{m,\eta,\gamma}^{(\alpha,s)}$
  6. Conclusion

Key Result

Lemma 1

(aktuglu) For the operators defined by (e2), we get

Figures (6)

  • Figure 1: Convergence of $\Re_{m,3,3}^{(0.9,4)}(\phi;z)$ to $\phi(z)=z (z-\frac{4}{7}) sin(\pi z)$ for $m=20, 30, 70$
  • Figure 2: Convergence of $\Re_{10,2,3}^{(\alpha,4)}(\phi;z)$ to $\phi(z)=(1-z) cos(2\pi z)$ for $\alpha=0.35,0.65,0.95$
  • Figure 3: Convergence of $\Re_{10,3,2}^{(0.75,s)}(\phi;z)$ to $\phi(z)=22z(z-0.9) (z-0.3)$ for $s=2,5,8$
  • Figure 4: Convergence of operators $\Re_{m_{1},m_{2},2,2,3,3}^{(0.9,0.9,2,2)}(\phi;z,y)$ to $\phi(;z,y)=(yz^2-1)\sin(2\pi y)$ for $m_{1}=m_{2}=10,30,90$
  • Figure 5: Convergence of operators $\Re_{15,15,2,2,2,2}^{(\alpha_{1},\alpha_{2},2,2)}(\phi;z,y)$ to $\phi(;z,y)=(yz+2)\cos(2\pi z)$ for $\alpha_{1}=\alpha_{2}=0.1,0.5,0.9$
  • ...and 1 more figures

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Corollary 1
  • Remark 1
  • ...and 18 more