A STUDY ON THE APPROXIMATION ORDER OF POSITIVE LINEAR OPERATORS BY CERTAIN APPROXIMATION METHODS

Abstract

Approximation theory, and in particular the study of positive linear operators, plays a fundamental role in both pure and applied mathematics. Classical operators such as those introduced by Bernstein and Stancu have provided powerful tools for approx- imating continuous functions on compact intervals. However, despite their elegance and historical significance, these operators come with certain well-known limitation- s. Their rate of convergence is generally quite slow, and in many cases they fail to reproduce even simple test functions such as quadratic polynomials, and sometimes even linear functions are not preserved. Moreover, while some attempts exist, they have not yet been systematically developed within more advanced frameworks such as fractional calculus, which accounts for memory effects, or fuzzy mathematics, which deals with uncertainty. These limitations highlight a research gap and provide strong motivation for the development of new families of operators with better approxima- tion properties, wider applicability and closer connections to real-world mathematical models. This thesis addresses these issues through the construction, analysis and appli- cation of several new classes of positive linear operators. Beginning with operators based on the Pólya-Eggenberger (contagion) distribution, parametric generalization- s are introduced to provide greater flexibility in capturing approximation behaviour. Variants of King-type, Kantorovich-type and genuine-type are then developed, which not only improve convergence but also preserve key test functions that classical op- erators do not. Inspired by kernels arising in partial differential equations, the thesis further introduces semi-exponential operators, carefully analyzing their central mo- ments, recurrence relations and generating functions. A Voronovskaya-type theorem is established, offering insights into the asymptotic behaviour of these operators. The study then moves into the fractional domain, where fractional versions of Bernstein-Kantorovich operators are proposed using Caputo’s fractional derivative. Their moments, using Laplace transforms, and convergence properties are derived, xiii xiv ACKNOWLEDGMENTS and their potential for solving fractional differential and fractional integro-differential equations is demonstrated. This represents a significant step toward connecting opera- tor theory with the modelling of systems that exhibit memory effects. Parallel to this, the theory of approximation is extended to the fuzzy domain by defining positive linear fuzzy operators and proving approximation results using the fuzzy Korovkin theorem. Higher-order constructions are also investigated in depth. Second and third or- der semi-exponential operators are defined and analyzed, revealing that their improved moments lead to better rates of convergence compared to their first order counter- parts. Numerical evidence supports these theoretical results, showing that there is an improvement in approximation as the order increases. Similarly, higher order Stancu- Bernstein operators based on the contagion distribution are developed, reducing the order of error from O(1/n) to O(1/n2). These operators are studied using Korovkin’s theorem, modulus of continuity and illustrative numerical examples, confirming that higher-order modifications are a powerful means of enhancing approximation accura- cy. In addition, sequence-based operators are constructed that avoid the use of deriva- tives, making them applicable to non-differentiable functions while still maintaining convergence. A careful comparison reveals that while all such operators converge u- niformly, their endpoint behaviour differs depending on the choice of sequences, with some operators interpolating the boundary values and others not. Taken together, the contributions of this thesis provide advancements in approx- imation theory using sequences of positive linear operators. By addressing the short- comings of classical operators, introducing higher-order and fractional variants, and extending the theory into fuzzy and non-differentiable settings, this work not only en- riches the theoretical foundations of approximation theory but also broadens its appli- cability to modern mathematical models. These results lay the groundwork for future investigations into the optimization of operator constructions, the study of higher-order generalizations and their application in fields such as numerical analysis, differential equations and uncertainty modelling.