Approximation by Kantorovich λ-Bernstein operators under shifted knot frameworks: numerical analysis and graphical study

This manuscript introduces a Kantorovich variant of λ -Bernstein operators with shifted knots, constructed using Bézier basis functions. After defining the operators, we provide a comprehensive approximation analysis utilizing Korovkin’s theorem, the modulus of continuity, and the Ditzian-Totik modulus of smoothness. Theoretical findings are further strengthened by direct theorems articulated through Peetre’s K-functional and Lipschitz-type conditions. To validate the proposed model, we present extensive numerical data and graphical evidence, with error norm tables confirming the efficiency and convergence rates of the operators.