Implementation of divided differences to shape preservation

Abstract Divided difference methods are used in many engineering and scientific applications to solve numerical interpolation problems. Since shape-preserving approximation has many applications in computer-based geometric design, image processing, geodesy, chemistry, and robotics, we aim to implement divided difference methods to study the shape-preserving properties of a class of blending-type operators, which are constructed via certain shape parameters and represented in terms of divided differences. Considering this new representation, we establish the shape-preserving properties of operators, such as linearity, positivity, end-point interpolation and specifically monotonicity and convexity-preserving properties in connection with a function on the interval $$\varvec{\left[ 0,1\right] }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mfenced> </mml:mrow> </mml:math> . Our theoretical, computational, and numerical results reveal that while operators preserve monotonicity entirely on the interval $$\varvec{\left[ 0,1\right] }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mfenced> </mml:mrow> </mml:math> for the defined intervals of shape parameters, they are unable to preserve the convexity of functions for some values of a shape parameter if the function has a monotonic behavior. Therefore, we give a modified convexity preservation result for the operators by imposing auxiliary assumptions on a function on the interval $$\varvec{\left[ 0,1\right] }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mfenced> </mml:mrow> </mml:math> . We conclude our analysis by providing a comparison of the convexity preservation abilities of operators equipped with the shape parameter $$\varvec{\lambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> </mml:mrow> </mml:math> .