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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{\left[ 0,1\right] }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="]" open="["> <mn mathvariant="bold">0</mn> <mo mathvariant="bold">,</mo> <mn mathvariant="bold">1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. Our theoretical, computational, and numerical results reveal that while operators preserve monotonicity entirely on the interval <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varvec{\left[ 0,1\right] }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="]" open="["> <mn mathvariant="bold">0</mn> <mo mathvariant="bold">,</mo> <mn mathvariant="bold">1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varvec{\left[ 0,1\right] }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="]" open="["> <mn mathvariant="bold">0</mn> <mo mathvariant="bold">,</mo> <mn mathvariant="bold">1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. We conclude our analysis by providing a comparison of the convexity preservation abilities of operators equipped with the shape parameter <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varvec{\lambda }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">λ</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Implementation of divided differences to shape preservation

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  • 发表日期：

    2025年6月30日

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  • Nezihe Turhan

摘要

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 \(\varvec{\left[ 0,1\right] }\) 0 , 1 . Our theoretical, computational, and numerical results reveal that while operators preserve monotonicity entirely on the interval \(\varvec{\left[ 0,1\right] }\) 0 , 1 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 \(\varvec{\left[ 0,1\right] }\) 0 , 1 . We conclude our analysis by providing a comparison of the convexity preservation abilities of operators equipped with the shape parameter \(\varvec{\lambda }\) λ .