Approximation by Some Positive Linear Operators and Applications to Generalized Bézier Curves

Abstract

An essential goal in approximation theory is simplifying analysis of complex functions by breaking them down into simpler and more elegant function representations. This simplification aids in gaining theoretical understanding, as well as providing a practical means of functional analysis. While the linear operator theory has a classical base of study extending over a hundred years, there has been increased focus on analyzing multiple linear operators simultaneously. This is due to the fact that modern computational resources are being integrated into scientific workflows, making operator theory an active, applied-discipline.

At the same time, Bézier and B-Spline curves have become essential in Computer-Aided Geometric Design (CAGD), Computer Generated Imagery (CGI), robotic path planning, and image processing. These curves are especially appreciated for their intuitive parameterization and local control in shaping smooth, flexible, and aesthetically coherent shapes. This thesis seeks to address the convergence of two paradigms: approximation theory of positive linear operators and geometric modeling of Bézier and B-spline-type curves, aiming to develop a framework that integrates theoretical rigor and practical applicability.

The thesis begins with an examination of the recently introduced families of positive linear operators. Many approximation properties are studied in both classical and weighted settings, including the derivation of Voronovskaja-type theorems alongside modulus of continuity and moment-generating functions. Apart from foundational results, the thesis integrates approximation properties of the differences of positive linear operators. This work focuses on the general quantitative estimates of the differences between newly defined composition-type positive linear operators and some classical operators, such as the Szász-Durrmeyer operators. The ́analysis is conducted within three frameworks: • operators with different basis functions but identical functionals; • operators differing in both basis functions and functionals; • operators with identical basis functions but differing functionals.

All three frameworks are accompanied by numerical examples and refinement using the weighted modulus of continuity which enhances understanding of the approximation processes. One notable angle of this work is the convergence behavior of certain operators with respect to collections of measures which expands the theory beyond Lebesgue integration. This is further generalized with the use of Choquet integrals which enable approximation within non-additive measure spaces, a direction relevant to decision theory and modeling under uncertainty.

Building on these theoretical frameworks, the dissertation goes on to define new classes of Bézier curves using positive linear operators. These curves are geometrically non-negative, ́exhibit partition of unity, and possess shape scaling which is advantageous for precise geometric modeling. A further generalization is proposed in the form of Quadratic Trigonometric B-spline (QTB-spline) curves constructed through a recursive approach. These curves expand their usefulness in sophisticated curve modeling and image processing because of the enhanced smoothness, parametric continuity, localized flexibility in design, and greater contrivance.

Thus the thesis advances the study of positive linear operators and their applications to geometric design in a unified framework. Beginning with discrete Laguerre-based operators, central moments and Voronovskaja-type results are derived alongside weighted direct estimates and numerical illustrations. The work then develops quantitative bounds for differences between newly introduced composition operators and classical counterparts, including Szász- type Durrmeyer variants. A further generalization is achieved through the construction of α-Bernstein-Durrmeyer operators with respect to measure collections, establishing convergence conditions that recover many well-known families. Extending this, α-Bernstein-Durrmeyer operators are defined via Choquet integrals over submodular set functions, with convergence properties linked to signal recovery sensitivity. In the geometric domain, a generalized blended trigonometric Bézier basis with a tunable shape parameter is proposed, preserving classical ́Bézier properties while enabling recursive construction and conditions for smooth joining of curve segments. This is complemented by quadratic trigonometric B-spline bases, extended to higher orders and analyzed for parametric continuity, with applications demonstrated through image processing and forensic curve reconstruction. Collectively, these contributions connect approximation theory with CAGD, highlighting both theoretical depth and practical utility.